# Volume 49, № 6, 1997

### Approximation of classes of convolutions by linear methods of summation of Fourier series

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 739–753

We consider a family of special linear methods of summation of Fourier series and establish exact equalities for the approximation of classes of convolutions with even and odd kernels by polynomials generated by these methods.

### On the best approximations and rate of convergence of decompositions in the root vectors of an operator

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 754–773

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator $A$ that acts in B. The corresponding estimates of the best approximations are expressed in terms of a *K*-functional associated with the operator *A*. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of *K*-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of *f* from B in the root vectors of the operator *A* to *f*

### Construction of approximations for a stationary solution of a system of singular ordinary differential equations

Romanyshyn I. M., Synyts’kyi L. A.

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 774–778

We propose a procedure for the construction of successive approximations of a stationary solution of a system of nonlinear ordinary differential equations with a small parameter with the derivative. We present sufficient conditions for the convergence of constructed approximations to the required stationary solution.

### Spectral problems for canonical systems of finite-difference equations on an axis

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 779–788

We reduce spectral problems on an axis to spectral problems on a semiaxis.

### Structure of locally graded nonnilpotent CDN[]-groups

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 789–798

We prove a theorem that gives a constructive description of locally graded nonnilpotent CDN []-groups.

### The existence of a bifurcation value of a parameter of a system of differential equations with deviating argument

Nasykhova L. V., Terekhin M. T.

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 799–805

We prove a theorem on the existence of a nonzero periodic solution of a system of differential equations with deviation that depends both on an unknown function and on its derivative. This result is obtained for the case where the matrix of linear approximation has zero and imaginary eigenvalues if the parameter takes a critical value.

### Conditional symmetry of the Navier-Stokes equations

Fushchich V. I., Serov N. I., Tulupova L. O.

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 806–813

We study the conditional symmetry of the Navier-Stokes equations and construct multiparameter families of exact solutions of the Navier-Stokes equations.

### Application of one constructive method for the construction of non-Lie solutions of nonlinear evolution equations

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 814–827

We propose a constructive method for the construction of exact solutions of nonlinear partial differential equations. The method is based on the investigation of a fixed nonlinear partial differential equation (system of partial differential equations) together with an additional condition in the form of a linear ordinary differential equation of higher order. By using this method, we obtain new solutions for nonlinear generalizations of the Fisher equation and for some nonlinear evolution systems that describe real processes in physics, biology, and chemistry.

### On the optimal renewal of bilinear functionals in linear Normed spaces

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 828–831

We study the problem of optimal renewal of bilinear functionals on the basis of optimal linear information in the general statement. We also represent some new results for special spaces of functions.

### Boundary-value problems for systems of difference equations

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 832–835

Boundary-value problems for systems of difference equations with discrete argument whose linear part is the Noetherian operator are considered. The necessary and sufficient conditions of the solvability of difference boundary-value problems of this sort are obtained.

### Conditions of nonoscillation of binomial systems of differential equations

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 836–841

We establish comparison theorems for solutions of the system of Kondrat’ev-type equations y^{(n)}+*P(t)y*=0.

### On ascending and subnormal subgroups of infinite factorized groups

De Glovanni F., Franclosi S., Sysak Ya. P.

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 842–848

We consider an almost hyper-Abellan group *G* of a finite Abelian sectional rank that is the product of two subgroups *A* and *B*. We prove that every subgroup *H* that belongs to the intersection *A* ∩ *B* and is ascending both in *A* and *B* is also an ascending subgroup in the group *G*. We also show that, in the general case, this statement is not true.

### Construction of nonnilpotent biprimary dispersible groups with metacyclic subgroups of nonprimary index

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 849–851

We give a constructive description of nonnilpotent biprimary dispersible groups with metacyclic subgroups of nonprimary index and classify them into 21 types.

### Congruences and quasiidentities on unars

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 852–856

We establish conditions under which the fact that all congruences of two unars (universal algebras with one unary operation) are convergent implies the convergence of manifolds and quasimanifolds generated by these unars.

### Stable difference scheme for a nonlinear Klein-Gordon equation

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 857–859

For a nonlinear Klein-Gordon equation, we obtain a stable difference scheme for large time intervals. We prove that this scheme has the sixth order of accuracy.

### A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 860–862

We establish that a pair *A, B*, of nonsingular matrices over a commutative domain *R* of principal ideals can be reduced to their canonical diagonal forms *D* ^{A} and *D* ^{B} by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices *U, V* _{A}, and *V* _{B} over *R* such that *UAV* _{a}=D^{A} and *UAV* _{B}=D^{B} if and only if the matrices *B* _{*}A and *D* _{*} ^{B} D^{A} where *B* _{*} 0 is the matrix adjoint to *B*, are equivalent.

### On the existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 863–871

We study the problem of existence and uniqueness of a solution of a linear stochastic differential equation with respect to a logarithmic process. For the conditional mathematical expectation of a solution, we obtain a partial differential equation.

### On the application of Ateb-functions to the construction of a solution of a nonlinear Klein-Gordon equation

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 872–877

For a nonlinear Klein-Gordon equation, we construct the first approximation of an asymptotic solution by using Ateb-functions. The resonance and nonresonance cases are considered.

### A remark on quasianalytic vectors for a pair of anticommuting operators

Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 878–880

We prove that two self-adjoint operators that anticommute on the dense invariant domain of their common quasianalytic vectors are strongly anticommuting.