Volume 49, № 6, 1997
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
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.
Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 779–788
We reduce spectral problems on an axis to spectral problems on a semiaxis.
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
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.
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.
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.
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.
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.
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.
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.
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=DA and UAV B=DB if and only if the matrices B *A and D * B DA 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.
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.