2017
Том 69
№ 7

All Issues

Volume 61, № 2, 2009

Article (Russian)

Approximation of unbounded operators by bounded operators in a Hilbert space

Babenko V. F., Bilichenko R. O.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 147-153

We determine the best approximation of an arbitrary power $A^k$ of an unbounded self-adjoint operator $A$ in a Hilbert space $H$ on the class $\{x ∈ D(A^r ) : ∥A^r x∥ ≤ 1\},\; k < r$.

Article (Russian)

Serial rings and tiled orders of width two

Bronitskaya N. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 154-159

We construct Artinian serial rings and tiled orders of width two with maximal finite global dimension.

Article (Russian)

Integral representations of generalized axially symmetric potentials in a simply connected domain

Grishchuk S. V., Plaksa S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 160-177

We obtain integral representations of generalized axially symmetric potentials via analytic functions of a complex variable that are defined in an arbitrary simply connected bounded domain symmetric with respect to the real axis. We prove that these integral representations establish a one-to-one correspondence between analytic functions of a complex variable that take real values on the real axis and generalized axially symmetric potentials of certain classes.

Article (Ukrainian)

State estimation for a dynamical system described by a linear equation with unknown parameters

Zhuk S. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 178-194

We investigate the state estimation problem for a dynamical system described by a linear operator equation with unknown parameters in a Hilbert space. In the case of quadratic restrictions on the unknown parameters, we propose formulas for a priori mean-square minimax estimators and a posteriori linear minimax estimators. A criterion for the finiteness of the minimax error is formulated. As an example, the main results are applied to a system of linear algebraic-differential equations with constant coefficients.

Article (Russian)

Mixed problem for the Petrovskii well-posed equation in a cylindrical domain

Guseinova É. S., Iskenderov B. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 214-230

We study the problem of existence and uniqueness of the solution of a mixed problem for the Petrovskii well-posed equation in a cylindrical domain and the behavior of this solution for large values of time.

Article (Ukrainian)

Improvement of one inequality for algebraic polynomials

Chaikovs'kyi A. V., Nesterenko A. N., Tymoshkevych T. D.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 231-242

We prove that the inequality $||g(⋅/n)||_{L_1[−1,1]}||P_{n+k}||_{L_1[−1,1]} ≤ 2||gP_{n+k}||_{L_1[−1,1]}$, where $g : [-1, 1]→ℝ$ is a monotone odd function and $P_{n+k}$ is an algebraic polynomial of degree not higher than $n + k$, is true for all natural $n$ for $k = 0$ and all natural $n ≥ 2$ for $k = 1$. We also propose some other new pairs $(n, k)$ for which this inequality holds. Some conditions on the polynomial $P_{n+k}$ under which this inequality turns into the equality are established. Some generalizations of this inequality are proposed.

Article (Russian)

Construction of continuous cocycles for the bicrossed product of locally compact groups

Chapovsky Yu., Podkolzin G. B.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 243-260

For locally compact groups $K, M$, and $N$ such that $M$ and $N$ are subgroups of $K, K = M ∙ N$ and $M ∩ N = \{e\}$, where $e$ is the identity of the group $K$, we give a complete description and propose a method for the construction of pairs of continuous cocycles used in the structure of bicrossed product with cocycles in terms of continuous 2-cocycles on the groups $M, N$, and $$K and 3-cocycles on the group $K$.

Article (Russian)

On the existence of a generalized asymmetric (α, β)-spline whose average values have equal minima at given points

Skorokhodov D. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 261-267

We solve the problem of existence of an asymmetric spline averaged in Steklov’s sense that takes equal minimum values at given points.

Article (Ukrainian)

Conditions for the existence and uniqueness of bounded solutions of nonlinear differential equations

Slyusarchuk V. Yu.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 268-279

We establish conditions required for the existence and uniqueness of bounded solutions of the nonlinear differential equation $f_1\left(\frac{dx(t)}{dt} \right) = f_2(x(t)), t ∈ ℝ$.

Brief Communications (Russian)

Solution of a countable system of quasilinear partial differential equations multiperiodic in a part of variables

Berzhanov A. B., Kurmangaliev E. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2009. - 61, № 2. - pp. 280-288

We establish sufficient conditions for the existence and uniqueness of solutions of a countable system of first-order quasilinear partial differential equations multiperiodic in a part of variables.