# Volume 47, № 4, 1995

### Asymptotic expansions associated with the jackknife functional. I

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 443–451

We obtain an asymptotic expansion of the functional of the jackknife method, which is used for the estimation of the variance of observational errors in a nonlinear regression model.

### Locally nilpotent groups of finite non-Abelian sectional rank

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 452–455

We introduce the notion of non-Abelian sectional rank of a group and study locally nilpotent non-Abelian groups of finite non-Abelian sectional rank. It is proved that the (special) rank of these groups is finite.

### Reduction of the self-dual Yang-Mills equations I. Poincaré group

Fushchich V. I., Lagno V. I., Zhdanov R. Z.

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 456–462

For the vector potential of the Yang-Mills field, we give a complete description of ansatzes invariant under three-parameter*P* (1, 3) -inequivalent subgroups of the Poincaré group. By using these ansatzes, we reduce the self-dual Yang-Mills equations to a system of ordinary differential equations.

### Singularly perturbed linear boundary-value problems with pulse effects and the regular reduced problem

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 463–467

A singularly perturbed linear boundary-value problem with pulse effect is considered. By using pseudo-inverse matrices, we construct an asymptotic solution with a single boundary layer.

### A constructive method for finding periodic solutions of differential systems

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 468–476

Coefficient conditions for the existence of periodic solutions of degenerate systems of differential equations are obtained. Iterative algorithms for finding these solutions are developed.

### Axially symmetric flow with free boundary

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 477–487

We prove the solvability of a boundary-value problem with the Bernoulli condition in the form of an inequality on a free boundary. By using the Rietz method, we construct an approximate solution that converges to an exact solution in the integral metric.

### Representations of *-algebras and many-dimensional dynamical systems

Ostrovskii V. L., Turovskaya L. B.

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 488–497

The procedure developed earlier for investigation of representations of *-algebras is generalized to a wider class of operator relations. In particular, the developed technique is used for description of all irreducible representations of a nonstandard three-dimensional quantum sphere.

### On the rate of convergence of projection-iterative methods for classes of weakly singular integral equations

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 498–505

For classes of weakly singular integral equations of the second kind whose kernels have a power singularity, we find the optimal order of the rate of convergence of projection-iterative methods. Moreover, iterative methods of the Sokolov type are considered and, for weakly singular equations with differentiable coefficients, we present estimates of the rate of convergence of such methods.

### Ideals of the semigroup of ultrafilters of a topological group

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 506–511

We construct two new series of closed left ideals of the semigroup \(\bar \tau \) of ultrafilters of a topological group (*G*, τ). The first series gives a disjunctive decomposition of τ-absorbing ultrafilters. Under certain restrictions on the topology of the group (*G*, τ), the second series gives a disjunctive decomposition of the semigroup of free ultrafilters. For a nondiscrete metrizable topological group (*G*, τ), we construct a large free subsemigroup of the semigroup \(\bar \tau \) .

### Vector measures of various smoothness classes and their limits

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 512–516

A relationship between different types of continuity with respect to direction and other types of smoothness is found for vector measures. The following problem is also studied: What vector measures can be represented as the limits of quasiinvariant, infinitely differentiable, analytic, and continuous measures in the topologies of convergence in variation, convergence in semivariation, and convergence on every measurable set.

### On the conditions of solvability and the resolvent of a second-order deferential boundary operator in a space of vector functions

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 517–524

A boundary differential operator generated by the Sturm-Liouville differential expression with bounded operator potential and nonlocal boundary conditions is considered. The conditions for a considered operator to be a Fredholm and solvable operator are established and its resolvent is constructed.

### On solutions of a quasilinear almost triangular system of difference equations

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 525–530

For an almost triangular system of difference equations, the problem of existence of particular solutions in the form of Fourier series with slowly varying coefficients and frequency is studied.

### On solutions of hyperbolic differential inclusions with nonconvex right-hand side

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 531–534

The existence of a generalized solution with continuous derivatives*u* _{ x },*u* _{ y } is proved for the differential inclusion*u* _{ xy }∈*F(x, y, u)* with a nonconvex right-hand side satisfying the Lipschitz conditionin*x, y*, and*u*.

### Approximation of a bounded solution of a linear difference equation on Z^{2} by solutions of corresponding boundary-value problems in a Banach space

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 535–541

We study the problem of existence and uniqueness of solutions of boundary-value difference problems in a Banach space that correspond to a certain difference equation on Z^{2}. We prove a theorem on approximation of the unique bounded solution of the considered equation by solutions of the corresponding boundary-value problems.

### Majorant estimates for solutions of quadratic difference systems with lag

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 542–545

Systems of differential equations with lag with quadratic right-hand sides are considered. A compact matrix form for representing these systems in the general form is suggested. Upper and lower bounds for solutions of systems of quadratic differential equations with arbitrary lag are obtained by using quadratic Lyapunov functions.

### On the multiplier of a group with nontrivial center

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 546–550

This paper is a sequel of [1]. On the basis of the results obtained therein, we establish inequalities that strengthen the known Green inequality for the order of the Schur multiplier of a*p*-group.

### On the stability of solutions of stochastic differential inclusions

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 551–554

We study the stability in probability of a solution of a stochastic differential inclusion in a finite-dimensional space with nonrandom coefficients and a maximally monotone operator in the drift coefficient.

### On local operators diagonal with respect to the system of Hermitian polynomials

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 555–561

We present necessary conditions for operators diagonal with respect to the system of Hermitian polynomials to be local.

### Automorphisms of groups of order $p^5$

Pechenyuk О. V., Pylyavs'ka O. S.

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 562-565

We study automorphisms of groups of order $p^5$ (p is an odd prime number). Groups without any automorphism of order 2 and groups with group automorphisms of order $p^6$ are found.

### Integrals of products of Bessel functions for applied electrodynamic problems

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 566–570

We determine definite integrals of products of Bessel functions of the first and second kind of different orders with various complicated arguments with finite limits of integration.

### On ideals of the group algebra of a free group of degree of freedom two over the field of complex numbers

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 571-572

In this paper, we prove the existence of an elementα of the group algebra $A=ℂF$ of a free group $F$ with two generatorsx andy over the field of complex numbers $C$ such that, for any complex $a$ and $b$ for which $¦a¦=¦b¦=1$, we have $A ∩ ϑ_{a,b} (α)A=0$, where $ϑ_{a,b}$ ($α$ is an automorphism of $A$ that maps $x,y$ into $a_x, b_y$, respectively. Thus, we give a negative answer to question 12.46 of P. A. Linnel from “Kourovka Notebook.”

### Estimates for information quantity in probability model in image encoding

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 573-576

Upper bounds are obtained for the amount of information in functional classes with initial uncertainty. These classes are similar to the classes of images with bounded variance. We use the probability model of the theory of information complexity. Unlike the case of classes $KH_{0}^{ α}$ , it turns out that, in this case, the method of differential pulse-code modulation does not give any advantage as compared to the deterministic case.