# Volume 45, № 12, 1993

### On the smoothness of solutions of differential equations with a discontinuous right-hand side

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1587–1594

A method for the investigation of differential equations with a nonclassical right-hand side [1] is applied to the study of the higher-order differentiability of solutions of differential equations with discontinuous right-hand sides with respect to the initial data. We use results from the theory of differential equations with pulse influence [2].

### Groups the lyapunov transformations

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1595–1600

We consider the groups of transformations that have basic asymptotic characteristics of differential equations as their invariants.

### Expansion of a bundle of fourth-order differential operators in a part of its eigenfunctions

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1601–1612

A bundle of differential operators $$\mathcal{L}(\lambda ),\lambda \in \mathbb{C}:\mathcal{L}(\lambda )y(x) = y^{(4)} (x) - 2\lambda ^2 y^{(2)} (x) + \lambda ^4 y(x),|x| \leqslant 1,y( \pm 1) = y\prime ( \pm 1) = 0,$$ is considered. In various function spaces, we establish the facts about the expansions of a pair of functions $f(x)$ and $g(x)$ in the system $\{y_k (x),\; iλ_k y_k (x)|}_{k=1}^{ ∞}$, where $y_k(x), k = 1,2,...,$ are the eigenfunctions of the bundle $L (λ)$ corresponding to the eigenvalues $λ_k$, with $\Im λ_k > 0$.

### Integral sets of systems of difference equations

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1613–1621

The problem of existence of integral sets of systems of difference equations is studied. We establish sufficient conditions for the existence of these sets and their stability. For the system under consideration, the behavior of trajectories that originate in a sufficiently small neighborhood of integral sets is investigated.

### On Gevrais classes of certain self-adjoint differential operators with degeneration

Izvekov I. G., Martynenko E. V.

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1622–1626

Classical spaces of ultra-diffcrentiable funetions on $[-1,1]$ are compared with the Gevrais classes of а self-adjoint differentiable operator whose eigenfunetions an the orthogonal Jacobi polynomials.

### Optimization of adaptive algorithms for the renewal of monotone functions from the class $H^ω$

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1627–1634

A problem of renewal of monotone functions $f(t) \in H^{\omega}[a, b]$ with fixed values at the ends of an interval is studied by using adaptive algorithms for calculating the values of $f(t)$ at certain points. Asymptotically exact estimates unimprovable on the entire set of adaptive algorithms are obtained for the least possible number $N(\varepsilon)$ of steps providing the uniform $ε$-error. For moduli of continuity of type $εα, 0 < α < 1$, the value $N(\varepsilon)$ has a higher order as $ε → 0$ than in the nonadaptive case for the same amount of information.

### On the limit distribution of the correlogram of a stationary Gaussian process with weak decrease in correlation

Leonenko N. N., Portnova A. Yu.

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1635–1641

An example of the non-Gaussian limit distribution of the statistical estimate of the correlation function of a stationary Gaussian process with unbounded spectral density (or with a nonintegrable correlation function) is given.

### Variational formulations of nonlinear boundary-value problems with a free boundary in the theory of interaction of surface waves with acoustic fields

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1642–1652

Variational problems equivalent to nonlinear evolutionary boundary-value problems with a free boundary are formulated. These problems arise in the theory of interaction of limited volumes of liquid, gas, and their interface with acoustic fields. It is proved that the principle of separation of motions can be applied to these variational problems. The problem of a capillary-acoustic equilibrium form is given in a variational formulation.

### On the reduction principle in the theory of stability of motion

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1653–1660

This paper deals with the development of Lyapunov's idea of reducing the problem of stability of the trivial solution of a system of higher-order differential equations to a similar problem for a system of lower order. Special attention is paid to the application of integral manifolds and approximate integral manifolds.

### Reducibility of linear systems of difference equations with almost periodic coefficients

Martynyuk D. I., Mitropolskiy Yu. A., Tynnyi V. I.

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1661–1667

For the linear systemof difference equations $x(t + 1) = Ax(t) + P(t)x(t)$, where the matrix $P(t)$ is almost periodic, sufficient conditions are given, which reduce it to a system with a constant matrix.

### The Poincare-Mel'nikov geometric analysis of the transversal splitting of manifolds of slowly perturbed nonlinear dynamical systems. I

Prykarpatsky A. K., Samoilenko A. M., Timchishin O. Ya.

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1668–1681

On the basis of the geometric ideas of Poincaré and Mel'nikov, we study sufficient criteria of the transversal splitting of heteroclinic separatrix manifolds of slowly perturbed nonlinear dynamical systems with a small parameter. An example of adiabatic invariance breakdown is considered for a system on a plane.

### On separatrix curves of a family of linear systems with pulse influence

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1682–1687

For a family of linear systems of differential equations with pulse influence, we establish conditions of existence of separatrix curves and present a method for finding these curves.

### On exact irreducible representations of locally normal groups

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1688–1694

We obtain a generalization of the Gaschutz criterion of existence of exact irreducible representations of finite groups to the class of normal groups.

### On the optimization of direct methods for solving fredholm integral equations of the second kind with infinitely smooth kernels

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1695–1701

We give a direct method, optimal in $L_2$, for solving the Fredholm integral equation of the second kind with operators acting into the space of functions harmonic in a disk or into the space of functions that can be analytically extended to an infinite strip. The exact order of the error of this method is determined.

### On the number of critical submanifolds of a function on a manifold

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1702–1705

We consider differentiable functions on a manifold such that the set of their critical points is a disconnected union of smooth submanifolds. A topological characteristic of the manifold is introduced; in terms of this characteristic, we estimate the least possible number of critical submanifolds of these functions.

### On the existence of vector fields with a given set of singular points on a two-dimensional closed oriented manifold

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1706–1709

We study the possibility of constructing locally gradient and arbitrary vector fields with a given set of singular points on a two-dimensional closed oriented manifold. The sum of the indices of the vector field at these points is equal to the Euler characteristic of the manifold.

### Limit functionals for a semicontinuous difference of renewal processes with discrete time

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1710–1712

For a difference, semicontinuous in discrete topology, of two renewal processes with discrete time, the distribution of principal limit functionals is found.

### Central limit theorem for centered frequencies of a countable ergodic markov chain

Moskal'tsova N. V., Shurenkov V. M.

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1713–1715

On the basis of results relating to the behavior of the potential of a countable ergodic Markov chain, for a certain class of functions, the asymptotic normality of a variable $\cfrac{1}{\sqrt{n}}\sum^{n-1}_{k=0}f(X_k)$ for $n \rightarrow \infty$ has been proved. The asymptotic normality of the centering frequencies has been obtained without using the finileness conditions for the time $M_0\tau^2$ of the first return into a chain state.

### Existence and uniqueness of solution of the Cauchy problem for singular systems of integro-differential equations

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1716–1720

Sufficient conditions are established for the existence of a unique solution of the Cauchy problem for singular systems of integro-differential equations.

### Index of volume 45 of „Ukrainian Mathematical Journal”

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1721-1727