# Volume 49, № 4, 1997

### On the optimization of approximate integration by Monte Carlo methods

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 475–480

We solve the problem of optimization of monte Carlo methods for approximate integration over an arbitrary absolutely continuous measure. We propose a convenient model of Monte Carlo methods which uses the notion of transition probability.

### Contraction operators, defect functions, and scattering theory

Boiko C. C., Dubovoi V. K., Fritzsche B., Kirstein B.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 481–489

We study functions that describe scattering through inner channels of a system (defect functions) and the corresponding contraction operators.

### Functional law of iterated logarithm for normalized integrals of processes with weak dependence

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 490–499

For normalized integrals of processes with weak dependence, we prove the law of iterated logarithm in the Strassen form. The results obtained are used for the construction of a curvilinear confidence region in which a solution of an equation with small parameter is sought.

### On differential properties of mappings into a Banach space

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 500–509

We prove that the Rieffel sharpness condition for a Banach space *E* is necessary and sufficient for an arbitrary Lipschitz function *f: [a, b]→E* to be differentiable almost everywhere on a segment [*a, b*]. We establish that, in the case where the sharpness condition is not satisfied, the major part (in the category sense) of Lipschitz functions has no derivatives at any point of the segment [*a, b*].

### Behavior of the extended spectral measure of a self-adjoint operator at infinity

Gorbachuk M. L., Gorbachuk V. I.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 510–516

We study the dependence of the rate of growth of the extended spectral measure of a self-adjoint operator at infinity on the order of singularity of the vector on which this measure is considered.

### On directional monogeneity sets

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 517–522

We introduce and investigate some new differential properties of functions by using geometrical properties of directional monogeneity sets.^{1}

### A sufficient condition for strong almost-periodicity of scalarly almost periodic representations of the group of real numbers

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 523–526

We prove the following theorem: If every separable subspace *Y* of a Banach space *X* has a separable weak sequential closure in *Y* ^{**}, then every scalarly almost periodic group acting in *X* is strongly almost periodic.

### Dual Appell system and Kondrat’ev spaces in analysis on Schwartz spaces

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 527–534

We study the biorthogonal Appell system and Kondrat’ev spaces in the case where the parameter of a μ-exponential is perturbed by holomorphic invertible functions. The results obtained are applied to the investigation of pseudodifferential equations.

### On structural transformations of equations of perturbed motion for a certain class of dynamical systems

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 535–539

We consider a general method for structural transformations of one class of dynamical systems with gyroscopic forces, which enables us to remove gyroscopic terms from the original equations of perturbed motion. Without changing the qualitative properties of these equations, this method simplifies their investigation.

### Systems of parabolic variational inequalities without initial conditions

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 540–547

We consider a parabolic variational inequality without initial conditions. We construct a class of existence and uniqueness for a solution of this inequality. This class is defined by the exponential decrease or increase of solutions as *t*→−∞, depending on the coefficients of the inequality.

### Analytical construction of the hierarchical matrix Lyapunov function for impulse systems

Begmuratov K. A., Martynyuk A. A.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 548–557

For impulse systems, we develop a method for the construction of the hierarchical matrix Lyapunov ffunction.

### A periodic problem for the inhomogeneous equation of string oscillations

Khoma G. P., Mitropolskiy Yu. A., Tsynalko P. V.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 558–565

We study a periodic problem for the equation *u* _{tt}−u_{xx}=g(x, t), u(x, t+T)=u(x, t), u(x+ω, t)= =u(x, t), ℝ^{2} and establish conditions of the existence and uniqueness of the classical solution.

### Some obligatory conditions and generalizations of the Fedorov theorem

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 566–572

We introduce a new type of conditions, *S*-obligatory conditions. We describe broad classes of groups with the Chernikov obligatory condition and some other conditions similar to the known Fedorov condition.

### Ideals and free Pairs in the semigroup β ℤ

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 573–580

We prove that the equations ξ+*x*=*m*ξ+*y*, *x*+ξ=*y*+*m*ξ have no solutions in the semigroup β ℤ for every free ultrafilter ξ and every integer *m*∈0, 1. We study semigroups generated by the ultrafilters ξ, *m*ξ. For left maximal idempotents, we prove a reduced hypothesis about elements of finite order in β ℤ.

### Boundary-value problems with parameters for a multifrequency oscillation system

Petryshyn Ya. R., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 581–589

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillation systems. We obtain estimates for the deviation of solutions of averaged problems from solutions of original problems.

### Ostap Stepanovich Parasiuk '75

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 590

### Rotary motions of autonomous systems with pulse influence

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 591–596

We study rotary motions for an autonomous second-order differential equation with pulse influence and periodic right-hand side and indicate some important properties of these motions. By using the numerical-analytic method, we establish sufficient conditions for the existence of rotary motions.

### On a representation of smooth invariants of Coxeter groups in terms of anisotropic spaces

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 597–600

A smooth function invariant under the action of the Coxeter group can be represented as a function of basic invariants. We propose to describe the latter in terms of special anisotropic spaces, which enables us to obtain more precise estimates of its smoothness.

### Periodic solutions of a weakly nonlinear system of partial differential equations with pulse influence

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 601–605

We establish conditions of the existence of solutions periodic in *t* with period *T* for a weakly nonlinear system of partial differential equations with pulse influence.

### Approximation of the classes $C_{β}^{ψ} H_{ω}$ by generalized de la Valiée-Poussin sums

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 606-610

On the classes of periodic functions $C_{β}^{ψ} H_{ω}$, we study approximating properties of trigonometric polynomials generated by methods for summation of Fourier series.

### International Conference “INAMTAP’96 (Computer Science, Computational and Applied Mathematics: Theory, Application, and Prospects)”

Lukovsky I. O., Makarov V. L., Prokhur M. Z.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 611–612

### Ukrainian school-seminar "Nonlinear boundary value problems of mathematical physics and applications"

Berezovsky A. A., Konet I. M., Lenyuk M. P., Samoilenko A. M., Teplinsky Yu. V.

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 613–617