# Volume 50, № 12, 1998

### On differentiability of open mappings

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1587–1600

The well-known Men’shov and Gehring-Lehto theorems on the differentiability of topological mappings of plane domains are generalized to the case of continuous open mappings of many-dimensional domains.

### Generalized and classical almost periodic solutions of Lagrangian systems convex on a compact set

Parasyuk I. O., Zakharin S. F.

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1601–1608

By using the variational method, we establish sufficient conditions for the existence of generalized Besicovitch almost (quasi)periodic solutions and classical quasiperiodic solutions of natural Lagrangian systems with force functions convex on a compact set.

### Random integral manifolds of systems of differential equations with unbounded operator in a Banach space

Kolomiyets V. G., Mel'nikov A. I.

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1609–1614

We establish existence conditions for a random integral manifold of a certain class of differential systems with unbounded sectorial operator and random right-hand side in a Banach space.

### On elements of the Lax-Phillips scattering scheme for $ρ$-perturbations of an abstract wave equation

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1615–1629

We give the definition of $ρ$-perturbations of an abstract wave equation. As a special case, this definition includes perturbations with compact support for the classical wave equation. By using the Lax-Phillips method, we study scattering of “$ρ$-perturbed” systems and establish some properties of corresponding scattering matrices.

### Strong summability of multiple Fourier series and Sidon-type inequalities

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1630–1635

We study different versions of strong summation of *N*-dimensional Fourier series over polyhedrons and related estimates for integral norms of linear means of the Dirichlet kernels (Sidon-type inequalities).

### Solutions of systems of nonlinear difference equations that are continuous and bounded on the entire real axis and their properties

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1636–1645

For a system of nonlinear difference equations, we establish conditions for the existence and uniqueness of a solution bounded on the entire real axis and study its properties.

### Irresolvable topologies on groups

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1646–1655

We prove that there exist *ZFC* models in which every nondiscrete topological Abelian group can be decomposed into countably many dense subsets. This statement is an answer to the question raised by Comfort and van Mill. We also prove that every submaximal left-topological Abelian group is σ-discrete.

### The theory of the numerical-analytic method: Achievements and new trends of development. IV

Ronto M. I., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1656–1672

We analyze the application of the numerical-analytic method proposed by Samoilenko in 1965 to autonomous systems of differential equations and impulsive equations.

### Perturbed Lamé equation and the Buslaev phase

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1673–1679

We investigate the perturbed Lamé equation with 1-zone potential and give an explicit description of the geometric phase that is contained in the leading term of the series of an asymptotic solution.

### Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1680–1685

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_{tt} - a^{2}u_{xx} = g(*x, t*) u(0, t) = u(π, *t*) = 0, u(*x, t + T*) *=* u(*x, t*), *x ∈* ℝ, *t* ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator.

### On a criterion of $NP$-completeness

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1686–1691

We consider the problem of construction of criteria of completeness of sets with respect to polynomially bounded reducibilities. We present a nonstandard description of sets from the class *NP*, a brief proof of an analog of the well-known Cook theorem, and a criterion of *NP*-completeness.

### Analyticity of a free boundary in one problem of axisymmetric flow

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1692–1700

We prove the solvability of a boundary-value problem in the case where the Bernoulli condition is given on a free boundary in the form of an inequality. We establish the analyticity of the free boundary.

### Filtration of components of processes of random evolution

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1701–1705

The problem of estimation of a nonobservable component θ_{t} for a two-dimensional process (θ_{t}, ξ_{t}) of random evolution (*θ* _{t},ξ_{t});x_{t}, 0≤t≤*T*, is investigated on the basis of observations of ξ_{s}. *s≤t*, where *x* _{t} is a homogeneous Markov process with infinitesimal operator *Q*. Applications to stochastic models of a *(B,S)*-market of securities is described under conditions of incomplete market.

### On samples of independent random vectors in spaces of infinitely increasing dimension

Ruzhilo M. Ya, Stepakhno V. I.

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1706–1711

We study the asymptotic behavior of a set of random vectors ξ_{i}, *i* = 1,..., *m*, whose coordinates are independent and identically distributed in a space of infinitely increasing dimension. We investigate the asymptotics of the distribution of the random vectors, the consistency of the sets *M* _{m} ^{(n)} = ξ_{1},..., ξ_{m} and *X* _{n} ^{λ} = x *∈ X* _{n}: ρ(*x*) ≤ λn, and the mutual location of pairs of vectors.

### Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1712–1714

We indicate criteria for the coincidence of the Knopp kernels *K(f) K(A* _{f}), and *K (R* _{f}) of bounded functions *f(t)*; here, $$R_f \left( t \right) = \frac{1}{{P\left( x \right)}}\int\limits_{\left[ {0;\left. t \right)} \right.} {f\left( x \right)dP and A_f \left( t \right)} = \frac{1}{{\int_0^\infty {e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP} }}\int\limits_0^\infty {f\left( x \right)} e^{{{ - x} \mathord{\left/ {\vphantom {{ - x} t}} \right. \kern-\nulldelimiterspace} t}} dP$$ . In Particular, we prove that *K(f) = K(A* _{f}) ⇔ *K(f) = K(R* _{f}).

### Random variables determined by the distributions of their digits in a numeration system with complex base

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1715–1720

We study the distributions of complex-valued random variables determined by the distributions of their digits in a numeration system with complex base. We establish sufficient conditions for the singularity of such random variables, in particular, in the cases where their spectrum has Lebesgue measure zero (*C*-type singular distribution) or is a rectangle (*S*-type singular distribution).

### Vladislav Kirillovich Dzyadyk

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1721

### Index of volume 50 of „Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1722-1728