# Volume 47, № 12, 1995

### Differential operators determining solutions of Elliptic equations

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1587–1592

We construct differential operators*Lg(z), Kg(z), Nf¯(z), Mf¯z)* which map arbitrary functions holomorphic in a simply connected domain*D* of the plane*z=x+iy* into regular solutions of the equation $$W_{z\bar z} + A(z,\bar z)W_{\bar z} + B(z,\bar z)W = 0$$ and present examples of the application of these differential operators to the solution of fundamental boundary-value problems in mathematical physics.

### Integral manifolds and exponential splitting of linear parabolic equations with rapidly varying coefficients

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1593–1608

We study linear parabolic equations with rapidly varying coefficients. It is assumed that the averaged equation corresponding to the source equation admits exponential splitting. We establish conditions under which the source equation also admits exponential splitting. It is shown that integral manifolds play an important role in constructing transformations that split the equations under consideration. To prove the existence of integral manifolds, we apply Zhikov's results on the justification of the averaging method for linear parabolic equations.

### Some properties of multiparameter random fields

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1609–1621

We study some properties of multiparameter random fields, namely, the problems of absolute continuity of measures and averaging in the multiparameter case. For a special stochastic system, we present inequalities of large deviations.

### Averaging in Volterra set-valued integral equations

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1622–1626

This paper is devoted to the justification of one averaging method for Volterra integral set-valued equations.

### A remark concerning the modulus of smoothness introduced by Ditzian and Totik

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1627–1638

For each function*f*(*x*) continuous on the segment [−1, 1], we set \(\tilde f(t) = f(\cos t)\) . We study the relationship between the ordinary*k*th modulus of continuity \(\omega _k (\tau ,\tilde f^{(r)} )\) of the*r*th derivative \(\tilde f^{(r)}\) of the function \(\tilde f\) and the*k*th modulus of continuity \(\bar \omega _{k,r} (\tau ,f^{(r)} )\) with weight ϕ_{ r } of the rth derivative*f* ^{(r)} of the function*f* introduced by Ditzian and Totik. Thus, if*r* is odd and*k* is even, we prove that these moduli are equivalent as*t*→0.

### Boundary-value problems for charged hyperbolic-parabolic equations with characteristic line of changing type

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1639–1652

We establish the unique solvability of some boundary-value problems for a mixed second-order hyperbolic-parabolic equation.

### Spectral problems with boundary and conjugating conditions depending on a parameter

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1653–1660

We establish generalized orthogonality conditions and prove a theorem on completeness of the collection of eigenfunctions of a spectral problem with boundary and conjugating conditions depending on a parameter for piecewise homogeneous bounded connected and disconnected domains.

### On stability of motion of a symmetric body placed on a vibrating base

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1661–1666

We consider the problem of stabilization of a symmetric solid body rotating about a fixed point and show that its unstable states can be stabilized by vertical vibration.

### Some approaches to the construction of approximate solutions of the generalized Goursat problem for systems of certain quasilinear partial differential equations with deviating argument

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1667–1675

We construct and investigate monotone and alternating rapidly convergent two-sided methods for the approximate integration of the generalized Goursat problem, prove the existence and uniqueness of its regular solution, establish theorems on differential inequality and comparison, and obtain sufficient conditions for the existence of solutions of the indicated problem of fixed sign in a given domain.

### Investigation of a discrete dynamical system in a neighborhood of the invariant torus

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1676–1685

We study the behavior of a discrete dynamical system in a neighborhood of the invariant torus for the case where the trajectories may have arbitrary structure on the torus and establish conditions under which the system can be reduced to the canonical form in the indicated neighborhood.

### Equilibrium in quantum systems of particles with magnetic interaction

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1686–1691

We compute density matrices determining equilibrium states in the thermodynamic limit for a class of quantum systems. Interaction in these systems is described as a collective electromagnetic field whose scalar potential is equal to zero.

### On isotopes of groups. II

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1692–1703

We select canonical decompositions of the isotopes of groups, show that they are unique, and establish relationships between them. We also obtain external characteristics of the identities which imply the linearity or alinearity of the isotopy and the commutativity of the corresponding group. We describe the identities of linear isotopes of Abelian groups, i.e., of*T*-quasigroups, and suggest a new method for the description of isotopic closures of classes of groups.

### Calculation of Bessel functions by using continued fractions

Kostinskii O. Ya., Valeyev K. G.

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1704–1705

We propose a new method for the calculation of Bessel functions of the first kind of integral order. By using the Laplace transformation, we solve a linear differential equation that defines the generating function for the Bessel functions expressed in terms of continued fractions.

### To the mathematical theory of representation of information in neural nets

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1706–1707

We study irreducible nonorthogonal resolutions of the identity. The results obtained show that, in contrast to traditional requirements of “independent measurements” type, the noncommutative approach gives a more precise description of information systems.

### Approximation of compressions of periodic functions in the space $L_p,\; p < 1$

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1708-1711

Behavior of the best approximation of function compressions by trigonometric polynomials in $L_p,\; p < 1$ is investigated.

### A criterion for Banach manifolds to be finite-dimensional

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1712–1713

We extend the results obtained in [1] to the case of arbitrary Banach spaces and manifolds. We give an example of a continuous bijective mapping with discontinuous inverse which acts in a Banach space and differs from the identical mapping only in an open unit ball. A criterion for a Banach manifold to be finite-dimensional is established in terms of the continuity of inverse operators.

### On the construction of asymptotic approximations for a nonautonomous wave equation

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1714–1716

For a nonautonomous wave equation with homogeneous boundary conditions, we construct one-frequency approximations of asymptotic solutions by using periodic Ateb-functions. Resonance and nonresonance cases are considered.

### Existence of a smooth solution of one boundary-value problem

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1717–1719

We study a periodic boundary-value problem for the quasilinear equation*u* _{tt}−u_{xx}=F[u, u_{t}], u(0, t)=u(π, t)=0,*u*(x, t+2π)=u(x, t). We establish conditions that guarantee the validity of the uniqueness theorem.

### Properties of restrictions of the operator of multiplication by a continuous function

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1720–1722

For the operator*A* of multiplication by a continuous function*a (t)* in the Hilbert space*L* _{2}[0, b]=H, we give a description of two sets of infinite-dimensional subspaces with infinite codimensions:*I(A)*={N⊂H:*A/N* is an isomorphism},*K(A)={M⊂H: A/M* is a compact mapping}. As an application, we consider the problem of determining whether the sequence {a(t)e_{n}(*t*)}, where {e_{n}(*t*)} is an orthonormal basis in L_{2}[0,*b*], is an unconditional basis.

### Index of volume 47 of "Ukrainian Mathematical Journal"

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1723-1729