# Volume 45, № 9, 1993

### Extreme modules over the Weyl algebra $A_n$

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1187-1197

Certain classes of simple modules over generalized Weyl algebras (extreme modules, polynomial modules, and modules with strong $D$-torsion) are classified. For these algebras, analogs of the Bernshtein and Stafford theorems are proved.

### Estimation of the discord time for a process of the Ornstein-Uhlenbeck type

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1198–1204

A consistent estimate is constructed for the discord time of a process of the Ornstein-Uhlenbeck type. The rate of the almost sure convergence of this estimate is investigated and the confidence interval is determined.

### Asymptotic normality of a projective estimator of an infinite-dimensional parameter of nonlinear regression

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1205–1214

A model of nonlinear regression is studied in infinite-dimensional space. Observation errors are equally distributed and have the identity correlation operator. A projective estimator of a parameter is constructed, and the conditions under which it is true are established. For a parameter that belongs to an ellipsoid in a Hilbert space, we prove that the estimators are asymptotically normal; for this purpose, the representation of the estimator in terms of the Lagrange factor is used and the asymptotics of this factor are studied. An example of the nonparametric estimator of a signal is examined for iterated observations under an additive noise.

### α-Stratified modules over the Lie algebra $\text{sl} (n, ℂ)$

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1215–1224

Properties of the generalized Weyl group are studied for α-stratified modules over the Lie algebra $\text{sl} (n, ℂ)$.

### On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1225–1231

Assume that (*X* _{n}) are independent random variables in a Banach space, (*b* _{n}) is a sequence of real numbers, S_{n}= ∑ _{1} ^{n} b_{i}X_{i}, and B_{n}=∑ _{1} ^{n} b _{i} ^{2} . Under certain moment restrictions imposed on the variables*X* _{n}, the conditions for the growth of the sequence (b_{n}) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (S_{n}/*B* _{n} ln ln B_{n})^{1/2}).

### The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation

Khoma L. G., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1232–1238

The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established.

### Stability of the trivial solution of a one-dimensional mathematical model of thermoelasticity

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1239–1252

Lyapunov stability is established for a one-dimensional physically linear mathematical model of thermoelasticity. For this purpose, the convergent iteration process is constructed; it consists of solving hyperbolic and parabolic problems successively by using new estimates for the solution of a mixed problem for the wave equation.

### On pairs of unbounded self-adjoint operators satisfying an algebraic relation

Ostrovskii V. L., Samoilenko Yu. S.

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1253–1258

Unbounded pairs of self-adjoint operators *A* and*B* satisfying the algebraic relation *F* _{1}(A)*B*=*BF* _{2}(A) are studied. For these relations, various definitions of “integrable” pairs of operators are presented and the class of “tame” relations is indicated; for the “tame” relations, the irreducible pairs are described and a structure theorem is presented.

### Investigation of the Cauchy problem for stochastic partial differential equations

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1259–1265

It is shown that the solutions of stochastic linear parabolic equations with Poisson perturbations are stabilized in the mean square. The problem of determining the reserve of stability for a rod under random perturbations is studied.

### Quantum oscillator in an infinite-particle harmonic system

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1266–1273

The behavior of a quantum oscillator in an infinite-particle system is studied for the case of linear interaction. The relation between the spectrum of the dynamic matrix of a complete system and oscillator damping is established. The dependence of the spectrum on the parameters of interaction is determined.

### On solvable groups, all proper factor groups of which have finite ranks

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1274–1281

This paper deals with finitely generated finitely approximable solvable groups of infinite special rank, all proper normal subgroups of which determine the factor groups of finite special ranks.

### Handle decompositions of simply-connected five-manifolds. II

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1282–1288

Handle decompositions of simply connected smooth or piecewise linear five-manifolds are considered. The basic notions and constructions necessary for proving further results are introduced.

### Boundary-value problems for the heat conduction equation with a fractional derivative in the boundary conditions. Difference methods for numerical realization of these problems

Berezovsky A. A., Kerefov A. A., Shkhanukov-Lafishev M. Kh.

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1289–1398

Boundary-value problems for the heat conduction equation are considered in the case where the boundary conditions contain a fractional derivative. Problems of this type arise when the heat processes are simulated by a nonstationary heat flow by using the one-dimensional thermal model of a two-layer system (coating — base). It is proved that the problem under consideration is correct. A one-parameter family of difference schemes is constructed; it is shown that these schemes are stable and convergent in the uniform metric.

### Correctness of multi-dimensional Darboux problems for the wave equation

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1299–1306

It is proved that multi-dimensional Darboux problems for the wave equation are correct in the domain \(D_3 \subset E_{m + 1} \) bounded by the surfaces ¦*x* ¦=*t + ɛ* and ¦*x* ¦=1*- t* and the plane *t =* 0, 0 ≤ ɛ < 1. The behavior of the solutions as ɛ → 0 is studied.

### On a nonlocal problem for a quasilinear first-order hyperbolic system with two independent variables

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1307–1311

A mixed problem with nonlocal conditions on a space variable is considered for a system of quasilinear first-order hyperbolic equations with two independent variables. The sufficient conditions of solvability of this system are given.

### On Green's function for the Helmholtz equation in a wedge

Mel'nik Yu. I., Podlipenko Yu. K.

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1312-1314

It is found that, in the spherical coordinate system, the fundamental solution of the Helmholtz equation in a wedge satisfies the Sommerfeld radiation conditions at infinity uniformly in angle coordinates.

### The McKean martingale for the homogeneous $R-D$-systems

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1315–1319

An analog of the McKean martingale is constructed for branching processes with a continuous phase space.