# Volume 45, № 7, 1993

### On the 80th birthday of Academician Aleksandr Yu. Ishlinskii

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 883-883

### Symmetrization and initial boundary-value problems for certain classes of nonlinear second order parabolic equations

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 884–892

Initial boundary-value problems are considered for equations of porous medium and $p$-Laplace types. By using the Schwarz symmetrization method, the $L_p$-estimates for the solutions of the original problems are obtained in terms of the analogous estimates for the corresponding symmetric solutions.

### Polynomials generating Hamming codes

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 893–897

A class of polynomials generating *q*-nary Hamming codes is studied. The criteria for a polynomial to belong to this class are established for the general case and for the case of prime polynomials. The conditions are determined under which reducible polynomials do not belong to the class of polynomials generating the *q*-nary Hamming codes.

### On the uniqueness of solutions to some boundary-value problems for differential equations in a domain with algebraic boundary

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 898–906

Classes of differential equations with constant coefficients admitting unique solutions of Dirichlet and Cauchy boundary-value problems are considered in a bounded domain with algebraic boundary. For the Dirichlet problem in a ball, the necessary and sufficient conditions for the uniqueness of the solution are obtained in the form of a countable sequence of inequalities polynomial in the coefficients of the equation.

### A quantum particle under the action of “white noise” type forces

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 907–914

We obtain the Schrödinger equation for the wave function of a particle interacting with the external field having a potential containing a component of the “white noise” type and the Kolmogorov equations for the distribution of a wave function.

### Some properties of polynomials orthogonal in a space with “intermediate” topology. Kernel function and extremal properties

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 915–923

A space of holomorphic functions is considered. A topology in this space is “intermediate” between the topology of uniform convergence and the topology of uniform convergence on compact sets. The properties of systems of orthonormal polynomials are studied in Hilbert spaces with this topology.

### The structure of Banach algebras of bounded continuous functions on the open disk that contain $H^{∞}$, the Hoffman algebra, and nontangential limits

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 924–931

Representable in the form $\mathcal{H}_B \bigcap G$, where $G = C(M(H^{\infty})) \overset{\rm def}{=} \text{alg}(H^{\infty}, \overline{H^{\infty}})$ and $\mathcal{H}_B$ is a closed subalgebra in $C(D)$ consisting of the functions that have nontangential limits almost everywhere on $\mathbb{T}$, and these limits belong to the Douglas algebra $B$. In this paper we describe the space $M(\mathcal{H}^G_B)$ of maximal ideals of the algebra $\mathcal{H}^G_B$ and prove that $M(\mathcal{H}^G_B) = M(B) \bigcup M(\mathcal{H}^G_0)$ and prove that $M(\mathcal{H}^G_0)$, where $\mathcal{H}^G_0$ is a closed ideal in $G$ consisting of functions having nontangential limits equal to zero almost everywhere on $\mathbb{T}$. Moreover, it is established that $H^{\infty \supset } [\overline Z ] \ne \mathcal{H}_{H^\infty + C}^G$ on the disk. The Chang-Marshall theorem is generalized for the Banach algebras $\mathcal{H}^G_B$. We also prove that $\mathcal{H}^G_B = {\rm alg}(\mathcal{H}^G_{H^{\infty}}, \overline{IB})$ for any Douglas algebra $B$, where $IB = \{u_{\alpha}\}_B$ are inner functions such that $\overline{u_{\alpha}} \in B$ on $\mathbb{T}$.

### On the existence of Lyapunov functions in problems of stability of integral sets

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 932–941

Розглянуто неавтономну систему звичайних диференціальних рівнянь. Доведено, що якщо ця система припускає рівномірно асимптотично стійку інтегральну множину, то в околі цієї множини існує функція, яка аналогічна функції Ляпунова.

### On nonlinear higher order elliptic systems positive in the first derivatives

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 942–947

Elliptic systems of a divergent type with a natural energy space $W^m_p \bigcap W^l_q$ are considered. Under certain restrictions imposed on the modulus of ellipticity of the lower part of the system, the Hölder property of the generalized solutions to this system and the Liouville theorem are established.

### On theg-convergence of nonlinear elliptic operators related to the dirichlet problem in variable domains

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 948–962

A notion of $G$-convergence of operators $A_s :\; W_s \rightarrow W_s^*$ to the operator $A:\; W \rightarrow W^*$ is introduced and studied under certain connection conditions for the Banach spaces $W_s,\; s = 1, 2, ... ,$ and the Banach space $W$. It has been established that the connection conditions for abstract space are satisfied by the Sobolev spaces $\overset{\circ}{W}^{k, m}(\Omega_s),\quad \overset{\circ}{W}^{k, m}(\Omega)$ ($\{\Omega_s\}$ is a sequence of perforated domains contained in a bounded domain $\Omega \subset \mathbb{R}^n$). Hence, the results obtained for abstract operators can be applied to the operators of Dirichlet problems in the domains $\Omega_s$.

### Averaging of randomly perturbed evolutionary equations

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 963–971

Evolutionary equations with coefficients perturbed by diffusion processes are considered. It is proved that the solutions of these equations converge weakly in distribution, as a small parameter tends to zero, to a unique solution of a martingale problem that corresponds to an evolutionary stochastic equation in the case where the powers of a small parameter are inconsistent.

### Asymptotic distinction of counting processes

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 972–979

A canonical representation is obtained for the logarithm of the likelihood ratio. Limit theorems describing its asymptotic behavior are proved. Using these theorems, we study the rate of decrease of the probability of an error of the second-kind in the Neyman-Pearson test.

### Limiting equilibrium of a plane with a circular hole

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 980–981

The elastoplastic state of a compressible isotropic plane with a circular hole is studied by the method of a small parameter. An unknown boundary separating the domain of limiting equilibrium and the elastic domain is determined. We construct the complex Kolosov-Muskhelishvili functions that describe the elastic state of a plane and compare these with the solutions of Galin's problem.

### Quasianalytic classes of functions on a plane

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 982–991

Quasianalytic classes of functions in a Jordan domain *G* are defined. We consider classes of functions defined by conditions imposed on the decrease rate of the best uniform polynomial approximations and investigate the dependence of the quasianalyticity of these classes on the geometric structure of a domain.

### On the $L_p -L_q$- estimates for the solutions of certain hyperbolic equations

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 992–1008

We establish conditions, which should be imposed on the exponentsp andq and the dimensionalityn of a space in order that the $L_p -L_q$-estimates should hold for the solutions of the Cauchy problem for a second-order hyperbolic equation with constant coefficients.

### Quasicqnformal mappings with restrictions in measure

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1009–1019

The principal result of the paper is a criterion of compactness for mappings quasiconformal in the mean. The semicontinuity of a deformation of homeomorphisms from the Sobolev class is also proved.

### On the Hölder property for functions from the class $B_{q, s}$

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1020–1028

A class of functions $B_{q, s}$ containing generalized solutions of some higher-order quasilinear parabolic equations is defined. We prove that $B_{q, s}$ is imbedded in the space of Hölder functions.

### Limiting behavior of the solutions to linear second-order parabolic equations

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1029–1038

Properties of solutions of parabolic equations in smooth cylindrical domains are studied. Conditions for existence of boundary non-tangents and $L_2$-limits as $t \rightarrow 0$ are found.

### Estimates of the solutions to the equations of motion for viscous fluid in a moving cavity with a porous damper

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1039–1044

Estimates for $L_2$-norms of solutions of equations of motion for viscous fluid in a moving elliptic cavity with a porous damper are obtained.

### Asymptotic behavior of the products of random matrices with values on a solvable Lie algebra

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1045–1048

The index of exponential growth is found for the products of random matrices with values on a solvable Lie algebra. The result is expressed via the eigenvalues of the terms.