# Volume 48, № 5, 1996

### Classification of modules of dimension $n$ and multiplicity 1 over the Weyl algebra $A_n$

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 579-587

Modules of the Gelfand - Kirillov dimension $n$ and the multiplicity 1 over the Weyl algebra $A_n$ are classified.

### Generalized Green operator of a boundary-value problem with degenerate pulse influence

Boichuk О. A., Chuiko E. V., Chuiko S. M.

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 588-594

We find necessary and sufficient conditions for solvability of nonhomogeneous linear boundary-value problems for systems of ordinary differential equations with impulsive force in a general case where the number of boundary-value conditions in not equal to the order of the differential systems (Noetherian problems). We construct a generalized Greens's operator for boundary-value problems, not every solution of which can be extended from the left end point to the right end point of the interval where the solution is defined.

### Reduction of differential equations and conditional symmetry

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 595-602

We determine conditions under which partial differential equations are reducible to equations with a smaller number of independent variables and show that these conditions are necessary and sufficient in the case of a single dependent variable.

### A noncompact variational problem in the theory of riesz potentials. II

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 603-613

We study some generalizations of the well-known problem of minimization of the Riesz energy on condensers. Under fairly general assumptions, we establish necessary and sufficient conditions for the existence of minimal measures.

### On the $Γ$-Convergence of integral functionals defined on sobolev weakly connected spaces

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 614-628

We introduce and study the concept of Γ-convergence of functionate*I* _{ s }:*W* ^{ k,m }(Ω)→ℝ,*s*=1,2,..., to a functional defined on (*W* ^{ k,m }(Ω))^{2} and describe the relationship between this type of convergence and the convergence of solutions of Neumann variational problems. For a sequence of integral functionate*I* _{ s }:*W* ^{ k,m }(Ω)→ℝ, we prove a theorem on the selection of a subsequence Γ-convergent to an integral functional defined on (*W* ^{ k,m }(Ω))^{2}.

### On the problem of strong summability on disks

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 629-634

We establish estimates for the sequence of norms of nonlinear functional appearing in the problem of strong summability on disks of the Fourier series of functions continuous in a two-dimensional torus.

### Asymptotic distinguishing of autoregressive processes

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 635-641

We study the behavior of the probability of errors of the Neumann-Pearson criterion under various null and alternative hypotheses by the results of observations of autoregressive processes.

### Exponential polystability of separable motions

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 642-649

We establish conditions of exponential*x* _{1}-stability and polystability for systems with separable motions. Stability conditions of these types are obtained by using the Lyapunov functions (scalar and matrix).

### The law of iterated logarithm for solutions of stochastic differential equations

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 650-655

We prove the law of iterated logarithm for solutions of stochastic differential equations with perturbed periodic coefficients.

### Optimal methods for specifying information in the solution of integral equations with analytic kernels

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 656-664

We determine the exact order of the minimum radius of information in the logarithmic scale for Fredholm integral equations of the second kind with periodic analytic kernels and free terms. We show that the information complexity of the solution of Fredholm equations with analytic kernels is greater in order than the complexity of the approximation of analytic functions. This distinguishes the analytic case from the case of finite smoothness.

### Criteria of convergence for quasiconformal mappings and their generalizations

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 665-674

We establish necessary and sufficient conditions for the convergence of normalized homeomorphisms of Sobolev class in terms of the Fourier transforms of complex characteristics in the case where the upper bound of dilations is exponentially bounded in measure. This allows us to construct various metrics generating locally uniform convergence of mappings.

### New conditions for averaging of nonlinear dirichlet problems in perforated domains

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 675-694

We study the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary. We consider a class of equations admitting degeneration with respect to the gradients of solutions. We prove a pointwise estimate for solutions of the model nonlinear boundary-value problem and construct an averaged boundary-value problem under new structural assumptions concerning perforated domains. In particular, it is not assumed that the diameters of cavities are small as compared to the distances between them.

### Symmetry reduction and exact solutions of the multidimensional monge-ampere equation

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 695-701

By using the subgroup structure of the generalized Poincare group*P*( 1, 4), we perform a symmetry reduction of the multidimensional Monge-Ampere equation to differential equations with a smaller number of independent variables. On the basis of solutions of the reduced equations, we construct some classes of exact solutions of the equation under consideration.

### On the well-posedness of derichlet problems for the many-dimensional wave equation and lavrent’ev-bitsadze equation

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 702-706

We prove the unique solvability of the Dirichlet problems for the many-dimensional wave equation and Lavrent’ev-Bitsadze equation.

### A noncommutattve analog of the cohen theorem

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 707-710

By using weakly primary right ideals, we prove an analog of the Cohen theorem for rings of principal right ideals.

### Approximation of measurable periodic functions in measure by step functions

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 711-715

For spaces defined by a function $ϕ$ of the type of modulus of continuity, we prove direct and inverse Jackson theorems for the approximation by step functions with uniform partition.

### On variations for mappings with restrictions on dilation in measure

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 716-718

We construct variations for classes of homeomorphisms with generalized derivatives in the case where restrictions in measure of general form are imposed on large values of dilation. We use the method for the construction of variations suggested by Gutlyanskii.

### On a nonlinear equation unsolved with respect to the levy laplacian

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 719-721

We propose a method for the solution of the nonlinear equation*f*(*U*(*x*),Δ*U*(*x*))=*F*(*x*) (Δ_{ L } is an infinite-dimensional Laplacian, Δ_{ L } *U*(*x*)=γ, γ≠0) unsolved with respect to the infinite-dimensional Laplacian, and for the solution of the Dirichlet problem for this equation.