# Volume 45, № 4, 1993

### Periodic Gibbs states

Barbulyak V. S., Kondratiev Yu. G.

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 451–458

Periodic Gibbs states for quantum lattice systems are investigated. We formulate the definition of the periodic Gibbs states and the measures associated with them. Theorems of existence are proved for these states. We also prove the existence of the critical temperature for the system of anharmonic quantum oscillators with pairwise interaction.

### On pseudoanalyticity of continuous functions with constant $σ$-extension

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 459–465

The theorem on pseudoanalyticity of continuous functions with constant $σ$-extension is proved; this is an analog of the well known results due to Bohr, Rademacher, Men'shov, and Trokhimchuk concerning the analyticity of functions with constant extension.

### The structure of linear extensions with the Favard type conditions II. Linear extensions with the additivity property of recurrent motions

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 466–471

We study the structure of linear extensions with external powers satisfying the condition of additivity of recurrent motions.

### Asymptotic normality of spherical means of nonlinear functionals of Gaussian random fields

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 472–480

The central limit theorem is proved for the integral-type functionals of nonlinear transformations of two-and three-dimensional uniform isotropic Gaussian random fields. A theorem on convergence of finite-dimensional distributions of these functionals to the corresponding distributions of the Wiener process is also established.

### On Galerkin's method for evolutionary equations with pulse influence

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 481–486

A new modification of Galerkin's approximation scheme is proposed for evolutionary equations with pulse influence and its convergence is proved. The result obtained is extended to the pulse evolutionary equations with deviating argument.

### Pasting of two processes with independent increments

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 487–491

A terminating stochastically continuous strictly Markov process is obtained as a result of pasting two nonterminating homogeneous stochastically continuous Markov processes with independent increments, one of which is semicontinuous. It is shown that this process can be extended to a complete homogeneous stochastically continuous strictly Markov Feller process. Previously, this problem has been solved by the author under stronger restrictions-both pasting processes were semicontinuous.

### On behavior of solutions of the quasilinear second-order parabolic equations in unbounded noncylindrical domains

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 492–499

The theorems of uniqueness of solutions are formulated in the classes of increasing functions for a mixed initial boundary value problem for the second-order degenerate quasiparabolic equations in unbounded noncylindrical domains. We present*a priori* estimates of a special kind, analogous to the Saint-Venant principle. The proofs are based on the method of introducing a parameter.

### Boundary-value problems for the helmholtz equation in an angular domain. II

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 500–519

We investigate boundary-value problems that appear in the study of the diffraction of acoustic waves on an infinite cylinder (with a cross section of an arbitrary shape) placed inside a wedge so that the axis of the cylinder is parallel to the edge of the wedge. The potential theory which enables one to reduce these boundary-value problems to integral equations is elaborated.

### Invariant tori of linear extensions which do not have Green's functions

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 520–524

We study the integral representation of invariant tori of linear extensions which do not have Green's functions.

### $G$-convergence of periodic parabolic operators with a small parameter by the time derivative

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 525–538

In this paper, we consider a sequence $\mathcal{P}^k$ of divergent parabolic operators of the second order, which are periodic in time with period $T = \text{const}$, and a sequence $\mathcal{P}^k_{\psi}$ of shifts of these operators by an arbitrary periodic vector function $ \psi \in X = \{L^2((0, T) \times \Omega)\}^n$ where $\Omega$ is a bounded Lipschitz domain in the space $\mathbb{R}^n$. The compactness of the family $\{P_{Ψ^k} ¦ Ψ \in X, k \in ℕ\}$ in $k$ with respect to strong $G$-convergence, the convergence of arbitrary solutions of the equations with the operator $\mathcal{P}^k_{\psi}$, and the local character of the strong $G$-convergence in $Ω$ are proved under the assumptions that the matrix of coefficients of $L^2$ is uniformly elliptic and bounded and that their time derivatives are uniformly bounded in the space $L^2(Ω; L^2(0,T))$.

### Symmetry and non-lie reduction of the nonlinear Schrödinger equation

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 539–551

The nonlinear Schrödinger-type equations invariant with respect to the extended Galilean group are described. We study the conditional symmetry of such equations, realize the reduction procedure, and construct the classes of exact solutions.

### Asymptotic expansions of solutions to singularly perturbed systems

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 552–560

Under the condition that a degenerate system has an exponentially stable integral manifold, an asymptotic expansion of the Cauchy problem that generalizes the well known Vasil'eva expansion is constructed for a perturbed system.

### Asymptotics of the general solution to a linear singularly perturbed system of second-order differential equations with degeneracy

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 561–575

By using the perturbation methods and Newton diagrams, we study the structure and construct the asymptotics of the general solution to a linear singularly perturbed system of ordinary second-order differential equations in the case where the matrix by the higher derivative is degenerate.

### Explicit relations for the reduced module and harmonic measure

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 576–578

Exact relations are established for the reduced modules of the families of curves separating certain connected subsets in arbitrary connected domains. New explicit expressions for the harmonic measure of a connected boundary subset are obtained as corollaries.

### On cubic formulas related to the mixed Hermite splines

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 579–581

A cubic formula containing partial integrals is considered on a class of functions of two variables. It is shown that the integral of a mixed Hermite spline gives the best cubic formula for the given class. The coincidence of cubic formulas, which are exact for odd and even mixed Hermite splines, is established.

### Two theorems on closeness of the set of Laplace-type transformations

Ukr. Mat. Zh. - 1993. - 45, № 4. - pp. 582–584

We formulate and prove two theorems on the limit of the Laplace transformations for random variables with values in the space of nonnegative measures.