# Volume 44, № 9, 1992

### An issue dedicated to the illustrious memory of Mykol Mykolayovych Bogolyubov

Bogolyubov N. N., Mitropolskiy Yu. A., Prykarpatsky A. K., Rudavsky Yu.K., Samoilenko A. M., Vakarchuk I. V.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1155-1156

### Tate-Shafarevich products in elliptic curves over pseudolocal fields with residue fields of characteristic 3

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1157–1165

Let $k$ be a general local field with pseudolocal residue field $x$, char $x = 3$, and $A$ an elliptic curve defined over $k$. It is proved that the Tate-Shafarevich product $H^1(k, A)×A_k→ Q/ℤ$ of the group $H^1(k, A)$ of principal homogeneous spaces of the curve $A$ over $k$ and the group $A_k$ of its $k$-rational points is left nondegenerate.

### Completions of functional spaces on Peano continua

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1165–1170

A description of the topology of the pair $(C(ΠxI), C(Π, I))$ for the Peano continuum $Π$, where $C(Π, I)$ is the closure in the hyperspace $\exp (ΠxI)$ of the image of the space of continuous functions $C(Π, I)$ under the natural embedding, is obtained.

### On discrete models on one-dimensional lattices with specified Lie algebra of symmetries

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1170–1174

Poisson realizations of the classical YBZF-algebras are constructed from special examples using the simplest rational solution of the classical Yang-Baxter equation, the Poisson realization of a Lie algebra, the moment mapping, and a generalization of $L$-operators. Sufficient conditions under which additional constraints on the image of the moment mapping are satisfied are established.

### Similitude operators generated by nonlocal problems for second-order elliptic equations

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1174–1181

The spectral properties and properties of the L^{2}-solutions of the nonlocal problem for second-order linear elliptic nondivergent-type equations that represent an isospectral disturbance of the Dirichlet problem are investigated.

### On the solvability of the initial- and boundary-value problem for the system of semilinear equations of magnetoelasticity

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1181–1186

An existence theorem is obtained for the system of semilinear equations of magnetoelasticity. The asymptotic behavior of the solutions over time is established.

### Analysis of dissipative structures based on the gauss variational principle

Hafiychuk V. V., Lubashevsky I.O.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1186–1192

The Gauss variational principle is suggested as a method of finding the solutions of dissipative systems. Using as an example a system of two reaction-diffusion equations, approximate solutions are found for the case of auto-solitons and periodic dissipative structures.

### On the Lyapunov convexity theorem with appications to sign-embeddings

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1192–1200

It is proved (Theorem 1) that for a Banach space $X$ the following assertions are equivalent:

(1) the range of every $X$- valued $σ$- additive nonatomic measure of finite variation possesses a convex closure;

(2) $L_1$ does not signembed in $X$.

### Method for separation of variables for bilinear matrix functional equation and its applications

Kalenyuk P. I., Nytrebych Z. M.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1201–1209

Necessary and sufficient conditions for the solvability of a bilinear matrix functional equation are presented. The conditions are applied in the construction of the solutions of systems of partial differential equations.

### Inverse problems of the theory of separately continuous mappings

Maslyuchenko V. K., Mykhailyuk V. V., Sobchuk V. S.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1209–1220

The present paper investigates the problem of constructing a separately continuous function defined on the product of two topological spaces that possesses a specified set of points of discontinuity and the related special problem of constructing a pointwise convergent sequence of continuous functions that possesses a specified set of points of nonuniform convergence and set of points of discontinuity of a limit function. In the metrizable case the former problem is solved for separable $F_σ$-sets whose projections onto every cofactor is of the first category. The second problem is solved for a pair of embedded $F_σ$.

### Reduction and geometric quantization

Mikityuk I. V., Prykarpatsky A. K.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1220–1228

A construction is created that makes it possible to geometrically quantize a reduced Hamiltonian system using the procedure of geometric quantization realized for a Hamiltonian system with symmetries (i.e., to find the discrete spectrum and the corresponding eigenfunctions, if these have been found for the initial system). The construction is used to geometrically quantize a system obtained by reduction of a Hamiltonian system that determines the geodesic flow on an $n$-dimensional sphere.

### Parallel factorizations of polynomial matrices

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1228–1233

Conditions are established under which suggested factorizations of polynomial matrices over a field are parallel to factorizations of their canonical diagonal forms. An existence criterion of these factorizations of polynomial matrices is indicated and a method of constructing them is suggested.

### On the set of singular points of the actions of finite groups on $(S^n)^k$

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1233-1237

Realizations of integral $D_3$-modules of rank 2 on $(S^n)^k$ for the dihedral groups $D_3$ are studied. Cohomologies of the sets of the singular points of the actions of the semidirect products $ℤ / pXℤ / q$ and the quaternion groups $Q$ on $(S^n)^k$ are investigated.

### On trivial differential equations in the spaces $L_p,\; 0 < p < 1$

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1238–1242

A description of the set $X_p$ of all solutions of the trivial Cauchy problem in $L_p, o< p <1$, is presented. The principal result is Theorem 2, which asserts that $X_p$ is a closed subspace of the $p$-Banach space $H_p$ of all curves in $L_p$ that satisfy a Hölder condition of order $p$ and emanate from O relative to the $p$-norm, which is equal to the minimal constant in the Hölder condition.

### Category of topological jet manifolds and certain applications in the theory of nonlinear infinite-dimensional dynamical systems

Fil' B. N., Prykarpatsky A. K.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1242–1256

A new category of topological jet manifolds is proposed for the purpose of investigating exact finite-dimensional approximations of nonlinear dynamical systems on infinite-dimensional functional manifolds. Differential geometry structures on these manifolds and their applications to the theory of integrability in quadratures of nonlinear dynamical Lax-type systems are studied.

### Hamiltonian analysis of exact integrability of the quantum 3-level superradiance Dicke model

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1256–1264

It is proved that the quantum 3- level superradiance Dicke model is exactly integrable. The Lax representation of the operator system of evolution equations is derived on the basis of a theory of Lie algebras of currents. The method employed in discussions of the quantum inverse scattering problem is applied to obtain quantum analogs of the action-angle variables. The spectra of the energy operator and of other quantum motion integrals as well as the exact one- and multiparticle excitation eigenstates of the model are constructed. It is shown that the model possesses states of constrained quasiparticles (quantum solitons) that induce superradiance pulses.

### Matrix solutions of the equation $\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x ,U] + 4cU:$ extension of the method of the inverse scattering problem

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1264-1275

Complex solution matrices of the nonlinear Schrödinger equation $\mathfrak{B}U_t = - U_{xx} + 2U^3 + \mathfrak{B}[U_x ,U] + 4cU:$ are found and the method of the inverse scattering problem is subjected to a natural extension. That is, for the nonself-conjugate $L-A$ Lax doublet that arises for this equation, the presence of chains of adjoint vectors for the operator $L$ is taken into account by means the corresponding normed chains. A uniqueness theorem for the Cauchy problem for the above Schrödinger equation is obtained. Here $$\mathfrak{B} = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right),[M,N] = MN - NM$$ and $c$ is a parameter.

### On the mean-square stability for a harmonic oscillator with random parameter

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1276–1278

Sufficient mean-square stability conditions of a harmonic oscillator whose random parameter is an Ornstein-Uhlenbeck process are obtained.

### Existence of Cesàro limit of bounded solution of evolution equation in banach space

Gorbachuk E. L., Yakons'ka N. O.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1279–1280

An existence criterion for the Cesàro limit $$\left( {\mathop {\lim }\limits_{t \to \infty } \frac{1}{t}\int\limits_0^t {y(\xi )d\xi } } \right)$$ of a bounded solution $y(t)$ of the problem $dy(t)/dt = Ay(t), y(0)=y_0, t ∈ [O, ∞)$, where $ A$ is a closed linear operator with dense domain of definition $D(A)$ in a reflexive Banach space $E$, is obtained under the condition that there exists a sufficiently small interval $(O, δ)$ belonging to the set of the regular points $ρ(A)$ of the operator $A$.

### Lower types of $δ$-subharmonic functions of fractional order

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1280–1284

It is proved that the lower types of functions $T(r, u)$ and $N(r, u) = N(r, u_1) + N(z, u_2)$ relative to the proximate order $ρ(r)$ of a function $u=U_1−u_2$ of fractional order $ρ δ$-subharmonic in $ℝ^m, m > - 2,$ coincide, that is, are simultaneously minimal or mean. In the case of an arbitrary proximate order $ρ(r)$, the assertion is, in general, false.

### Application of variational methods in the theory of parabolic equations

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1284–1287

The basic mathematical principles in the application of variational methods in the theory of parabolic equations are set forth.

### Multiplication operator on a matrix polynomial

Al'-Tundzhi M., Mikityuk Ya. V.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1287–1289

It is proved that the study of a perturbed multiplication operator on a matrix polynomial in the space $L_2(ℝ, ℂ^n)$ may be reduced to the study of a perturbed multiplication operator with independent variable in the space $L_2(ℝ, ω, ℂ^N)$ with weight $ω$ satisfying the Mackenhaupt condition.

### Lifting functors to Eilenberg-Moore category of monad generated by functor $C_p C_p$

Pikhurko О. B., Zarichnyi M. M.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1289–1291

The second iteration of the contravariant functor of spaces of continuous functions in the pointwise convergence topology is a functorial part of a monad (triple) on the category of Tikhonov spaces. The problem of lifting functors to the Eilenberg-Moore category of this monad is investigated.

### Structure of integrable supersymmetric nonlinear dynamical systems on reduced invariant submanifolds

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1292–1295

Based on an analysis of a supersymmetric extension of the algebra of pseudodifferential operators on $ℝ^1$ an infinite hierarchy of supersymmetric Lax-integrable nonlinear dynamical systems is constructed by means of the Yang-Baxter $ℛ$-equation method. The structure of these systems on reduced invariant submanifolds specified by a natural invariant Lax-type spectral problem is investigated.

### Minimum of modulus of dirichlet multisequence

Lutsyshyn M. R., Skaskiv O. B.

Ukr. Mat. Zh. - 1992. - 44, № 9. - pp. 1295–1297

Conditions are established under which the following relation is satisfied: $$M(x) = (1 + o(1))m(x) = (1 + o(1))\mu (x)$$ as $|x |→ + ∞$ outside a sufficiently small set, for an entire function $F(z)$ of several complex variables $z ∈ ℂ_p,p ≥ 2$, represented by a Dirichlet series. Here $M(x) = \sup \{|F(x+iy) |: y ∈ ℝ^p\}$ and $m(x) = \inf \{ |F(x+iy) |:y ∈ ℝ^p,$ with $μ(x)$ the maximal term of the Dirichlet series, $x ∈ ℝ^p$.