2019
Том 71
№ 11

All Issues

Volume 44, № 9, 1992

Obituaries (Ukrainian)

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.

Full text (.pdf)

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

Article (Ukrainian)

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

Andrychuk V. I.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

Completions of functional spaces on Peano continua

Bazilevich L. E.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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

Balinsky A. I.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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

Baranetskij Ya. O.

↓ Abstract   |   Full text (.pdf)

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

The spectral properties and properties of the L2-solutions of the nonlocal problem for second-order linear elliptic nondivergent-type equations that represent an isospectral disturbance of the Dirichlet problem are investigated.

Article (Ukrainian)

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

Botsenyuk А. N.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

Analysis of dissipative structures based on the gauss variational principle

Hafiychuk V. V., Lubashevsky I.O.

↓ Abstract   |   Full text (.pdf)

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.

Article (English)

On the Lyapunov convexity theorem with appications to sign-embeddings

Kadets V. М., Popov M. M.

↓ Abstract   |   Full text (.pdf)

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 $X$.

Article (Ukrainian)

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

Kalenyuk P. I., Nytrebych Z. M.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

Inverse problems of the theory of separately continuous mappings

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

↓ Abstract   |   Full text (.pdf)

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 $F_σ$.

Article (Ukrainian)

Reduction and geometric quantization

Mikityuk I. V., Prykarpatsky A. K.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

Parallel factorizations of polynomial matrices

Petrichkovich V. M.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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

Plakhta L. P.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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

Popova L. V.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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

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

↓ Abstract   |   Full text (.pdf)

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.

Article (English)

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

Samuliak R. V.

↓ Abstract   |   Full text (.pdf)

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.

Article (Ukrainian)

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

Syroyid I. -P. P.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Ukrainian)

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

Bobrik R. V.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Ukrainian)

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

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

↓ Abstract   |   Full text (.pdf)

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 $A$.

Brief Communications (Ukrainian)

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

Zabolotskii N. V.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Ukrainian)

Application of variational methods in the theory of parabolic equations

Zelenyak T. I.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Ukrainian)

Multiplication operator on a matrix polynomial

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

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (English)

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

Pikhurko О. B., Zarichnyi M. M.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Ukrainian)

Structure of integrable supersymmetric nonlinear dynamical systems on reduced invariant submanifolds

Kuybida V. S., Pritula N. N.

↓ Abstract   |   Full text (.pdf)

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.

Brief Communications (Ukrainian)

Minimum of modulus of dirichlet multisequence

Lutsyshyn M. R., Skaskiv O. B.

↓ Abstract   |   Full text (.pdf)

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, $x ∈ ℝ^p$.