# Volume 46, № 6, 1994

### Reduction of the multidimensional d’Alembert equation to two-dimensional equations

Barannik L. F., Barannyk A. F., Fushchich V. I.

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 651–662

We give a classification of the maximal subalgebras of rank $n - 1$ for the extended Poincare algebra $A\bar P (1.n)$, which is realized on the set of solutions of the d'Alembeit equation $\square u + \lambda u^k = 0$. These subalgebras are used for constructing the anzatses reducing this equation to differential equations with two invariant variables.

### Cauchy problem for an essentially infinite-dimensional parabolic equation with variable coefficients

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 663–670

The Cauchy problem for the equation $\cfrac{\partial u}{\partial t} = \mathcal{L}_x u = j(x) (u_x'')$ with positive essentially infinite-dimensional functionals $j(x)$ is studied in a certain Banach space of functions on an infinite-dimensional separable real Hilbert space.

### On the isomorphism of wreath products of groups

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 671–679

The well-known Neumann theorem on the isomorphism of standard wreath products is generalized to the wreath products of an arbitrary transitive permutation group and an abstract group.

### Extremal properties of nondifferentiable convex functions on euclidean sets of combinations with repetitions

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 680–691

A general approach is suggested for studying extremal properties of nondifferentiable convex functions on Euclidean combinatorial sets. On the basis of this approach, by solving the linear optimization problem on a set of combinations with repetitions, we obtain estimates of minimum values of convex and strongly convex objective functions in optimization problems on sets of combinations with repetitions and establish sufficient conditions for the existence of the corresponding minima.

### Normal structure of the jonquere group over a field of characteristic zero

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 692–698

The lattice of normal divisors of the Jonquere group over a field of characteristic zero is described.

### On the existence of a mixed sum of additive systems

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 699–707

A representation is obtained for a mixed sum of additive systems with values in a Banach ring *X* with identity and norm.

### On necessary conditions of equivalence of gaussian measures corresponding to homogeneous random fields

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 708–719

We give necessary conditions of equivalence of probability measures corresponding to generalized homogenous Gaussian fields. For the most part, the results are also new for standard homogenous fields and even in the one-dimensional case, i.e., for stationary processes.

### $T$-differentiable functionals and ther critical points

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 720–728

The critical points of the functionals $F:\; D \subset X \rightarrow \mathbb{R}$ defined on "nonlinear" sets $D$ in the topological vector spaces $X$ are studied. A construction of a $T$-derivative is suggested for these functionals and compared with to known constructions. The concept of a weak critical point is introduced and Coleman's principle is justified for $T$-differentiable functionals.

### Control of singular distributed parabolic systems with discontinuous nonlinearities

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 729–736

We present the statement of control problems for singular distributed parabolic systems with discontinuous nonlinearties. Sufficient conditions for the existence of the optimal “control-state” pair are established under the assumption that the set of admissible “control-state” pairs is nonempty. The problem of existence of serniregular solutions is studied for the equation of state of a distributed system. It is not assumed that the nonlinearity of this equation increases sublinearly in the phase variable or possesses a bounded variation on any segment of the straight line.

### Solutions of systems of nonlinear functional-differential equations bounded in the entire real axis and their properties

Pelyukh G. P., Samoilenko A. M.

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 737–747

For a system of nonlinear functional-differential equations with a linearly transformed argument, we establish the existence and uniqueness conditions for a solution bounded in the entire real axis and study the properties of this solution.

### Dependence of Green’s function of an $e$-dichotomous differential equation with matrix projector in the space M on parameters

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 748–753

In the space $\mathfrak{M}$ of bounded numeric sequences, we consider dichotomonly differential equation with a matrix projector. We prove that Green's function of this equation exists, and give the conditions, under which it is continuous and differentiable with respect to the parameters in the equation.

### Nontrivial periodic solutions of a nonautonomous system of second-order differential equations

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 754–759

We prove theorem, in which basic conditions for the existence of nontrivial periodic solutions are formulated in terms of the properties of the elements of the matrix of a linear approximation to a system.

### Refined estimates for the ε-entropy of the classes $KH_0^α$

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 760–764

By the methods of differential pulse-code modulation and “generalized” polygonal lines, we obtain almost exact estimates for the ɛ-entropy of classes simulating signals of various types. The complexity of coding and reconstruction of functions from the classes under consideration is investigated. We present a numerical solution of the problem of minimization of constants in the order-of-magnitude inequality for the ɛ-entropy of the classes $KH_0^α$.

### Limiting process for integral functionals of a wiener process on a cylinder

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 765–768

We prove that integral functionals, whose integrands are bounded functions of a Wiener process on a cylinder, weakly converge to the process $w_1(τ(t)),\, τ(t) = β_1 t + (β_2 − β_1) \text{mes} \{s:w 2(s) ≥ 0,\, s < t\}$, where $w_1(t)$ and $w_2(t)$ are independent one-dimensional Wiener processes, $β_1$ and $β_2$ are nonrandom values, and $β_2 ≥ β_1 ≥ 0$.

### On a set of partial limits of a sequence of weighted sums of independent random variables

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 769–775

We give the complete description of the set of partial limits for a large class of sequences of weighted sums of independent random variables with triangular matrices of coefficients.

### Circular $m$-functions

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 776–781

Circular $m$-functions are introduced on smooth manifolds with boundary. We study the distribution of their critical circles and construct an example of a four-dimensional manifol $dM^4$ with boundary $∂M^4$ that satisfies the condition $ξ(∂M 4) = ξ(M^4,∂M^4) = 0$ but does not contain any circularm-function. We prove that a manifold with boundary $M^n (n ≥ 5)$ such that $ξ(∂M^n , ∂M^n ) = 0$ always contains a circularm-function without critical points in the interior manifold.

### On a method for constructing one-frequency solutions of a nonlinear wave equation

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 782–784

A method for constructing one-frequency solutions of nonlinear wave equations is suggested. This approach is based on a modified representation of asymptotic expansions by using special periodic Atebfunctions. This method makes it possible to obtain approximate solution of the problem under consideration without difficulty.

### Necessary and sufficient conditions of harmonicity of functions of infinitely many variables (Jacobian case)

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 785–788

A criterion of harmonicity of functions in a Hilbert space is given in the case of weakened mutual dependence of the second derivatives.

### Strict quasicomplements and the operators of dense imbedding

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 789–792

A quasicomplement $М$ ofasubspace $N$ of a Banach space $X$ is called strict if $M$ does not contain an infinite-dimensional subspace $M_1$, such that the linear manifold $N + M_1$, is closed. It is proved that if $X$ is separable, then $N$ always has a strict quasicomplement. We study the properties of the dense imbedding operator restricted to infinite-dimensional closed subspaces of the space, where it is defined.