# Volume 47, № 3, 1995

### Noncommutative central limit theorem for Gibbs temperature states

Antoniouk A. Val., Antoniouk A. Vict., Kondratiev Yu. G.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 299–306

For Gibbs temperature states, the scheme of the proof of the noncommutative central limit theorem is given by using the commutative central limit theorem for corresponding Euclidean measures. Applications are constructed for the model of a temperature-anharmonic crystal and the generalized Ising model with compact continuous configuration space.

### Control over linear pulse systems

Akhmetov M. U., Perestyuk N. A., Tleubergenov M. I.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 307–314

Rank conditions for control of linear pulse systems are established. The Pontryagin maximum principle is obtained in sufficient form. An example of control synthesis in a problem for linear pulse systems is given.

### A theorem about snakes for weak Cartesian systems

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 315–321

It is shown that Karlin's theorem about snakes remains true for polynomials in any weak Cartesian system of continuous functions on a compact set without any change in the statement, and the existence part is valid for arbitrary weak Chebyshev systems.

### Integral representation of an analytic function in a ring and its applications

Cherskii Yu. I., Kerekesha P. V.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 322–329

An integral representation of an analytic function in a ring with corresponding limit values on the boundary is obtained. A new singular integral equation is suggested and solved in quadratures by using an integral representation and by complete investigation of the Cárleman problem for a ring (in the normal case).

### Non-Gaussian limit distributions of solutions of the many-dimensional Bürgers equation with random initial data

Leonenko N. N., Li Zhanbing, Rybasov K. V.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 330–336

Limit distributions of solutions of the multidimensional Bürgers equation are found in the case where an initial condition is a random field of type χ^{2} of degree*k* with a long-range dependence.

### Construction of analogs of the Lyapunov equation for a matrix polynomial

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 337–343

We develop a method for localization of the eigenvalues of a matrix polynomial. This method is related to a generalization and solution of the Lyapunov equation.

### Separately continuous functions on products of compact sets and their dependence on $\mathfrak{n}$ variables

Maslyuchenko V. K., Mykhailyuk V. V.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 344-350

By using the theorem on the density of the topological product and the generalized theorem on the dependence of a continuous function defined on a product of spaces on countably many coordinates, we show that every separately continuous function defined on a product of two spaces representable as products of compact spaces with density $≤ \mathfrak{n}$ depends on n variables. In the case of metrizable compact sets, we obtain a complete description of the sets of discontinuity points for functions of this sort.

### Approximate solution of the Fokker-Planck-Kolmogorov equation

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 351–361

For the Fokker-Planck-Kolmogorov equation, the higher approximations are constructed by using the Bogolyubov averaging method.

### Iterative reduction of the Dirichlet problem to the Neumann problem

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 362–369

An iteration procedure for reduction of the Dirichlet boundary-value problem to the Neumann problem is suggested. Estimates of the rate of convergence are established.

### Periodic solutions of nonlinear differential equations with pulse influence in a banach space

Perestyuk N. A., Slyusarchuk V. Yu.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 370–380

Rank conditions for control of linear pulse systems are established. An example of control synthesis in a problem for linear pulse systems is given.

### Necessary and sufficient conditions for oscillation of solutions of nonlinear pulse systems with multiplicatively separated right-hand side

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 381–389

Necessary and sufficient conditions for oscillation of solutions of nonlinear differential equations with fixed times of pulse influence and multiplicatively separated right-hand sides are established.

### On the Radii of univalence of Gel'fond-Leont'ev derivatives

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 390–399

Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions $$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$ analytic in $\{ z: |z| < R \}$, and let $$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ be a formal power series. We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing sequence, $f \in A(R)$, and $|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, then the sequence $(\rho_n)$ of radii of univalence of the Gel'fondLeont'ev derivatives satisfies the relation $$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$ The case where the condition $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$, is not satisfied is also considered.

### Boundary-value problem in an infinite layer

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 400–402

We establish necessary and sufficient conditions for a nonlocal two-point boundary-value problem in an infinite layer for the equation $$\frac{{\partial ^2 u(x,t)}}{{\partial t^2 }} + P\left( {\frac{\partial }{{\partial x}}} \right)\frac{{\partial u(x + h_1 ,t)}}{{\partial t}} + Q\left( {\frac{\partial }{{\partial x}}} \right)u(x + h_2 ,t) = 0,$$ where*P(s)* and*Q(s)* are polynomials in*s*∈ℂ^{ m } with constant coefficients, to have infinite type and be degenerate.

### Generalization of some extremal properties of splines

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 403–407

We generalize well-known inequalities for the norms of the derivatives of periodic splines with minimal defect, perfect splines, and monosplines.

### Descriptive classes of sets and topological functors

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 408–410

It is proved that the image of a normal functor from the Stone-Cech compactification of the projective class of sets also belongs to this class.

### On the invertibility of differential operators of the second order

Baskakov A. G., Chernyshov M. K.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 411–413

We present sufficient conditions for the invertibility of a second-order differential operator with variable coefficients in the space L_{p}.

### Description of the operators and isomorphisms of the space of continuous functions commuting with the operator of multiplication

Berezovsky N. I., Linchuk S. S.

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 414–418

We propose the description of the operators and isomorphisms of the space *C[a, b]* commuting with the operator of multiplication by a continuous strictly piecewise monotone function.

### Dependence of the Green's function of a linear extension of a dynamical system on a torus upon a parameter

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 419–421

We study the dependence of the Green's function of a linear extension of a dynamical system on a torus upon a parameter.

### On an equality equivalent to the Riemann hypothesis

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 422–423

We prove that the Riemann hypothesis on zeros of the zeta function ζ(*s*) is equivalent to the equality $$\int\limits_0^\infty {\frac{{1 - 12t^2 }}{{(1 + 4t^2 )^3 }}dt} \int\limits_{1/2}^\infty {\ln |\varsigma (\sigma + it)|d\sigma = \pi \frac{{3 - \gamma }}{{32}},}$$ where $$\gamma = \mathop {\lim }\limits_{N \to \infty } \left( {\sum\limits_{n = 1}^N {\frac{1}{n} - \ln N} } \right)$$ is the Euler constant.

### $ℂ$-differentiability of mappings of Hilbert spaces

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 424–428

The process of differentiation is studied for a class of mappings of regions of a Hilbert space. We introduce the concept of ℂ*Cl*-differentiability of a mapping at a point and establish sufficient conditions of ℂ*Cl*-differentiability.

### Kotel'nikov-Shannon formula for Fourier transforms of distributions with compact supports

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 429-431

For the Fourier transforms of finite distributions, we establish an analog of the well-known Kotel'nikov formula.

### On the existence of the Stieltjes integral for functions of bounded variation

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 432-435

We obtain sufficient conditions of existence of the Stieltjes integral $$\int\limits_s^t {f(\tau )} d\mathcal{F}(\tau ) = \mathop {\lim }\limits_{\delta _n \to 0} \sum\limits_{k = 1}^{m_n } {f(\xi _k )(\mathcal{F}(t_k^n ) - \mathcal{F}(t_{k - 1}^n ))}$$ for functions of bounded variation taking values in a Banach algebra with identity regardless of the choice of points $ξ_k \in [t_{k−1}, t_k]$.

### On one property of hypercyclic groups

Ukr. Mat. Zh. - 1995. - 47, № 3. - pp. 436-438

We prove that the condition of periodicity of all normal Abelian subgroups of a group with increasing normal series with cyclic factors such that infinite factors are central is preserved on passing to subgroups of finite index. We present an example which demonstrates that the requirement that all infinite cyclic factors must be central cannot be omitted.