# Volume 46, № 7, 1994

### Boundary-value problems for hyperbolic equations with constant coefficients

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 795–802

By using a metric approach, we study the problem of well-posedness of boundary-value problems for hyperbolic equations of ordern $(n ≥ 2)$ with constant coefficients in a cylindrical domain. Conditions of existence and uniqueness of solutions are formulated in number-theoretic terms. We prove a metric theorem on lower estimates of small denominators that appear when constructing solutions.

### Dirichlet problem for the Poisson equation with an essentially infinite-dimensional elliptic operator

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 803–808

In a special class of domains in an infinite-dimensional Hilbert space, the solvability of the Dirichlet problem for the Poisson equation with an elliptic operator of the form $(Lu)(x)=j(x)(u''(x))$ vanishing on cylindrical functions is proved.

### On the behavior of solutions of operator-differential equations at infinity

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 809–813

The existence of limits at the infinity, generalized in the Abel sense, is established for bounded solutions of the operator-differential equation $y'(t) = Ay(t)$ in a reflexive Banach space.

### Exact values of mean $n$-widths for the classes of functions analytic in the upper half plane in the Hardy space

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 814–824

In the Hardy space $H_2 ℝ_+^2$ of functions analytic in the upper half plane such that $$\sup \left\{ {\int\limits_\mathbb{R} {|f(x + iy)|^2 dx: 0< y< \infty } } \right\}< \infty ,$$ we determine mean $N$-widths and find their exact values for numerous classes of functions.

### Convergence of an algorithm for constructing snakes

Dzyadyk V. K., Kovtunets V. V.

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 825–832

We investigate an algorithm for constructing snakes (extremal polynomials introduced by S. Karlin) suggested by Dzyadyk. It is proved that, in the general case, this algorithm is linearly convergent. In the case where the basis functions of the Chebyshev system belong to the class $C^2$, this algorithm is quadratically convergent.

### On conditions for oscillation and nonoscillation of the solutions of a semilinear second-order differential equation

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 833–841

We establish sufficient conditions for oscillation and nonoscillation of regular solutions of the secondorder differential equation $y'' + p(t)|y|^{1-\lambda} |y'|^{\lambda} \,{\text {\rm sign }}\, y = 0$, where $λ < 1$ and $p: [a,ω[→ ℝ,\, −∞ < a < ω ≤ +∞$ is a locally summable function.

### On differential properties of real functions

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 842–848

We study the categorical analog of the Stepanov theorem and new criteria of asymptotic differentiability of real functions.

### Generalized Green's matrix for linear pulse boundary-value problems

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 849–856

We establish an algebraic criterion of solvability, study the structure of general solutions of linear boundary-value problems for systems of differential equations with pulse effects, and construct the generalized Green's matrix.

### On the evolution operators for some equations of mathematical physics with variable coefficients

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 857–869

It is shown that, with the help of a relatively simple operator technique, it is possible to solve, from a common point of view, the Cauchy problem for many important equations of mathematical physics with variable coefficients. This result is applied to the equations of kinetic theory, and diffusion and heat conduction equations. We discuss the problem of equivalence of different schemes of expansion according to the Hausdorff formula.

### Limiting distributions of the solutions of the many-dimensional Bürgers equation with random initial data. I

Leonenko N. N., Orsingher E., Rybasov K. V.

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 870–877

We find Gaussian limiting distributions of the solutions of the many-dimensional Bürgers equation with the initial condition given by a homogeneous isotropic Gaussian random field with strong dependence.

### Essential normality of some classes of operators in tensor products of Hilbert spaces

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 878–885

We establish general criteria for the domains of definition of spectral integrals to be essential. On the basis of these criteria, we prove that some classes of operators are essentially normal in tensor products of Hilbert spaces.

### Characteristics of the sum of a multidimensional series

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 886–892

We study the relationship between the asymptotic behavior of coefficients of a multidimensional series of exponents and the asymptotic behavior of its sum near a point on the boundary of the domain of convergence. Growth characteristics, an order $\rho_Q(a)$, and a type $\sigma_{Q \beta}(a)$ in an octant $Q(a)$ are determined. The dependence of growth characteristics on the coordinates of points of the boundary of the domain of convergence is established.

### On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 893–903

An exact complexity exponent is found for an approximate solution of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm realizing an optimal order are given. As a consequence, we find an exact complexity exponent for an approximate solution of Fredholm integral equations of second kind with kernels and free terms having square integrable ψ-derivatives.

### Reduction and coisotropic invariant tori of Hamiltonian systems with non-poisson commutative symmetries. II

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 904–914

Haamiltonian systems of mechanical type are considered on a twisted cotangent stratification of a manifold admitting a smooth free torus action. In the case where these systems possess non-Poisson symmetries generated by the torus action, the Lee-Cartan reduction scheme is described and the structure of a reduced phase space and reduced Hamiltonians is clarified.

### On Kolmogorov widths of classes $B^r_{p, \theta}$ of periodic functions of many variables with low smoothness in the space $L_q$

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 915–926

We study the Kolmogorov widths of Besov classes $B^r_{p, \theta}$ of periodic functions of many variables with low smoothness in the space $L_q, 1 < q < ∞$. We also investigate the behavior of widths of such classes with critical indices of smoothness.

### Selection of regular direct summands from matrix polynomials

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 927–934

We study the problem of reducibility of matrix polynomials to a quasidiagonal form with regular diagonal blocks by a similarity transformation. The results obtained are applied to solving matrix algebraic equations of Riccati type.

### Handle decompositions of simply connected five-manifolds. III

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 935–940

We prove the existence of exact handle decomposition of a simply connected smooth or piecewise linear (PL) five-manifold with a standard simply connected boundary of signature zero, the triviality of a five-dimensional *h*- cobordism with ends of such type, and the uniqueness up to diffeomorphism (PL-isomorphism) of a smooth (PL)*h* -cobordism between a given simply connected four-manifold and the corresponding standard manifold.

### Story of rational approximation for the class of Stieltjes functions: From Stieltjes to recent optimal estimations of errors

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 941–943

We present the history of inequalities for Padé approximant errors in the Stieltjes case. Inequalities optimal in order are obtained for these approximants by using results of J. Vinuesa and A. P. Magnus.

### Boundedness of solutions and asymptotic properties of some systems of differential equations

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 944–946

We establish conditions under which solutions of some systems of differential equations are bounded and study their asymptotic properties.

### Representations of solutions of nonlocal boundary-value problems for parabolic equations

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 947–951

We study two-point boundary-value problems for parabolic equations whose solutions are representable in terms of Green's functions of the Cauchy problem.

### Polynomial approximations of solutions of higher-order operator-differential equations

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 952–955

We study the Cauchy problem for higher-order operator-differential equations in a Banach space and construct polynomial approximations of its solutions.

### Coordinated approximation method for nonlinear ill-posed problems

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 956-961

A generalization of the method of coordinated approximation suggested by Yu. Gaponenko [1] for the space $L_2(0, 1)$ is developed for abstract Hilbeit spaces. In particular, it is shown that, for $L_2(0, 1)$, some assumptions concerning ал exact solution can be weaken.

### Properties of nonstationary boundary operators in a full scale of Sobolev-type spaces

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 962–965

Properties of boundary-value operators are established for the system of equations of the theory of elasticity in the complete scale of spaces of Sobolev type.