# Volume 53, № 2, 2001

### Three-Term Recurrence Relation for Polynomials Orthogonal with Respect to Harmonic Measure

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 147-155

We prove that a three-term recurrence relation for analytic polynomials orthogonal with respect to harmonic measure in a simply connected domain *G* exists if and only if ∂*G* is an ellipse.

### On the Manifolds of Eigenvectors of Linear and Quasilinear Finite-Dimensional Self-Adjoint Operators. I

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 156-167

We investigate the vector bundle of the manifold of normalized eigenvectors of self-adjoint operators and its stratification with respect to the numbers and multiplicities of eigenvalues.

### Extremal Problems in the Theory of Capacities of Condensers in Locally Compact Spaces. I

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 168-189

The present paper is the first part of a work devoted to the development of the theory of κ-capacities of condensers in a locally compact space ** X**; here, κ:

**×**

*X***→ (−∞, +∞] is a lower-semicontinuous function. Condensers are understood in a generalized sense. We investigate the corresponding problem on the minimum of energy on fairly general classes of normalized signed Radon measures. We describe potentials of minimal measures, establish their characteristic properties, and study the uniqueness problem. (The subsequent two parts of this work are devoted to the problem of existence of minimal measures in the noncompact case and to the development of the corresponding approaches and methods.) As an auxiliary result, we investigate the continuity of the mapping $$\left( {x,{\mu }} \right) \mapsto \int {\kappa \left( {x,y} \right)} d{\mu }\left( y \right),\quad \left( {x,{\mu }} \right) \in X \times \mathfrak{M}^ + \left( X \right),$$ where \(\mathfrak{M}^ +\) is the cone of positive measures in**

*X***equipped with the topology of vague convergence.**

*X*### An A Priori Estimate for the Modulus of Continuity of a Generalized Solution of a Parabolic Equation of Divergent Form with Degeneracy

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 190-202

We study parabolic equations of divergent form with degeneracy λ(*x*) with respect to the space variable. We establish an a priori estimate of the Hölder norm of generalized solutions. The problem is considered in parabolic cylinders having a special time dimension induced by the degeneracy λ(*x*).

### Asymptotic Characteristics of Compact Sets and Stepanets Classes

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 203-210

We obtain estimates for the ε-entropy and ε-capacity of sets of periodic functions with mean value zero that have a (ψ, β)-derivative belonging to the space *L* ^{2}(0, 2π).

### Symmetric Equivalence of Matrix Polynomials and Isolation of a Common Unital Divisor in Matrix Polynomials

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 211-219

We find necessary and sufficient conditions for the existence of common unital divisors with given Smith forms of nonsingular matrix polynomials and common factorization of symmetric matrices over rings of polynomials with involution. We obtain conditions for the symmetric equivalence of such matrices.

### Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

Brzychczy S., Prikarpatskii A. K., Samoilenko V. G.

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 220-228

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrödinger equation, and a hydrodynamic model.

### Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 229-237

In the Banach space of functions analytic in a Jordan domain \(\Omega \subset \mathbb{C}\) , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in Ω by Cauchy-type integrals along the rectifiable curve Γ = ∂Ω and can be analytically continued to Ω′ ⊃ Ω or to \(\mathbb{C}\) .

### Pointwise Inequalities of Landau–Kolmogorov Type for Functions Defined on a Finite Segment

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 238-243

For arbitrary *t* ∈ [0, 1], *s* ∈ [1, ∞], and *A* ≥ 2, we determine the unimprovable constant *B* for the inequality $$\left| {x\prime \left( t \right)} \right| \leqslant A\left\| x \right\|_{L_\infty \left[ {0,1} \right]} + B\left\| {x} \right\|_{L_s \left[ {0,1} \right]} .$$ .

### A Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Data on the Entire Boundary of a Domain

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 244-249

In a domain that is the Cartesian product of a segment and a *p*-dimensional torus, we investigate a boundary-value problem for weakly nonlinear hyperbolic equations of higher order. For almost all (with respect to Lebesgue measure) parameters of the domain, we establish conditions for the existence of a unique solution of the problem.

### Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 250-254

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function *f*(*z*) from the Hardy space *H* _{p}, *p* ≥ 1, in disks of radii ρ, ρ_{1}, and ρ_{2}, 0 < ρ_{1} < ρ < ρ_{2} < 1.

### On Completeness of a System of Functions in an Angular Domain

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 255-257

We show that the system {*e* ^{−λz }/(1 + *z* ^{2}) : λ > 0} is complete in a class of functions analytic in an angle.

### On the Solution of a Singular Cauchy Problem for a First-Order Differential Equation Unsolved with Respect to the Derivative of an Unknown Function

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 258-262

For a first-order ordinary differential equation, we establish conditions under which a singular Cauchy problem has a unique continuously differentiable solution with required asymptotic behavior.

### Singularities of Toric Manifolds

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 263-267

By using methods of toric geometry, we investigate compactifications of *F*-theory on the elliptic Calabi–Yau threefolds.

### On One Class of Separable Dedekind Groups

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 269-273

We describe locally solvable groups *G* such that all their infinite subgroups that do not belong to a certain proper subgroup of the group under consideration are normal.

### On Multiplicativity of Canonical Diagonal Forms of Matrices over the Domain of Principal Ideals. II

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 274-277

We investigate the structure of matrices over the domain of principal ideals that possess the property of multiplicativity of canonical diagonal forms.

### A New Morera-Type Theorem on a Unit Disk

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 278-281

We present a new Morera-type theorem on a unit disk.

### Investigation of Invariant Sets of Itô Stochastic Systems with the Use of Lyapunov Functions

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 282-285

By using Lyapunov functions, we obtain conditions for the invariance and stochastic stability of invariant sets of Itô-type systems.

### An Exact Estimate for the Measure of the Exceptional Set in the Borel Relation for Entire Functions

Ukr. Mat. Zh. - 2001. - 53, № 2. - pp. 286-288

We obtain an exact estimate for the measure of the exceptional set in the Borel relation for entire functions.