# Volume 53, № 9, 2001

### On a Two-Point Version of Transfinite Diameter

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1155-1160

We study properties of a two-point version of the transfinite diameter of a set. By using relations obtained for its calculation, we prove a two-point version of the well-known Pólya theorem on an estimate from above for the Hankel determinants of a holomorphic function.

### Interval Oscillation Criteria for Second-Order Nonlinear Differential Equations

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1161-1173

We present new interval oscillation criteria for certain classes of second-order nonlinear differential equations, which are different from the most known ones in the sense that they are based only on information on a sequence of subintervals of [*t* _{0}, ∞) rather than on the whole half-line. We also present several examples that demonstrate wide possibilities of the results obtained.

### On Moduli Spaces, Equidistribution, Estimates, and Rational Points of Algebraic Curves

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1174-1183

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic *p*. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime *p*, we study rational points on curves *y* ^{2} = *f*(*x*). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.

### Asymptotics of Solutions of an Infinite System of Linear Algebraic Equations in Potential Theory

Gomilko A. M., Koval'chuk V. F.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1184-1193

For bounded solutions *x* _{k} of an infinite system of linear algebraic equations arising in potential theory in the course of investigation of an axially symmetric problem in the exterior of two spheres with equal radii, we obtain asymptotic formulas with respect to a parameter that characterizes the approach of the spheres to one another and for *k* → ∞.

### On the Asymptotic Behavior of Solutions of a Singular Cauchy Problem

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1194-1203

We consider a singular Cauchy problem for a nonlinear differential equation unsolved with respect to the derivative of the unknown function. We prove the existence of continuously differentiable solutions, investigate their asymptotic behavior near the initial point, and determine their number.

### On an Adaptive Estimator of the Least Contrast in a Model with Nonlinear Functional Relations

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1204-1209

We consider an implicit nonlinear functional model with errors in variables. On the basis of the concept of deconvolution, we propose a new adaptive estimator of the least contrast of the regression parameter. We formulate sufficient conditions for the consistency of this estimator. We consider several examples within the framework of the *L* _{1}- and *L* _{2}-approaches.

### Estimates for a Group of Deviations in Generalized Hölder Metric

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1210-1217

We present order relations for a group of deviations of a function *f*(·) ∈ *H* _{ω} in terms of partial Fourier sums of this function in a generalized Hölder metric defined in a generalized Hölder space *H* _{ω*} ⊃ *H* _{ω}.

### Algebra of Bergman Operators with Automorphic Coefficients and Parabolic Group of Shifts

Chernetskii V. A., Mozel’ V. A.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1218-1223

UDC 517.983

We study the algebra of operators with the Bergman kernel extended by isometric weighted shift operators. The coefficients of the algebra are assumed to be automorphic with respect to a cyclic parabolic group of fractional-linear transformations of a unit disk and continuous on the Riemann surface of the group. By using an isometric transformation, we obtain a quasiautomorphic matrix operator on the Riemann surface with properties similar to the properties of the Bergman operator. This enables us to construct the algebra of symbols, devise an efficient criterion for the Fredholm property, and calculate the index of the operators of the algebra considered.

### Estimates for Approximation Characteristics of the Besov Classes $B_ r^{p,θ}$ of Periodic Functions of Many Variables in the Space $L_q. I$

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1224-1231

We obtain order estimates for the approximation of the classes $B_ r^{p,θ}$ of periodic functions of many variables in the space $L_q$ by using operators of orthogonal projection and linear operators satisfying certain conditions.

### On Reduction Principle in Stability Theory for Systems with Random Perturbations

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1232-1240

For stochastic systems, we obtain an analog of the reduction principle that enables one to reduce the analysis of the stability of a system with random perturbations to the analysis of the stability of a deterministic system.

### On the Smoothness of the Invariant Torus of a Countable System of Difference Equations with Parameters

Marchuk N. A., Teplinsky Yu. V.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1241-1250

We establish sufficient conditions for the differentiability of the invariant torus of a countable system of linear difference equations defined on a finite-dimensional torus with respect to an angular variable and the parameter of the original system of equations.

### Stochastic Semigroups and Random Mass Transfer

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1251-1259

We consider the problem of random mass transfer on a metric compactum defined by a purely discontinuous stochastic semigroup \(T_t^s\) . We give a description of this semigroup based on a Markov process with random transition probability. We present conditions for the independence of measure-valued processes of the form \(T_t^0 {\mu}_{0}\) , depending on the initial mass μ_{0}.

### Generalized Hardy Transformation and Toeplitz Operators in BMOA-Type Spaces

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1260-1271

We study the action of the generalized Hardy transformation in the BMOA classes in a half-disk and present a criterion for the boundedness of the Toeplitz operators acting in a BMOA-type space in the unit disk.

### Optimization of Linear Functions at the Vertices of a Permutation Polyhedron with Additional Linear Constraints

Valuiskaya O. A., Yakovlev S. V.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1272-1280

We propose an approximate polynomial method that enables one to determine with given accuracy the extremum of a function on a permutation polyhedron with additional linear constraints.

### Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Variable Coefficients

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1281-1286

We establish conditions for the unique solvability of a boundary-value problem for a weakly nonlinear hyperbolic equation of order 2*n*, *n* > (3*p* + 1)/2, with coefficients dependent on the space coordinates and data given on the entire boundary of a cylindric domain \(D \subset \mathbb{R}^{p + 1}\) . The investigation of this problem is connected with the problem of small denominators.

### On the Strong Law of Large Numbers for Multivariate Martingales with Continuous Time

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1287-1291

We prove the strong law of large numbers for vector martingales with arbitrary operator normalizations. From the theorem proved, we deduce several known results on the strong law of large numbers for martingales with continuous time.