### On extension of continuous functions defined on a circle

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1011–1017

We consider the problem of the extension of a continuous function defined on a circle to the interior of the disk without critical points.

### Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1018-1029

We consider the problem of estimates for the powers of the Cayley transform $V = (А + I)(А - I)^{-1}$ of the generator of a uniformly bounded $C_0$-semigroup of operators $e^{tA} , t \geq 0$, that acts in a Hilbert space $H$. In particular, we establish the estimate $\sup_{n \in N}\left(||V^n||/\ln(n + 1)\right) < \infty$. We show that the estimate $\sup_{n ∈ N} ∥V^n∥ < ∞$ is true in the following cases: (a) the semigroups $e^{tA}$ and $e^{tA^{−1}}$ are uniformly bounded; (b) the semigroup etA uniformly bounded for $t ≥ ∞$ is analytic (in particular, if the generator of the semigroup is a bounded operator).

### Groups with handles of order different from three

Kozulin S. N., Senashov V. I., Shunkov V. P.

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1030–1042

We obtain a test for the unsimplicity of an infinite group.

### LI-Yorke sensitivity and other concepts of chaos

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1043–1061

We give a survey of the theory of chaos for topological dynamical systems defined by continuous maps on compact metric spaces.

### On the set of extremal functions in certain Kolmogorov-type inequalities

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1062–1075

We determine the sets of all extremal functions in certain Kolmogorov-type and Bohr-Favard-type inequalities.

### Reducibility of a nonlinear oscillation system with pulse influence in the neighborhood of an integral manifold

Dudnytskyi P. M., Petryshyn R. I., Samoilenko A. M.

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1076–1094

In the neighborhood of an asymptotically stable integral manifold of a multifrequency system with pulse influence at fixed times, we perform a decomposition of the equations for angular and position variables.

### Asymptotic behavior of solutions of a nonlinear difference equation with continuous argument

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1095–1100

We consider the difference equation with continuous argument $$x(t + 2) - 2\lambda x(t + 1) + \lambda ^2 x(t) = f(t,x(t)),$$ where λ > 0, *t* ∈ [0, ∞), and *f*: [0, ∞) × **R** → **R**. Conditions for the existence and uniqueness of continuous asymptotically periodic solutions of this equation are given. We also prove the following result: Let *x*(*t*) be a real continuous function such that $$\mathop {\lim }\limits_{t \to \infty } (x(t + 2) - (1 - \alpha )x(t + 1) - \alpha x(t)) = 0$$ for some α ∈ **R**. Then it always follows from the boundedness of *x*(*t*) that $$\mathop {\lim }\limits_{t \to \infty } (x(t + 1) - x(t)) = 0$$ *t* → ∞ if and only if α ∈ **R** {1}.

### Correction of nonlinear orthogonal regression estimator

Fazekas L., Kukush A. G., Zwanzig S.

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1101–1118

For any nonlinear regression function, it is shown that the orthogonal regression procedure delivers an inconsistent estimator. A new technical approach to the proof of inconsistency based on the implicit-function theorem is presented. For small measurement errors, the leading term of the asymptotic expansion of the estimator is derived. We construct a corrected estimator, which has a smaller asymptotic deviation for small measurement errors.

### Criterion for the uniqueness of a solution of the Darboux-Protter problem for multidimensional Hyperbolic equations with Chaplygin operator

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1119–1127

We obtain a criterion for the uniqueness of a regular solution of the Darboux-Protter problem for multidimensional hyperbolic equations with Chaplygin operator. We also prove a theorem on the uniqueness of solutions of the dual problem.

### Point spectrum of the schrödinger operator with point interactions at the vertices of regular *N*-gons

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1128–1134

We present a complete description of the point spectrum of the Laplace operator perturbed by point potentials concentrated at the vertices of regular polygons. We prove a criterion for the absence of points of the point spectrum of a singular perturbed positive self-adjoint operator with the property of cyclicity of defect vectors.

### On the degree of holomorphic mappings from an annulus into an annulus

Nguyen Doan Tuan, Pham Viet Duc

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1135–1138

We establish an estimate for the degree of a holomorphic mapping *f* : *K* 1 → *K* _{2} (here, *K* _{1} and *K* _{2} are doubly-connected domains) in terms of the modulus of a family of curves in *K* _{2}. This estimate generalizes a result obtained by Kobayashi.

### Piecewise-maximum closed Geodesic trajectories on bounded Toroidal manifolds

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1139–1142

We use toroidal coordinates for the investigation of maximum geodesic trajectories for arbitrary parameters of a torus. Conditions under which trajectories are located in a bounded part of the toroidal manifold are considered. Using global invariants, we construct closed piecewise-maximum geodesic trajectories.

### Generalized golden sections and a new approach to the geometric definition of a number

↓ Abstract

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1143–1150

We consider applications of generalized golden sections to the geometric definition of a number and establish new properties of natural numbers that follow from this approach.

### Anatolii Yakovych Dorogovtsev

Buldygin V. V., Gorodnii M. F., Gusak D. V., Korolyuk V. S., Samoilenko A. M.

Ukr. Mat. Zh. - 2004νmber=8. - 56, № 8. - pp. 1151-1152