2017
Том 69
№ 4

All Issues

Volume 56, № 8, 2004

Article (Russian)

On extension of continuous functions defined on a circle

Andriyuk E. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Article (Russian)

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

Gomilko A. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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).

Article (Russian)

Groups with handles of order different from three

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

↓ Abstract   |   Full text (.pdf)

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

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

Article (Ukrainian)

LI-Yorke sensitivity and other concepts of chaos

Kolyada S. F.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Article (Russian)

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

Kofanov V. A.

↓ Abstract   |   Full text (.pdf)

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

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

Article (Ukrainian)

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   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Article (English)

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

Stevic S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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, ∞) × RR. 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}.

Article (English)

Correction of nonlinear orthogonal regression estimator

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

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Brief Communications (Russian)

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

Aldashev S. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Brief Communications (Ukrainian)

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

Dudkin M. Ye.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Brief Communications (English)

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

Nguyen Doan Tuan, Pham Viet Duc

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Brief Communications (Russian)

Piecewise-maximum closed Geodesic trajectories on bounded Toroidal manifolds

Romanov S. S.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Brief Communications (Russian)

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

Stakhov A. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 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.

Obituaries (Ukrainian)

Anatolii Yakovych Dorogovtsev

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

Full text (.pdf)

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