Том 71
№ 11

All Issues

Volume 64, № 9, 2012

Article (Ukrainian)

On convolution of functions in angular domains

Dilnyi V. M.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1155-1164

We obtain analogs of the Parseval theorem, convolution theorem, and some other properties of the convolution of functions from the Hardy – Smirnov spaces in an arbitrary convex unbounded polygon.

Article (Russian)

Asymptotic representations of solutions of essentially nonlinear systems of ordinary differential equations with regularly and rapidly varying nonlinearities

Evtukhov V. M., Shlepakov O. R.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1185

We obtain asymptotic representations for one class of solutions of systems of ordinary differential equations more general than systems of the Emden – Fowler type.

Article (Russian)

Homotopic types of right stabilizers and orbits of smooth functions on surfaces

Maksimenko S. I.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1204

Let $\mathcal{M}$ be a smooth connected compact surface, $P$ be either the real line $\mathbb{R}$ or a circle $S^1$. For a subset $X ⊂ M$, let $\mathcal{D}(M, X)$ denote the group of diffeomorphisms of $M$ fixed on $X$. In this note, we consider a special class F of smooth maps $f : M → P$ with isolated singularities that contains all Morse maps. For each map $f ∈ \mathcal{F}$, we consider certain submanifolds $X ⊂ M$ that are “adopted” with $f$ in a natural sense, and study the right action of the group $\mathcal{D}(M, X)$ on $C^{∞}(M, P)$. The main result describes the homotopy types of the connected components of the stabilizers $S(f)$ and orbits $\mathcal{O}(f)$ for all maps $f ∈ \mathcal{F}$. It extends previous results of the author on this topic.

Article (Ukrainian)

Approximation of the classes $B^{\Omega}_{p, \theta}$ of periodic functions of many variables by Fourier sums in the space $L_p$ with $p = 1, \infty$

Myronyuk V. V.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1204-1213

We obtain an exact-order estimate for the deviation of Fourier sums of periodic functions of many variables from the classes $B^{\Omega}_{p, \theta}$ in the space $L_p$ for $p = 1, \infty$.

Article (Russian)

Smoothness of functions in the metric spaces Lψ

Pichugov S. A.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1214-1232

Let $L_0(T)$ be thе set of real-valued periodic measurable functions, let $\psi : R^+ \rightarrow R^+$ be a modulus of continuity $(\psi \neq 0)$ , and let $$L_{\psi} \equiv L_{\psi}(T ) = \left\{f \in L_0 (T ): ||f||_{\psi} := \int_T \psi( |f (x)| ) dx < \infty \right\}.$$ The following problems are investigated: Relationship between the rate of approximation of $f$ by trigonometric polynomials in $L_{\psi}$ and smoothness in $L_1$. Correlation between the moduli of continuity of $f$ in $L_{\psi}$ and $L_1$, and theorems on imbedding of the classes $\text{Lip} (\alpha, \psi)$ in $L_1$. Structure of functions from the class $\text{Lip}(1, \psi)$.

Article (English)

Comparison theorems and necessary/sufficient conditions for existence of nonoscillatory solutions of forced impulsive delay differential equations

Cheng Sui Sun, Shao Yuan Huang

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1233-1248

In 1997, A. H. Nasr provided necessary and sufficient conditions for the oscillation of the equation $$x''(t) + p(t) |x(g(t))|^{\eta} \text{sgn} (x(g(t))) = e(t),$$ where $\eta > 0$, $p$, and $g$ are continuous functions on $[0, \infty)$ such that $p(t) \geq 0,\;\; g(t) \leq t,\;\; g'(t) \geq \alpha > 0$, and $\lim_{t \rightarrow \infty} g(t) = \infty$ It is important to note that the condition $g'(t) \geq \alpha > 0$ is required. In this paper, we remove this restriction under the superlinear assumption $\eta > 0$. Infact, we can do even better by considering impulsive differential equations with delay and obtain necessary and sufficient conditions for the existence of nonoscillatory solutions and also a comparison theorem that enables us to apply known oscillation results for impulsive equations without forcing terms to yield oscillation criteria for our equations.

Article (Russian)

Best approximations of periodic functions in generalized lebesgue spaces

Chaichenko S. O.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1249-1265

In generalized Lebesgue spaces with variable exponent, we determine the order of the best approximation on the classes of $(\psi, \beta)$-differentiable $2\pi$-periodic functions. We also obtain an analog of the well-known Bernstein inequality for the $(\psi, \beta)$-derivative, with the help of which the converse theorems of approximation theory are proved on the indicated classes.

Article (Ukrainian)

Representations of canonical anticommutation relations with orthogonality condition

Yakymiv R. Ya.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1266-1272

We study the class of Hilbert space representations of the ∗-algebra $A^{(d)}_0$ generated by relations of the form $$A^{(d)}_0 = \mathbb{C}\langle a_j, a_j^{*} | a_j^{*} a_j = 1 - a_j a_j^{*},\; a_j, a_j^{*} = 0, i \neq j,\; i, j = 1,...,d\rangle,$$ Namely, we describe the classes of unitary equivalence of irreducible representations of $A^{(d)}_0$ such that there exists $j = 1,...,d$ for which $a^2_j \neq 0$.

Brief Communications (Russian)

Geodesic spaces tangent to metric spaces

Bilet V. V.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1273-1281

We investigate the geometry of spaces tangent and pretangent to general metric spaces with marked point. We find a sufficient condition under which every separable tangent space is geodesic. This condition is almost exact in the sense that it is necessarily satisfied if all spaces pretangent to a given metric space are geodesic.

Brief Communications (Russian)

On one Shemetkov problem

Semenchuk V. N., Velesnitskii V. F.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1282-1288

This work is devoted to the investigation of the structure of superradical formations.

Brief Communications (English)

Generalized Weyl's theorem and tensor product

Rashid M. H. M.

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Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1289-1296

We give necessary and/or sufficient conditions ensuring the passage of generalized a-Weyl theorem and property $(gw)$ from $A$ and $B$ to $A \otimes B$.