2019
Том 71
№ 1

All Issues

Maslyuchenko V. K.

Articles: 18
Article (Ukrainian)

Construction of intermediate differentiable functions

Maslyuchenko V. K., Mel'nik V. S.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 5. - pp. 672-681

For given upper and lower semicontinuous real-valued functions $g$ and $h$, respectively, defined on a closed parallelepiped $X$ in $R^n$ and such that $g(x) < h(x)$ on $X$ and points $x_0 \in X$ and $y_0 \in (g(x_0), h(x_0))$, we construct a smooth function $f : X \rightarrow R$ such that $f(x_0) = y_0$ and $g(x) < f(x) < h(x)$ on $X$. We also present similar constructions for functions defined on separable Hilbert spaces and Asplund spaces.

Article (Ukrainian)

Haar’s condition and joint polynomiality of separate polynomial functions

Kosovan V. M., Maslyuchenko V. K., Voloshyn H. A.

↓ Abstract

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 17-27

For systems of functions $F = \{ f_n \in K^X : n \in N\}$ and $G = \{ g_n \in K^Y : n \in N\}$ we consider an $F$ -polynomial $f = \sum^n_{k=1}\lambda_k f_k$, a $G$-polynomial $h = \sum^n_{k,j=1} \lambda_{k,j} f_k \otimes g_j$, and an $F \otimes G$-polynomial $(f_k\otimes g_j)(x, y) = = f_k(x)g_j(y)$, where $(f_k\otimes g_j)(x, y) = f_k(x)g_j(y)$. By using the well-known Haar’s condition from the approximation theory we study the following question: under what assumptions every function $h : X \times Y \rightarrow K$, such that all $x$-sections $h^x = h(x, \cdot )$ are $G$-polynomials and all $y$-sections $h_y = h(\cdot , y)$ are $F$ -polynomials, is an $F \otimes G$-polynomialy. A similar problem is investigated for functions of $n$ variables.

Article (Ukrainian)

Sequential closure of the space of jointly continuous functions in the space of separately continuous functions

Maslyuchenko V. K., Voloshyn H. A.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 156-161

Given compact spaces $X$ and $Y$, we study the space $S(X \times Y )$ of separately continuous functions $f : X \times Y \rightarrow R$ endowed with the locally convex topology generated by the seminorms $|| f||^x = \mathrm{max}_{y \in Y} |f(x, y)|,\; x \in X$, and $|| f||_y = \mathrm{max}_{x \in X} |f(x, y)|,\; y \in Y$. Under the assumption that the compact space $X$ is metrizable, we prove that a separately continuous function $f : X \times Y \rightarrow R$ is the limit of a sequence $(f_n)^{\infty}_{n=1}$ of jointly continuous function $f_n : X \times Y \rightarrow R$ in $S(X \times Y )$ provided that the set $D(f)$ of discontinuity points of $f$ has countable projections on $X$.

Article (Ukrainian)

Properties of the Ceder Product

Maslyuchenko O. V., Maslyuchenko V. K., Myronyk O. D.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2015. - 67, № 6. - pp. 780-787

We study properties of the Ceder product $X ×_b Y$ of topological spaces $X$ and $Y$, where $b ∈ Y$, recently introduced by the authors. Important examples of the Ceder product are the Ceder plane and the Alexandroff double circle. In particular, for $i = 0, 1, 2, 3$ we establish necessary and sufficient conditions for the Ceder product to be a $T_i$ -space. We prove that the Ceder product $X ×_b Y$ is metrizable if and only if the spaces $X$ and $\overset{.}{Y}=Y\backslash \left\{b\right\}$ are metrizable, $X$ is $σ$-discrete, and the set $\{b\}$ is closed in $Y$. If $X$ is not discrete, then the point $b$ has a countable base of closed neighborhoods in $Y$.

Article (Ukrainian)

Points of joint continuity and large oscillations

Maslyuchenko V. K., Nesterenko V. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 791–800

For topological spaces $X$ and $Y$ and a metric space $Z$, we introduce a new class $N(X × Y,Z)$ of mappings $f:\; X × Y → Z$ containing all horizontally quasicontinuous mappings continuous with respect to the second variable. It is shown that, for each mapping $f$ from this class and any countable-type set $B$ in $Y$, the set $C_B (f)$ of all points $x$ from $X$ such that $f$ is jointly continuous at any point of the set $\{x\} × B$ is residual in $X$: We also prove that if $X$ is a Baire space, $Y$ is a metrizable compact set, $Z$ is a metric space, and $f ∈ N(X×Y,Z)$, then, for any $ε > 0$, the projection of the set $D^{ε} (f)$ of all points $p ∈ X × Y$ at which the oscillation $ω_f (p) ≥ ε$ onto $X$ is a closed set nowhere dense in $X$.

Article (Ukrainian)

Joint continuity of $K_h C$-functions with values in moore spaces

Filipchuk O. I., Maslyuchenko V. K., Mykhailyuk V. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2008. - 60, № 11. - pp. 1539 – 1547

We introduce a notion of a categorical cliquish mapping and prove that, for each $K_h C$-mapping $f : X \times Y \rightarrow Z$ (here, $X$ is a topological space, $Y$ is a first countable space, and $Z$ is a Moore space) with categorical cliquish horizontal $y$-sections $f_y$ , the sets $C_y (f)$ are residual $G_\delta$-sets in $X$ for each $y \in Y.$

Article (Ukrainian)

Separately continuous mappings with values in nonlocally convex spaces

Karlova O. O., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1639–1646

We prove that the collection $(X, Y, Z)$ is the Lebesgue triple if $X$ is a metrizable space, $Y$ is a perfectly normal space, and $Z$ is a strongly $\sigma$-metrizable topological vector space with stratification $(Z_m)^{\infty}_{m=1}$, where, for every $m \in \mathbb{N}$, $Z_m$ is a closed metrizable separable subspace of $Z$ arcwise connected and locally arcwise connected.

Article (Ukrainian)

Constancy of upper-continuous two-valued mappings into the Sorgenfrey line

Fotii O. H., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1034–1039

By using the Sierpiński continuum theorem, we prove that every upper-continuous two-valued mapping of a linearly connected space (or even a c-connected space, i.e., a space in which any two points can be connected by a continuum) into the Sorgenfrey line is necessarily constant.

Article (Ukrainian)

Direct and Inverse Problems of Baire Classification of Integrals Depending on a Parameter

Banakh T. O., Kutsak S. M., Maslyuchenko O. V., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 11. - pp. 1443-1457

We study the problem of the Baire classification of integrals g (y) = (If)(y) = ∫ X f(x, y)dμ(x), where y is a parameter that belongs to a topological space Y and f are separately continuous functions or functions similar to them. For a given function g, we consider the inverse problem of constructing a function f such that g = If. In particular, for compact spaces X and Y and a finite Borel measure μ on X, we prove the following result: In order that there exist a separately continuous function f : X × Y → ℝ such that g = If, it is necessary and sufficient that all restrictions g| Y n of the function g: Y → ℝ be continuous for some closed covering { Y n : n ∈ ℕ} of the space Y.

Brief Communications (Ukrainian)

Separately continuous functions with respect to a variable frame

Herasymchuk V. H., Maslyuchenko O. V., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1281-1286

We show that the set D(f) of discontinuity points of a function f : R 2R continuous at every point p with respect to two variable linearly independent directions e 1(p) and e 2(p) is a set of the first category. Furthermore, if f is differentiable along one of directions, then D(f) is a nowhere dense set.

Brief Communications (Ukrainian)

Lebesgue–Cech Dimensionality and Baire Classification of Vector-Valued Separately Continuous Mappings

Kalancha A. K., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2003. - 55, № 11. - pp. 1576-1579

For a metrizable space X with finite Lebesgue–Cech dimensionality, a topological space Y, and a topological vector space Z, we consider mappings f: X × YZ continuous in the first variable and belonging to the Baire class α in the second variable for all values of the first variable from a certain set everywhere dense in X. We prove that every mapping of this type belongs to the Baire class α + 1.

Brief Communications (Ukrainian)

Joint Continuity and Quasicontinuity of Horizontally Quasicontinuous Mappings

Maslyuchenko V. K., Nesterenko V. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1711-1714

We show that if Xis a topological space, Ysatisfies the second axiom of countability, and Zis a metrizable space, then, for every mapping f: X× YZthat is horizontally quasicontinuous and continuous in the second variable, a set of points xXsuch that fis continuous at every point from {x} × Yis residual in X. We also generalize a result of Martin concerning the quasicontinuity of separately quasicontinuous mappings.

Article (Ukrainian)

New Generalizations of the Scorza-Dragoni Theorem

Gaidukevich O. L., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 7. - pp. 881-888

We consider Carathéodory functions f : T × XY, where T is a topological space with regular σ-finite measure, the spaces X and Y are metrizable and separable, and X is locally compact. We show that every function of this sort possesses the Scorza-Dragoni property. A similar result is also established in the case where the space T is locally compact and X = ℝ.

Article (Ukrainian)

Characterization of the sets of discontinuity points of separately continuous functions of many variables on the products of metrizable spaces

Maslyuchenko V. K., Mykhailyuk V. V.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 6. - pp. 740–747

We show that a subset of the product ofn metrizable spaces is the set of discontinuity points of some separately continuous function if and only if this subset can be represented in the form of the union of a sequence ofF σ-sets each, of which is locally projectively a set of the first category.

Article (Ukrainian)

Construction of a separately continuous function with given oscillation

Maslyuchenko O. V., Maslyuchenko V. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1998. - 50, № 7. - pp. 948–959

We investigate the problem of construction of a separately continuous function f whose oscillation is equal to a given nonnegative function g. We show that, in the case of a metrizable Baire product, the problem under consideration is solvable if and only if g is upper semicontinuous and its support can be covered by countably many sets, which are locally contained in products of sets of the first category.

Article (Ukrainian)

A property of partial derivatives

Maslyuchenko V. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 4. - pp. 529–531

Article (Ukrainian)

Embeddability conditions for some spaces (a categorical approach)

Maslyuchenko V. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 3. - pp. 316 - 321

Article (Ukrainian)

Conditions for the inclusions of intersections and unions of spaces Lp ( μ ) with a weight

Maslyuchenko V. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1982. - 34, № 4. - pp. 518—522