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 Carpathian Mathematical Publications   [1 followers]  Follow       Open Access journal    ISSN (Print) 2075-9827    Published by Vasyl Stefanyk Precarpathian National University  [1 journal]
• Approximation of capacities with additive measures

• Authors: O. R. Nykyforchyn, I. D. Hlushak
Abstract: For a space of non-additive regular measures on a metric compactum with the Prokhorov-style metric, it is shown that the problem of approximation of arbitrary measure with an additive measure on a fixed finite subspace reduces to linear optimization problem with parameters dependent on the values of the measure on a finite number of sets.

An algorithm for such an approximation, which is more efficient than the straighforward usage of simplex method, is presented.
PubDate: 2017-06-23
Issue No: Vol. 9 (2017)

• On the growth of a klasss of Dirichlet series absolutely convergent in
half-plane

• Authors: L. V. Kulyavetc', O. M. Mulyava
Abstract: In terms of generalized orders it is investigated a relation between the growth of a Dirichlet series $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with the abscissa of asolute convergence $A\in (-\infty,+\infty)$ and the growth of Dirichlet series $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$, $1\le j\le 2$, with the same abscissa of absolute convergence, if the coefficients $a_n$ are connected with the coefficients $a_{n,j}$ by correlation $$\beta\left(\frac{\lambda_n}{\ln\,\left( a_n e^{A\lambda_n}\right)}\right)=(1+o(1)) \prod\limits_{j=1}^{m}\beta\left(\frac{\lambda_n} {\ln\,\left( a_{n,j} e^{A\lambda_n}\right)}\right)^{\omega_j},\quad n\to\infty,$$ where $\omega_j>0$ $(1\le j\le m)$, $\sum\limits_{j=1}^{m}\omega_j=1$, and $\alpha$ - positive slowly increasing function on $[x_0, +\infty)$.
PubDate: 2017-06-21
Issue No: Vol. 9 (2017)

• Topology on the spectrum of the algebra of entire symmetric functions of
bounded type on the complex $L_\infty$

• Authors: T. V. Vasylyshyn
Abstract: It is known that the so-called elementary symmetric polynomials $R_n(x) = \int_{[0,1]}(x(t))^n\,dt$ form an algebraic basis in the algebra of all symmetric continuous polynomials on the complex Banach space $L_\infty,$ which is dense in the Fr\'{e}chet algebra $H_{bs}(L_\infty)$ of all entire symmetric functions of bounded  type on $L_\infty.$ Consequently, every continuous homomorphism $\varphi: H_{bs}(L_\infty) \to \mathbb{C}$ is uniquely determined by the sequence $\{\varphi(R_n)\}_{n=1}^\infty.$ By the continuity of the homomorphism $\varphi,$ the sequence $\{\sqrt[n]{ \varphi(R_n) }\}_{n=1}^\infty$ is bounded. On the other hand, for every sequence $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C},$ such that the sequence $\{\sqrt[n]{ \xi_n }\}_{n=1}^\infty$ is bounded,  there exists  $x_\xi \in L_\infty$ such that $R_n(x_\xi) = \xi_n$ for every $n \in \mathbb{N}.$ Therefore, for the point-evaluation functional $\delta_{x_\xi}$ we have $\delta_{x_\xi}(R_n) = \xi_n$ for every $n \in \mathbb{N}.$ Thus, every continuous complex-valued homomorphism of $H_{bs}(L_\infty)$ is a point-evaluation functional at some point of $L_\infty.$ Note that such a point is not unique. We can consider an equivalence relation on $L_\infty,$ defined by $x\sim y \Leftrightarrow \delta_x = \delta_y.$ The spectrum (the set of all continuous complex-valued homomorphisms) $M_{bs}$ of the algebra $H_{bs}(L_\infty)$ is one-to-one with the quotient set $L_\infty/_\sim.$ Consequently, $M_{bs}$ can be endowed with the quotient topology. On the other hand, it is naturally to identify $M_{bs}$ with the set of all sequences $\{\xi_n\}_{n=1}^\infty \subset \mathbb{C}$ such that the sequence $\{\sqrt[n]{ \xi_n }\}_{n=1}^\infty$ is bounded.We show that the quotient topology is Hausdorffand that $M_{bs}$ with the operation of coordinate-wise addition of sequences forms an abelian topological group.
PubDate: 2017-06-19
Issue No: Vol. 9 (2017)

• Points of narrowness and uniformly narrow operators

• Authors: A. I. Gumenchuk, I. V. Krasikova, M. M. Popov
Abstract: It is known that the sum of every two narrow operators on $L_1$ is narrow, however the same is false for $L_p$ with $1 < p < \infty$. The present paper continues numerous investigations of the kind. Firstly, we study narrowness of a linear and orthogonally additive operators on Kothe function spaces and Riesz spaces at a fixed point. Theorem 1 asserts that, for every Kothe Banach space $E$ on a finite atomless measure space there exist continuous linear operators $S,T: E \to E$ which are narrow at some fixed point but the sum $S+T$ is not narrow at the same point. Secondly, we introduce and study uniformly narrow pairs of operators $S,T: E \to X$, that is, for every $e \in E$ and every $\varepsilon > 0$ there exists a decomposition $e = e' + e''$ to disjoint elements such that $\ S(e') - S(e'')\ < \varepsilon$ and $\ T(e') - T(e'')\ < \varepsilon$. The standard tool in the literature to prove the narrowness of the sum of two narrow operators $S+T$ is to show that the pair $S,T$ is uniformly narrow. We study the question of whether every pair of narrow operators with narrow sum is uniformly narrow. Having no counterexample, we prove several theorems showing that the answer is affirmative for some partial cases.
PubDate: 2017-06-15
Issue No: Vol. 9 (2017)

• Properties of power series of analytic in a bidisc functions of bounded
$\mathbf{L}$-index in joint variables

• Authors: A. I. Bandura, N. V. Petrechko
Abstract: We generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in a bidisc functions, where $\mathbf{L}(z)=(l_1(z_1,z_2),$ $l_{2}(z_1,z_2)),$ $l_j:\mathbb{D}^2\to \mathbb{R}_+$ is a continuous function, $j\in\{1,2\},$ $\mathbb{D}^2$ is a bidisc $\{(z_1,z_2)\in\mathbb{C}^2: z_1 <1, z_2 <1\}.$ The propositions describe a behaviour of power series expansion on a skeleton of a bidisc. We estimated power series expansion by a dominating homogeneous polynomial with the degree that does not exceed some number depending only from radii of bidisc. Replacing universal quantifier by existential quantifier for radii of bidisc, we also proved sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for analytic functions which are weaker than necessary conditions.
PubDate: 2017-06-08
Issue No: Vol. 9 (2017)

• Convergence criterion for branched contіnued fractions of the special
form with positive elements

• Authors: D. I. Bodnar, I. B. Bilanyk
Abstract: In this paper the problem of convergence of the important type of a multidimensional generalization of continued fractions, the branched continued fractions with independent variables, is considered. This fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. When variables are fixed these fractions are called the branched continued fractions of the special form. Their structure is much simpler then the structure of general branched continued fractions. It has given a possibility to establish the necessary and sufficient conditions of convergence of branched continued fractions of the special form with the positive elements. The received result is the multidimensional analog of Seidel's criterion for the continued fractions. The condition of convergence of investigated fractions is the divergence of series, whose elements are continued fractions. Therefore, the sufficient condition of the convergence of this fraction which has been formulated by the divergence of series composed of partial denominators of this fraction, is established. Using the established criterion and Stieltjes-Vitali Theorem the parabolic theorems of branched continued fractions of the special form with complex elements convergence, is investigated. The sufficient conditions gave a possibility to make the condition of convergence of the branched continued fractions of the special form, whose elements lie in parabolic domains.
PubDate: 2017-06-08
Issue No: Vol. 9 (2017)

• Superextensions of three-element semigroups

• Authors: V. M. Gavrylkiv
Abstract: A family $\mathcal{A}$ of non-empty subsets of a set $X$ is called an {\em upfamily} if for each set $A\in\mathcal{A}$ any set $B\supset A$ belongs to $\mathcal{A}$. An upfamily $\mathcal L$ of subsets of $X$ is said to be {\em linked} if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $X$ is {\em maximal linked} if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $X$ that contains $\mathcal M$. The {\em superextension} $\lambda(X)$ consists of all maximal linked upfamilies on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $\circ: \lambda(X)\times\lambda(X)\to\lambda(X)$ by the formula $\mathcal L\circ\mathcal M=\Big\langle\bigcup_{a\in L}a*M_a:L\in\mathcal L,\;\{M_a\}_{a\in L}\subset\mathcal M\Big\rangle$ for maximal linked upfamilies $\mathcal{L}, \mathcal{M}\in\lambda(X)$. In the paper we describe superextensions of all three-element semigroups up to isomorphism.
PubDate: 2017-06-08
Issue No: Vol. 9 (2017)

• Parabolic systems of Shilov-type with coefficients of bounded smoothness
and nonnegative genus

• Authors: V. A. Litovchenko, G. M. Unguryan
Abstract: The Shilov-type parabolic systems are parabolically stable systems for changing its coefficients unlike of parabolic systems by Petrovskii. That's why the modern theory of the Cauchy problem for class by Shilov-type systems is developing abreast how the theory of the systems with constant or time-dependent coefficients alone. Building the theory of the Cauchy problem for systems with variable coefficients is actually today. A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with variable coefficients that includes a class of the Shilov-type systems with time-dependent coefficients and non-negative genus is considered in this work. A main part of differential expression concerning space variable $x$ of each such system is parabolic (by Shilov) expression. Coefficients of this expression are time-dependent, but coefficients of a group of younger members may depend also a space variable. We built the fundamental solution of the Cauchy problem for systems from this class by the method of sequential approximations. Conditions of  minimal smoothness on  coefficients of the systems by  variable $x$ are founded, the smoothness of solution is investigated and estimates of derivatives of this solution are obtained.  These results are important for  investigating  of the correct solution of the Cauchy problem for this systems in different functional spaces, obtaining forms of description of the solution of this problem and its properties.
PubDate: 2017-06-08
Issue No: Vol. 9 (2017)

• Boundary problem for the singular heat equation

• Authors: O. V. Makhnei
Abstract: The scheme for solving of a mixed problem with general boundary conditions is proposed for a heat equation $a(x)\frac{\partial T}{\partial \tau}= \frac{\partial}{\partial x} \left(\lambda(x)\frac{\partial T}{\partial x}\right)$ with coefficient $a(x)$ that is thegeneralized derivative of a function of bounded variation, $\lambda(x)>0$, $\lambda^{-1}(x)$ is a bounded and measurable function. The boundary conditions have the form $$\left\{ \begin{array}{l}p_{11}T(0,\tau)+p_{12}T^{[1]}_x (0,\tau)+ q_{11}T(l,\tau)+q_{12}T^{[1]}_x (l,\tau)= \psi_1(\tau),\\p_{21}T(0,\tau)+p_{22}T^{[1]}_x (0,\tau)+ q_{21}T(l,\tau)+q_{22}T^{[1]}_x (l,\tau)= \psi_2(\tau),\end{array}\right.$$ where by $T^{[1]}_x (x,\tau)$ we denote the quasiderivative $\lambda(x)\frac{\partial T}{\partial x}$. A solution of this problem seek by thereduction method in the form of sum of two functions $T(x,\tau)=u(x,\tau)+v(x,\tau)$. This method allows to reduce solving of proposed problem to solving oftwo problems: a quasistationary boundary problem with initialand boundary conditions for the search of the function $u(x,\tau)$ and a mixed problem with zero boundaryconditions for some inhomogeneous equation with an unknown function $v(x,\tau)$. The first of these problems is solved through the introduction of the quasiderivative. Fourier method andexpansions in eigenfunctions of some boundary value problem forthe second-order quasidifferential equation $(\lambda(x)X'(x))'+ \omega a(x)X(x)=0$ are used for solving of the second problem. The function $v(x,\tau)$ is represented as a series in eigenfunctions of this boundary value problem. The results can be used in the investigation process of heat transfer in a multilayer plate.
PubDate: 2017-06-08
Issue No: Vol. 9 (2017)

• Bogdan Yosypovych Ptashnyk (obituary)

• Authors: I. Ya. Ivasiuk
Abstract: 
PubDate: 2017-06-08
Issue No: Vol. 9 (2017)

• An example of a non-Borel locally-connected finite-dimensional topological
group

• Authors: I. Banakh, T. Banakh, M. Vovk
Abstract: According to a classical theorem of Gleason and Montgomery, every finite-dimensional locally path-connected topological group is a Lie group. In the paper for every $n\ge 2$ we construct a locally connected subgroup $G\subset{\mathbb R}^{n+1}$ of dimension $\dim(G)=n$, which is not locally compact. This answers a question posed by S. Maillot on MathOverflow and shows that the local path-connectedness in the result of Gleason and Montgomery can not be weakened to the local connectedness.
PubDate: 2017-06-07
Issue No: Vol. 9 (2017)

• The structure of solutions of the matrix linear unilateral polynomial
equation with two variables

• Authors: N. S. Dzhaliuk, V. M. Petrychkovych
Abstract: We investigate the structure of solutions of the matrix linear polynomial equation $A(\lambda)X(\lambda)+B(\lambda)Y(\lambda)=C(\lambda),$ in particular, possible degrees of the solutions. The solving of this equation is reduced to the solving of the equivalent matrix polynomial equation with matrix coefficients in triangular forms with invariant factors on the main diagonals, to which the matrices $A (\lambda), B(\lambda)$ \ and \ $C(\lambda)$ are reduced by means of semiscalar equivalent transformations. On the basis of it, we have pointed out the bounds of the degrees of the matrix polynomial equation solutions. Necessary and sufficient conditions for the uniqueness of a solution with a minimal degree are established. An effective method for constructing minimal degree solutions of the equations is suggested. In this article, unlike well-known results about the estimations of the degrees of the solutions of the matrix polynomial equations in which both matrix coefficients are regular or at least one of them is regular, we have considered the case when the matrix polynomial equation has arbitrary matrix coefficients $A(\lambda)$ and $B(\lambda).$
PubDate: 2017-06-07
Issue No: Vol. 9 (2017)

• Poincare series for the algebras of joint invariants and covariants of $n$

• Authors: N. B. Ilash
Abstract: We consider one of the fundamental problems of classical invariant theory - the research of Poincare series for an algebra of invariants of Lie group $SL_2$. The first two terms of the Laurent series expansion of Poincare series at the point $z = 1$ give us important information about the structure of the algebra $\mathcal{I}_{d}.$ It was derived by Hilbert for the algebra ${\mathcal{I}_{d}=\mathbb{C}[V_d]^{\,SL_2}}$ of invariants for binary $d-$form (by $V_d$ denote the vector space over $\mathbb{C}$ consisting of all binary forms homogeneous of degree $d$). Springer got this result, using explicit formula for the Poincare series of this algebra. We consider this problem for the algebra of joint invariants $\mathcal{I}_{2n}=\mathbb{C}[\underbrace{V_2 \oplus V_2 \oplus \cdots \oplus V_2}_{\text{n times}}]^{SL_2}$ and the algebra of joint covariants $\mathcal{C}_{2n}=\mathbb{C}[\underbrace{V_2 {\oplus} V_2 {\oplus} \cdots {\oplus} V_2}_{\text{n times}}{\oplus}\mathbb{C}^2 ]^{SL_2}$ of $n$ quadratic forms. We express the Poincare series $\mathcal{P}(\mathcal{C}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{C}_{2n})_{j}\, z^j$ and $\mathcal{P}(\mathcal{I}_{2n},z)=\sum_{j=0}^{\infty }\dim(\mathcal{I}_{2n})_{j}\, z^j$ of these algebras in terms of Narayana polynomials.   Also, for these algebras we calculate the degrees and asymptotic behavious of the degrees, using their Poincare series.
PubDate: 2017-06-07
Issue No: Vol. 9 (2017)

• A note on the necessity of filtering mechanism for polynomial
observability of time-discrete wave equation

• Authors: Z. Hajjej
Abstract: The problem of uniform polynomial observability was recently analyzed. It is shown that, when the continuous model is uniformly polynomially observable, it is sufficient to filter initial data to derive uniform polynomial observability inequalities for suitable time-discretization schemes. In this note, we prove that a filtering mechanism of high frequency modes is necessary to obtain uniform polynomial observability.More precisely, we give a counterexample which proves that this latter fails without filtering the initial data for time semi-discrete approximations of the wave equation.
PubDate: 2017-06-07
Issue No: Vol. 9 (2017)

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