Matematychni Studii最新文献

{"title":"The reverse Holder inequality for an elementary function","authors":"A.O. Korenovskii","doi":"10.30970/ms.56.1.28-38","DOIUrl":"https://doi.org/10.30970/ms.56.1.28-38","url":null,"abstract":"For a positive function $f$ on the interval $[0,1]$, the power mean of order $pinmathbb R$ is defined by \u0000smallskipcenterline{$displaystyle|, f,|_p=left(int_0^1 f^p(x),dxright)^{1/p}quad(pne0),qquad|, f,|_0=expleft(int_0^1ln f(x),dxright).$} \u0000Assume that $0<A<B$, $0<theta<1$ and consider the step function$g_{A<B,theta}=Bcdotchi_{[0,theta)}+Acdotchi_{[theta,1]}$, where $chi_E$ is the characteristic function of the set $E$. \u0000Let $-infty<p<q<+infty$. The main result of this work consists in finding the term \u0000smallskipcenterline{$displaystyleC_{p<q,A<B}=maxlimits_{0lethetale1}frac{|,g_{A<B,theta},|_q}{|,g_{A<B,theta},|_p}.$} \u0000smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $theta_{p<q,A<B}$ with respect to $beta=B/Ain(1,+infty)$.The cases $p=0$ or $q=0$ are considered separately. \u0000The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41665831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Finite M/M/1 retrial model with changeable service rate","authors":"M. Bratiichuk, A. A. Chechelnitsky, I. Usar","doi":"10.30970/ms.56.1.96-102","DOIUrl":"https://doi.org/10.30970/ms.56.1.96-102","url":null,"abstract":"The article deals with M/M/1 -type retrial queueing system with finite orbit. It is supposedthat service rate depends on the loading of the system. The explicit formulae for ergodicdistribution of the number of customers in the system are obtained. The theoretical results areillustrated by numerical examples.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49306842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces","authors":"S. Halushchak","doi":"10.30970/ms.56.1.106-112","DOIUrl":"https://doi.org/10.30970/ms.56.1.106-112","url":null,"abstract":"The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces. Let $X$ and $Y$ be complex Banach spaces. Let $mathbb{A}= {A_1, A_2, ldots, A_n, ldots}$ and $mathbb{P}={P_1, P_2,$ ldots, $P_n, ldots }$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $|A_n|_1=|P_n|_1=1$ and $deg A_n=deg P_n=n,$ $nin mathbb{N}.$ We consider the subalgebras $H_{bmathbb{A}}(X)$ and $H_{bmathbb{P}}(Y)$ of the Fr'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $mathbb{A}$ and $mathbb{P}$, respectively. It is easy to see that $H_{bmathbb{A}}(X)$ and $H_{bmathbb{P}}(Y)$ are the Fr'{e}chet algebras as well. In this paper we investigate conditions of isomorphism of the topological algebras $H_{bmathbb{A}}(X)$ and $H_{bmathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{infty})$ of entire functions of bounded type on $L_{infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $ell_{infty}.$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42099418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The boundedness of a class of semiclassical Fourier integral operators on Sobolev space $H^{s}$","authors":"O. F. Aid, A. Senoussaoui","doi":"10.30970/ms.56.1.61-66","DOIUrl":"https://doi.org/10.30970/ms.56.1.61-66","url":null,"abstract":"We introduce the relevant background information thatwill be used throughout the paper.Following that, we will go over some fundamental concepts from thetheory of a particular class of semiclassical Fourier integraloperators (symbols and phase functions), which will serve as thestarting point for our main goal. \u0000Furthermore, these integral operators turn out to be bounded on$Sleft(mathbb{R}^{n}right)$ the space of rapidly decreasingfunctions (or Schwartz space) and its dual$S^{prime}left(mathbb{R}^{n}right)$ the space of temperatedistributions. \u0000Moreover, we will give a brief introduction about$H^s(mathbb{R}^n)$ Sobolev space (with $sinmathbb{R}$).Results about the composition of semiclassical Fourier integraloperators with its $L^{2}$-adjoint are proved. These allow to obtainresults about the boundedness on the Sobolev spaces$H^s(mathbb{R}^n)$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47834563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Asymptotic vectors of entire curves","authors":"Y. Savchuk, Andriy Ivanovych Bandura","doi":"10.30970/ms.56.1.48-54","DOIUrl":"https://doi.org/10.30970/ms.56.1.48-54","url":null,"abstract":"We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors. \u0000A non-zero vector $vec{a}=(a_1,a_2,ldots,a_p)in mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $vec{G}(z)=(g_1(z),g_2(z),ldots,g_p(z))$ if there exists a continuous curve $L: mathbb{R}_+to mathbb{C}$ given by an equation $z=zleft(tright)$, $0le t<infty $, $left|zleft(tright)right|<infty $, $zleft(tright)to infty $ as $tto infty $ such that$$limlimits_{stackrel{ztoinfty}{zin L}} frac{vec{G}(z)vec{a} }{big|vec{G}(z)big|}=limlimits_{ttoinfty} frac{vec{G}(z(t))vec{a} }{big|vec{G}(z(t))big|} =0,$$ where $big|vec{G}(z)big|=big(|g_1(z)|^2+ldots +|g_p(z)|^2big)^{1/2}$, $vec{G}(z)vec{a}=g_1(z)cdotbar{a}_1+g_2(z)cdotbar{a}_2+ldots+g_p(z)cdotbar{a}_p$. A non-zero vector $vec{a}=(a_1,a_2,ldots,a_p)in mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $vec{G}(z)$ if the function $vec{G}(z)vec{a}$ has a finite number of zeros in $left{left|zright|<infty right}$. \u0000We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com-po-nents and without common zeros is an asymptotic vector.Here we de-mon-stra-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44293117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the algebraic dimension of Riesz spaces","authors":"N. Baziv, O. B. Hrybel","doi":"10.30970/ms.56.1.67-71","DOIUrl":"https://doi.org/10.30970/ms.56.1.67-71","url":null,"abstract":"We prove that the algebraic dimension of an infinite dimensional $C$-$sigma$-complete Riesz space (in particular, of a Dedekind $sigma$-complete and a laterally $sigma$-complete Riesz space) with the principal projection property which either has a weak order unit or is not purely atomic, is at least continuum. A similar (incomparable to ours) result for complete metric linear spaces is well known.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47923989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fundamentals of metric theory of real numbers in their $overline{Q_3}$-representation","authors":"I. Zamrii, V. Shkapa, H. Vlasyk","doi":"10.30970/ms.56.1.3-19","DOIUrl":"https://doi.org/10.30970/ms.56.1.3-19","url":null,"abstract":"In the paper we were studied encoding of fractional part of a real number with an infinite alphabet (set of digits) coinciding with the set of non-negative integers. The geometry of this encoding is generated by $Q_3$-representation of real numbers, which is a generalization of the classical ternary representation. The new representation has infinite alphabet, zero surfeit and can be efficiently used for specifying mathematical objects with fractal properties. \u0000We have been studied the functions that store the \"tails\" of $overline{Q_3}$-representation of numbers and the set of such functions,some metric problems and some problems of probability theory are connected with $overline{Q_3}$-representation.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69301827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal recovery of operator sequences","authors":"V. Babenko, N. Parfinovych, D. Skorokhodov","doi":"10.30970/ms.56.2.193-207","DOIUrl":"https://doi.org/10.30970/ms.56.2.193-207","url":null,"abstract":"In this paper we solve two problems of optimal recovery based on information given with an error. First is the problem of optimal recovery of the class $W^T_q = {(t_1h_1,t_2h_2,ldots),colon ,|h|_{ell_q}le 1}$, where $1le q < infty$ and $t_1ge t_2ge ldots ge 0$ are given, in the space $ell_q$. Information available about a sequence $xin W^T_q$ is provided either (i) by an element $yinmathbb{R}^n$, $ninmathbb{N}$, whose distance to the first $n$ coordinates $left(x_1,ldots,x_nright)$ of $x$ in the space $ell_r^n$, $0 < r le infty$, does not exceed given $varepsilonge 0$, or (ii) by a sequence $yinell_infty$ whose distance to $x$ in the space $ell_r$ does not exceed $varepsilon$. We show that the optimal method of recovery in this problem is either operator $Phi^*_m$ with some $minmathbb{Z}_+$ ($mle n$ in case $yinell^n_r$), where \u0000smallskipcenterline{$displaystyle Phi^*_m(y) = Big{y_1left(1 - frac{t_{m+1}^q}{t_{1}^q}Big),ldots,y_mBig(1 - frac{t_{m+1}^q}{t_{m}^q}Big),0,ldotsright},quad yinmathbb{R}^ntext{ or } yinell_infty,$} \u0000smallskipnoior convex combination $(1-lambda) Phi^*_{m+1} + lambdaPhi^*_{m}$. \u0000The second one is the problem of optimal recovery of the scalar product operator acting on the Cartesian product $W^{T,S}_{p,q}$ of classes $W^T_p$ and $W^S_q$, where $1 < p,q < infty$, $frac{1}{p} + frac{1}{q} = 1$ and $s_1ge s_2ge ldots ge 0$ are given. Information available about elements $xin W^T_p$ and $yin W^S_q$ is provided by elements $z,win mathbb{R}^n$ such that the distance between vectors $left(x_1y_1, x_2y_2,ldots,x_ny_nright)$ and $left(z_1w_1,ldots,z_nw_nright)$ in the space $ell_r^n$ does not exceed $varepsilon$. We show that the optimal method of recovery is delivered either by operator $Psi^*_m$ with some $min{0,1,ldots,n}$, where \u0000smallskipcenterline{$displaystyle Psi^*_m = sum_{k=1}^m z_kw_kBig(1 - frac{t_{m+1}s_{m+1}}{t_ks_k}Big),quad z,winmathbb{R}^n,$} \u0000smallskipnoior by convex combination $(1-lambda)Psi^*_{m+1} + lambdaPsi^*_{m}$. \u0000As an application of our results we consider the problem of optimal recovery of classes in Hilbert spaces by the Fourier coefficients of its elements known with an error measured in the space $ell_p$ with $p > 2$.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42191413","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On asymorphisms of finitary coarse spaces","authors":"I. Protasov","doi":"10.30970/ms.56.2.212-214","DOIUrl":"https://doi.org/10.30970/ms.56.2.212-214","url":null,"abstract":"We characterize finitary coarse spaces X such that every permutation of X is an asymorphism.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46918662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On entire functions from the Laguerre-Polya I class with non-monotonic second quotients of Taylor coefficients","authors":"Thu Hien Nguyen, A. Vishnyakova","doi":"10.30970/ms.56.2.149-161","DOIUrl":"https://doi.org/10.30970/ms.56.2.149-161","url":null,"abstract":"For an entire function $f(z) = sum_{k=0}^infty a_k z^k, a_k>0,$ we define its second quotients of Taylor coefficients as $q_k (f):= frac{a_{k-1}^2}{a_{k-2}a_k}, k geq 2.$ In the present paper, we study entire functions of order zerowith non-monotonic second quotients of Taylor coefficients. We consider those entire functions for which the even-indexed quotients are all equal and the odd-indexed ones are all equal:$q_{2k} = a>1$ and $q_{2k+1} = b>1$ for all $k in mathbb{N}.$We obtain necessary and sufficient conditions under which such functions belong to the Laguerre-P'olya I class or, in our case, have only real negative zeros. In addition, we illustrate their relation to the partial theta function.","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44038967","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}