The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{ipsi_n},$ $-pi/2+eta
Blaschke积是具有有界Nevanlinna特征的单位圆盘上解析函数的一个重要子类，除了0的累加点$B(z)$外，也是$mathbb{C}$上的亚纯函数。亚纯函数的对数导数的渐近性和估计在数学的各个领域中起着重要的作用。特别是，Goldberg a.a.， Korenkov n.e.， Hayman w.k.， Miles j在Nevanlinna的值分布理论和Chyzhykov i.e.， Strelitz sh . i在微分方程解析理论中研究了这类问题。设$z_0=1$为零的唯一边界点 $(a_n)$ %=1-r_ne^{ipsi_n},$ $-pi/2+eta
{"title":"LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS","authors":"Y. Gal, M. Zabolotskyi, M. Mostova","doi":"10.31861/bmj2021.01.13","DOIUrl":"https://doi.org/10.31861/bmj2021.01.13","url":null,"abstract":"The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $mathbb{C}$ except for the accumulation points of zeros $B(z)$.\u0000Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I.\u0000\u0000Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{ipsi_n},$ $-pi/2+eta<psi_n<pi/2-eta,$ $r_nto0+$ as $nto+infty,$\u0000of the Blaschke product $B(z);$ $Gamma_m=bigcuplimits_{j=1}^{m}{z:|z|<1,mathop{text{arg}}(1-z)=-theta_j}=bigcuplimits_{j=1}^{m}l_{theta_j},$ $-pi/2+eta<theta_1<theta_2<ldots<theta_m<pi/2-eta,$ be a finite system of rays, $0<eta<1$; $upsilon(t)$ be continuous on $[0,1)$, $upsilon(0)=0$, slowly increasing at the point 1 function, that is $upsilon(t)simupsilonleft({(1+t)}/2right),$ $tto1-;$ $n(t,theta_j;B)$ be a number of zeros $a_n=1-r_ne^{itheta_j}$ of the product $B(z)$ on the ray $l_{theta_j}$ such that $1-r_nleq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-ivarphi}to1,$ $-pi/2<varphi<pi/2,$ $varphineqtheta_j,$ under the condition that zeros of $B(z)$ lay on $Gamma_m$ and $n(t,theta_j;B)sim Delta_jupsilon(t),$ $tto1-,$ for all $j=overline{1,m},$ $0leqDelta_j<+infty.$ We also considered the inverse problem for such $B(z).$","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115782562","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 LOCALIZATION PRINCIPLE FOR FORMAL FOURIER SERIES SUMMARIZED BY GAUSS-WEIERSTRASS METHOD","authors":"O. Martynyuk, V. Gorodetskyi","doi":"10.31861/bmj2019.02.030","DOIUrl":"https://doi.org/10.31861/bmj2019.02.030","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115032071","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}
The goal of this paper is to study evolution equations of the parabolic type with operators $displaystyle varphiBig(i frac{partial}{partial x}Big)$ built according to certain functions (different from polynomials), in particular, with operators of fractional differentiation. It is found that the restriction of such operators to certain $S$-type spaces match with pseudo-differential operators in such spaces constructed by these functions, which are multipliers in spaces that are Fourier transforms of $S$-type spaces. The well-posedness of the nonlocal multipoint by time problem is proved for such equations with initial functions that are elements of spaces of generalized functions of $S$-type. The properties of the fundamental solutions of the specified problem, the behavior of the solution at $tto +infty$ in spaces of $S'$-type (weak stabilization) were studied. We found conditions under which the solution stabilizes to zero uniformly on $mathbb{R}$.
{"title":"MULTIPOINT BY TIME PROBLEM FOR A CLASS OF EVOLUTION EQUATIONS IN S TYPE SPACE","authors":"V. Horodetskii, N. Shevchuk, R. Kolisnyk","doi":"10.31861/bmj2022.02.07","DOIUrl":"https://doi.org/10.31861/bmj2022.02.07","url":null,"abstract":"The goal of this paper is to study evolution equations of the parabolic type with operators $displaystyle varphiBig(i frac{partial}{partial x}Big)$ built according to certain functions (different from polynomials), in particular, with operators of fractional differentiation. It is found that the restriction of such operators to certain $S$-type spaces match with pseudo-differential operators in such spaces constructed by these functions, which are multipliers in spaces that are Fourier transforms of $S$-type spaces. The well-posedness of the nonlocal multipoint by time problem is proved for such equations with initial functions that are elements of spaces of generalized functions of $S$-type. The properties of the fundamental solutions of the specified problem, the behavior of the solution at $tto +infty$ in spaces of $S'$-type (weak stabilization) were studied. We found conditions under which the solution stabilizes to zero uniformly on $mathbb{R}$.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"11 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120895590","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}
We study the following question: "Let $f: mathbb{C}to mathbb{C}$ be an entire function of bounded $l$-index, $Phi: mathbb{C}^nto mathbb{C}$ be a slice entire function, $ngeq2,$ $l:mathbb{C}to mathbb{R}_+$ be a continuous function.We study the following question: "Let $f: mathbb{C}to mathbb{C}$ be an entire function of bounded $l$-index, $Phi: mathbb{C}^nto mathbb{C}$ be a slice entire function, $ngeq2,$ $l:mathbb{C}to mathbb{R}_+$ be a continuous function.What is a positive continuous function $L:mathbb{C}^nto mathbb{R}_+$ and a direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ such that the composite function $f(Phi(z))$ has bounded $L$-index in the direction~$mathbf{b}$?". In the present paper, early known results on boundedness of $L$-index in direction for the composition of entire functions$f(Phi(z))$ are generalized to the case where $Phi: mathbb{C}^nto mathbb{C}$ is a slice entire function, i.e.it is an entire function on a complex line ${z^0+tmathbf{b}: tinmathbb{C}}$ for any $z^0inmathbb{C}^n$ andfor a given direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$.These slice entire functions are not joint holomorphic in the general case. For~example, it allows consideration of functions which are holomorphic in variable $z_1$ and continuous in variable $z_2.$
{"title":"COMPOSITION OF SLICE ENTIRE FUNCTIONS AND BOUNDED L-INDEX IN DIRECTION","authors":"O. Skaskiv, Andriy Ivanovych Bandura","doi":"10.31861/bmj2021.01.02","DOIUrl":"https://doi.org/10.31861/bmj2021.01.02","url":null,"abstract":"We study the following question: \"Let $f: mathbb{C}to mathbb{C}$ be an entire function of bounded $l$-index, $Phi: mathbb{C}^nto mathbb{C}$ be a slice entire function, $ngeq2,$ $l:mathbb{C}to mathbb{R}_+$ be a continuous function.We study the following question: \"Let $f: mathbb{C}to mathbb{C}$ be an entire function of bounded $l$-index, $Phi: mathbb{C}^nto mathbb{C}$ be a slice entire function, $ngeq2,$ $l:mathbb{C}to mathbb{R}_+$ be a continuous function.What is a positive continuous function $L:mathbb{C}^nto mathbb{R}_+$ and a direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$ such that the composite function $f(Phi(z))$ has bounded $L$-index in the direction~$mathbf{b}$?\". In the present paper, early known results on boundedness of $L$-index in direction for the composition of entire functions$f(Phi(z))$ are generalized to the case where $Phi: mathbb{C}^nto mathbb{C}$ is a slice entire function, i.e.it is an entire function on a complex line ${z^0+tmathbf{b}: tinmathbb{C}}$ for any $z^0inmathbb{C}^n$ andfor a given direction $mathbf{b}inmathbb{C}^nsetminus{mathbf{0}}$.These slice entire functions are not joint holomorphic in the general case. For~example, it allows consideration of functions which are holomorphic in variable $z_1$ and continuous in variable $z_2.$","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122871229","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}
The problem with two nodes on the selected variable $t$ and periodicity conditions in other coordinates $x_1,ldots,x_p$ for linear partial differential equations is investigated. The conditions of solvability problem in the spaces of smooth functions with exponential behavior of Fourier coefficients are established. The estimates for characteristic determinants of the problem are proved.
{"title":"TWO-POINT PROBLEM FOR LINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS","authors":"M. Symotiuk","doi":"10.31861/bmj2019.02.086","DOIUrl":"https://doi.org/10.31861/bmj2019.02.086","url":null,"abstract":"The problem with two nodes on the selected variable $t$ and periodicity conditions in other coordinates $x_1,ldots,x_p$ for linear partial differential equations is investigated. The conditions of solvability problem in the spaces of smooth functions with exponential behavior of Fourier coefficients are established. The estimates for characteristic determinants of the problem are proved.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"331 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123311509","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}
The dependence of continuous mappings on a certain number of coordinates was intensively studied in the works of many mathematicians in the middle of the 20th century. It has become a convenient tool in the study of properties of continuous mappings. The most general results in this direction were obtained in [5], where the necessary and sufficient conditions for the dependence of continuous functions on products from a certain number of coordinates were obtained. Starting with [8] the dependence of separately continuous mappings on a certain number of coordinates became the subject of research at the Chernivtsi University. For functions of two variables the most general results were obtained in [10]. The dependence on a certain number of coordinates of separately continuous functions of three or more variables was studied in [7], where the necessary and sufficient conditions were established only in the case of metrizability of all factors, which leaves a lot of room for further research. We obtain necessary and sufficient conditions of dependence on countable many of coordinates of functions on the product of three spaces each of which is the product of a family of compact Kempisty spaces.
{"title":"DEPENDENCE ON COUNTABLE MANY OF COORDINATES OF SEPARATELY CONTINUOUS FUNCTIONS OF THREE VARIABLES","authors":"V. Mykhaylyuk","doi":"10.31861/bmj2022.02.14","DOIUrl":"https://doi.org/10.31861/bmj2022.02.14","url":null,"abstract":"The dependence of continuous mappings on a certain number of coordinates was intensively studied in the works of many mathematicians in the middle of the 20th century. It has become a convenient tool in the study of properties of continuous mappings. The most general results in this direction were obtained in [5], where the necessary and sufficient conditions for the dependence of continuous functions on products from a certain number of coordinates were obtained.\u0000\u0000Starting with [8] the dependence of separately continuous mappings on a certain number of coordinates became the subject of research at the Chernivtsi University. For functions of two variables the most general results were obtained in [10]. The dependence on a certain number of coordinates of separately continuous functions of three or more variables was studied in [7], where the necessary and sufficient conditions were established only in the case of metrizability of all factors, which leaves a lot of room for further research.\u0000\u0000We obtain necessary and sufficient conditions of dependence on countable many of coordinates of functions on the product of three spaces each of which is the product of a family of compact Kempisty spaces.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"164 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123163665","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}
A basis for the mathematical modeling of non-isothermal gas competitive adsorption in a porous solid using Langmuir equilibrium is given. High-performance analytical solutions of considered adsorption models based on the Heaviside operating method and Landau’s decom- position and linearization approach of Langmuir equilibrium by expanding into a convergent series in the temperature phase transition point are proposed. Numerical experiments results based on high-speed computations on multicore computers are presented.
{"title":"NONLINEAR MODEL OF THE THREE-COMPONENTS COMPETITIVE ADSORPTION USING LANGMUIR EQUILIBRIUM","authors":"I. Boyko, M. Petryk, M. Shynkaryk, O. Petryk","doi":"10.31861/bmj2021.01.06","DOIUrl":"https://doi.org/10.31861/bmj2021.01.06","url":null,"abstract":"A basis for the mathematical modeling of non-isothermal gas competitive adsorption in a\u0000porous solid using Langmuir equilibrium is given. High-performance analytical solutions of\u0000considered adsorption models based on the Heaviside operating method and Landau’s decom-\u0000position and linearization approach of Langmuir equilibrium by expanding into a convergent\u0000series in the temperature phase transition point are proposed.\u0000Numerical experiments results based on high-speed computations on multicore computers\u0000are presented.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130589632","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":"Inverse problem for a 2d strongly degenerate heat equation with integral overdetermination conditions","authors":"V. Vlasov, M. Ivanchov","doi":"10.31861/bmj2019.01.032","DOIUrl":"https://doi.org/10.31861/bmj2019.01.032","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129342599","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}
In this paper we continue the study of interconnections between separately continuous function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .
在本文中，我们继续研究由V. K. Maslyuchenko开始的独立连续函数之间的相互关系。如果g≤h，且g是上半连续函数，h是下半连续函数，则拓扑空间上的函数对(g, h)称为Hahn对。我们说一对Hahn (g, h)是由一个函数f生成的，它依赖于两个变量，如果f对第二个变量的最小值和最大值分别等于g和h。证明了对于任意完全正规空间X和非伪紧空间Y, X上的每一对哈恩函数都是由X X Y上的一个连续函数生成的。我们还得到了对于任何完全正规空间X和具有非分散紧化的空间Y, X上的任何哈恩函数对都是由X X Y上的一个单独的连续函数生成的。
{"title":"PAIRS OF HAHN AND SEPARATELY CONTINUOUS FUNCTION","authors":"O. Maslyuchenko, A. Kushnir","doi":"10.31861/bmj2021.01.18","DOIUrl":"https://doi.org/10.31861/bmj2021.01.18","url":null,"abstract":"In this paper we continue the study of interconnections between separately continuous\u0000function which was started by V. K. Maslyuchenko. A pair (g, h) of functions on a topological space is called a pair of Hahn if g ≤ h, g is an upper semicontinuous function and h is a lower semicontinuous function. We say that a pair of Hahn (g, h) is generated by a function f, which depends on two variables, if the infimum of f and the supremum of f with respect to the second variable equals g and h respectively. We prove that for any perfectly normal space X and non-pseudocompact space Y every pair of Hahn on X is generated by a continuous function on X x Y . We also obtain that for any perfectly normal space X and for any space Y having non-scattered compactification any pair of Hahn on X is generated by a separately continuous function on X x Y .","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":"19 7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128903984","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}
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