In 1932 Sierpi'nski proved that every real-valued separately continuous function defined on the plane $mathbb R^2$ is determined uniquely on any everywhere dense subset of $mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $mathbb R^2$, then they are equal at each point of the plane. Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:Xtimes Yto mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $Xtimes Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $Xtimes Y$. Henriksen and Woods proved that for an infinite cardinal $aleph$, an $aleph^+$-Baire space $X$ and a topological space $Y$ with countable $pi$-character every separately continuous function $f:Xtimes Yto mathbb R$ is also determined uniquely on everywhere dense subset of $Xtimes Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $pi$-character and Urysohn space $Z$. Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $pi$-character, $Z$ is Urysohn space, $Asubseteq Xtimes Y$ is everywhere dense set, $f:Xtimes Yto Z$ and $g:Xtimes Yto Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$. In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi'nsi-type theorems.
{"title":"A GENERALIZATION OF SIERPINSKI THEOREM ON UNIQUE DETERMINING OF A SEPARATELY CONTINUOUS FUNCTION","authors":"V. Mykhaylyuk, O. Karlova","doi":"10.31861/bmj2021.01.21","DOIUrl":"https://doi.org/10.31861/bmj2021.01.21","url":null,"abstract":"In 1932 Sierpi'nski proved that every real-valued separately continuous function defined on the plane $mathbb R^2$ is determined uniquely on any everywhere dense subset of $mathbb R^2$. Namely, if two separately continuous functions coincide of an everywhere dense subset of $mathbb R^2$, then they are equal at each point of the plane.\u0000\u0000Piotrowski and Wingler showed that above-mentioned results can be transferred to maps with values in completely regular spaces. They proved that if every separately continuous function $f:Xtimes Yto mathbb R$ is feebly continuous, then for every completely regular space $Z$ every separately continuous map defined on $Xtimes Y$ with values in $Z$ is determined uniquely on everywhere dense subset of $Xtimes Y$. Henriksen and Woods proved that for an infinite cardinal $aleph$, an $aleph^+$-Baire space $X$ and a topological space $Y$ with countable $pi$-character every separately continuous function $f:Xtimes Yto mathbb R$ is also determined uniquely on everywhere dense subset of $Xtimes Y$. Later, Mykhaylyuk proved the same result for a Baire space $X$, a topological space $Y$ with countable $pi$-character and Urysohn space $Z$.\u0000\u0000Moreover, it is natural to consider weaker conditions than separate continuity. The results in this direction were obtained by Volodymyr Maslyuchenko and Filipchuk. They proved that if $X$ is a Baire space, $Y$ is a topological space with countable $pi$-character, $Z$ is Urysohn space, $Asubseteq Xtimes Y$ is everywhere dense set, $f:Xtimes Yto Z$ and $g:Xtimes Yto Z$ are weakly horizontally quasi-continuous, continuous with respect to the second variable, equi-feebly continuous wuth respect to the first one and such that $f|_A=g|_A$, then $f=g$.\u0000\u0000\u0000In this paper we generalize all of the results mentioned above. Moreover, we analize classes of topological spaces wich are favorable for Sierpi'nsi-type theorems.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133486994","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 establish the convergence rate to exponential distribution in a limit theorem for extreme values of birth and death processes. Some applications of this result are given to processes specifying queue length.). We establish uniform estimates for the convergence rate in the exponential distribution in a limit theorem for extreme values of birth and death processes. This topic is closely related to the problem on the time of first intersection of some level u by a regenerating process. Of course, we assume that both time t and level u grow infinitely. The proof of our main result is based on an important estimate for general regenerating processes. Investigations of the kind are needed in different fields: mathematical theory of reliability, queueing theory, some statistical problems in physics. We also provide with examples of applications of our results to extremal queueing problems M/M/s. In particular case of queueing M/M/1, we show that the obtained estimates have the right order with respect to the probability q(u) of the exceeding of a level u at one regeneration cycle, that is, only improvement of the corresponding constants is possible.
{"title":"ON EXTREME VALUES OF BIRTH AND DEATH PROCESSES","authors":"I. Matsak","doi":"10.31861/bmj2021.01.20","DOIUrl":"https://doi.org/10.31861/bmj2021.01.20","url":null,"abstract":"We establish the convergence rate to exponential distribution in a limit theorem for extreme values of birth and death processes. Some applications of this result are given to processes specifying queue length.).\u0000We establish uniform estimates for the convergence rate in the exponential distribution in a limit theorem for extreme values of birth and death processes. This topic is closely related to the problem on the time of first intersection of some level u by a regenerating process. Of course, we assume that both time t and level u grow infinitely. The proof of our main result is based on an important estimate for general regenerating processes. Investigations of the kind are needed in different fields: mathematical theory of reliability, queueing theory, some statistical problems in physics. We also provide with examples of applications of our results to extremal queueing problems M/M/s. In particular case of queueing M/M/1, we show that the obtained estimates have the right order with respect to the probability q(u) of the exceeding of a level u at one regeneration cycle, that is, only improvement of the corresponding constants is possible.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132385364","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 control in a nonlocal boundary value problem with integral conditions for parabolic equation with degeneration","authors":"I. Pukal’skii, B. Yashan","doi":"10.31861/bmj2019.01.082","DOIUrl":"https://doi.org/10.31861/bmj2019.01.082","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116382797","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 problems for Eidelman type equations and systems of equations are considered in this paper. They were the large part of scientific interests for Prof. Ivasyshen S.D. The results of investigations of Cauchy problem, initial-boundary and the inverse problems for this type of equations in bounded or unbounded domains are given. The results are represented as the estimates of the solutions, the integral representations of solutions, theorems of the existence, uniqueness and stability of solutions.
{"title":"ON PROBLEMS FOR EIDELMAN TYPE EQUATIONS AND SYSTEM OF EQUATIONS","authors":"N. Protsakh, H. Ivasiuk, T. Fratavchan","doi":"10.31861/bmj2022.02.17","DOIUrl":"https://doi.org/10.31861/bmj2022.02.17","url":null,"abstract":"The problems for Eidelman type equations and systems of equations are considered in this paper. They were the large part of scientific interests for Prof. Ivasyshen S.D. The results of investigations of Cauchy problem, initial-boundary and the inverse problems for this type of equations in bounded or unbounded domains are given. The results are represented as the estimates of the solutions, the integral representations of solutions, theorems of the existence, uniqueness and stability of solutions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132980987","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 a nonlocal problem for parabolic type equations","authors":"O. Martynyuk, V. Gorodetskiy, R. Kolisnyk","doi":"10.31861/bmj2019.01.014","DOIUrl":"https://doi.org/10.31861/bmj2019.01.014","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122398599","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 weight Holder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solution of the Cauchy problem for degenerated $overrightarrow{2b}$-parabolic equation of Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weight Holder spaces. The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weight Holder spaces.
{"title":"PROPERTIES OF INTEGRALS WHICH HAVE THE TYPE OF DERIVATIVES OF VOLUME POTENTIALS FOR DEGENERATED $overrightarrow{2lowercase{b}}$ - PARABOLIC EQUATION OF KOLMOGOROV TYPE","authors":"V. Dron’, I. Medyns’kyi","doi":"10.31861/bmj2021.02.01","DOIUrl":"https://doi.org/10.31861/bmj2021.02.01","url":null,"abstract":"In weight Holder spaces it is studied the smoothness of integrals, which have the structure\u0000and properties of derivatives of volume potentials which generated by fundamental solution of the Cauchy problem for degenerated $overrightarrow{2b}$-parabolic equation of Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weight Holder spaces.\u0000The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weight Holder spaces.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127117794","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 group classification of one class of (2+1)-dimensional linear equations of Asian options pricing was carried out. As a result, the kernel of maximal invariance algebras and continuous equivalence transformations of this class of equations were found. Using equivalence transformations, all non-equivalent subclasses of equations that have an invariance algebra wider than the kernel of maximal invariance algebras are selected. For each such subclass of equations, Lie algebras of symmetry operators of dimensions four, five, and eight are found.
{"title":"GROUP CLASSIFICATION OF ONE CLASS (2+1)-DIMENSIONAL LINEAR EQUATIONS OF ASIAN OPTIONS PRICING","authors":"S. Spichak, V. Stogniy, I. Kopas","doi":"10.31861/bmj2022.02.19","DOIUrl":"https://doi.org/10.31861/bmj2022.02.19","url":null,"abstract":"A group classification of one class of (2+1)-dimensional linear equations of Asian options pricing was carried out. As a result, the kernel of maximal invariance algebras and continuous equivalence transformations of this class of equations were found. Using equivalence transformations, all non-equivalent subclasses of equations that have an invariance algebra wider than the kernel of maximal invariance algebras are selected. For each such subclass of equations, Lie algebras of symmetry operators of dimensions four, five, and eight are found.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128043964","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 present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-functions analytic in the unit ball, where L = (l1, l2) : B → R+ is a positive continuous vector-function, B = {z ∈ C : |z| = √ |z1| + |z2| ≤ 1}. These conditions describe local behavior of homogeneous polynomials (so-called a main polynomial) with power series expansion for analytic vector-valued functions in the unit ball. These results use a bidisc exhaustion of a unit ball.
{"title":"ON EXISTENCE OF MAIN POLYNOMIAL FOR ANALYTIC VECTOR-VALUED FUNCTIONS OF BOUNDED L-INDEX IN THE UNIT BALL","authors":"Andriy Ivanovych Bandura, V. Baksa, O. Skaskiv","doi":"10.31861/bmj2019.02.006","DOIUrl":"https://doi.org/10.31861/bmj2019.02.006","url":null,"abstract":"In this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-functions analytic in the unit ball, where L = (l1, l2) : B → R+ is a positive continuous vector-function, B = {z ∈ C : |z| = √ |z1| + |z2| ≤ 1}. These conditions describe local behavior of homogeneous polynomials (so-called a main polynomial) with power series expansion for analytic vector-valued functions in the unit ball. These results use a bidisc exhaustion of a unit ball.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127599953","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 find conditions for a singular point O(0, 0) of a center or a focus type to be a center, in a cubic differential system with one irreducible invariant conic. The presence of a center at O(0, 0) is proved by constructing integrating factors.
{"title":"CENTER CONDITIONS FOR A CUBIC DIFFERENTIAL SYSTEM WITH AN INVARIANT CONIC","authors":"D. Cozma","doi":"10.31861/bmj2022.01.02","DOIUrl":"https://doi.org/10.31861/bmj2022.01.02","url":null,"abstract":"We find conditions for a singular point O(0, 0) of a center or a focus type to be a center,\u0000in a cubic differential system with one irreducible invariant conic. The presence of a center at O(0, 0) is proved by constructing integrating factors.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124589654","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 article is an essay about the life and work of an outstanding mathematician, talented teacher, doctor of physical and mathematical sciences, professor S. D. Ivasyshen. The article consists of two interconnected parts. The first part is actually a description of the life path, and the second part is a description and brief anal is of the main areas of scientific research. The whole life of S. D. Ivasyshen was closely related to the mathematics: preparing for classes, writing articles, conducting research and obtaining new results-not a day without mathematics. Being a highly educated and talented mathematician, scientist and teacher, he constantly worked hard, realizing himself through work and respectful attitude towards people.
这篇文章是关于一位杰出的数学家、天才教师、物理与数学科学博士、S. D. Ivasyshen教授的生活和工作的文章。这篇文章由两个相互关联的部分组成。第一部分实际上是对生命路径的描述，第二部分是对主要科学研究领域的描述和简要介绍。S. D. Ivasyshen的一生都与数学密切相关:备课、写文章、做研究、得到新的结果——没有一天是没有数学的。作为一名受过高等教育的天才数学家、科学家和教师，他不断努力工作，在工作中实现自我，尊重他人。
{"title":"IVASYSHEN STEPAN DMYTROVYCH: LIFE AND CREATIVE PATH","authors":"I. Medynsky, H. Pasichnyk","doi":"10.31861/bmj2022.02.01","DOIUrl":"https://doi.org/10.31861/bmj2022.02.01","url":null,"abstract":"The article is an essay about the life and work of an outstanding mathematician, talented teacher, doctor of physical and mathematical sciences, professor S. D. Ivasyshen. The article consists of two interconnected parts. The first part is actually a description of the life path, and the second part is a description and brief anal is of the main areas of scientific research. The whole life of S. D. Ivasyshen was closely related to the mathematics: preparing for classes, writing articles, conducting research and obtaining new results-not a day without mathematics. Being a highly educated and talented mathematician, scientist and teacher, he constantly worked hard, realizing himself through work and respectful attitude towards people.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121673925","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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