Let 𝑅 be a prime ring and 𝐿 a non-central Lie ideal of 𝑅. In this paper, we aim to classify the generalized derivations of 𝑅 satisfying some algebraic identities with power values on 𝐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra.
{"title":"Identities with generalized derivations on Lie ideals and Banach algebras","authors":"Abderrahman Hermas, Lahcen Oukhtite","doi":"10.1515/gmj-2023-2101","DOIUrl":"https://doi.org/10.1515/gmj-2023-2101","url":null,"abstract":"Let 𝑅 be a prime ring and 𝐿 a non-central Lie ideal of 𝑅. In this paper, we aim to classify the generalized derivations of 𝑅 satisfying some algebraic identities with power values on 𝐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"120 4 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce a new class of derivations that generalizes skew derivations and semi-derivations, and we call it skew semi-derivation. Furthermore, we present a study of the conditions under which this type of multiplicative derivation becomes additive.
{"title":"Additivity of multiplicative (generalized) skew semi-derivations on rings","authors":"Sk Aziz, Arindam Ghosh, Om Prakash","doi":"10.1515/gmj-2023-2100","DOIUrl":"https://doi.org/10.1515/gmj-2023-2100","url":null,"abstract":"In this paper, we introduce a new class of derivations that generalizes skew derivations and semi-derivations, and we call it skew semi-derivation. Furthermore, we present a study of the conditions under which this type of multiplicative derivation becomes additive.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"37 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we thoroughly study the Nakayama property and some related concepts. Also, we describe multiplication modules that, among other things, satisfy the Nakayama property. Next, we show that a ring 𝑅 is a Max ring if and only if all modules that can be generated by a finite or countable set have the weak Nakayama property. We prove that a ring 𝑅 is a perfect ring if and only if every module that can be generated by a finite or countable set has the Nakayama property. Finally, we present some categorical results on the aforementioned properties.
{"title":"Concerning the Nakayama property of a module","authors":"Somayeh Karimzadeh, Esmaeil Rostami, Somayeh Hadjirezaei","doi":"10.1515/gmj-2023-2102","DOIUrl":"https://doi.org/10.1515/gmj-2023-2102","url":null,"abstract":"In this paper, we thoroughly study the Nakayama property and some related concepts. Also, we describe multiplication modules that, among other things, satisfy the Nakayama property. Next, we show that a ring 𝑅 is a Max ring if and only if all modules that can be generated by a finite or countable set have the weak Nakayama property. We prove that a ring 𝑅 is a perfect ring if and only if every module that can be generated by a finite or countable set has the Nakayama property. Finally, we present some categorical results on the aforementioned properties.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"174 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the problem with data on the boundary of the infinite layer {(t,x):t∈(0,h),x∈Rs},h>0,s∈N,{(t,x):tin(0,h),,xinmathbb{R}^{s}},quad h>0,,sinmathbb{N}, for the system of two differential equations of the second order in the time variable 𝑡 with shifts in the spatial variables x1,x2,…,xsx_{1},x_{2},ldots,x_{s}. We propose a differential-symbol method of constructing a solution of the problem and identify a class of vector functions in which the obtained solution is unique. The method of solving the Dirichlet problem in the layer is illustrated by examples.
在本文中，我们研究了无限层 { ( t , x ) : t∈ ( 0 , h ) , x∈ R s } 边界上的数据问题。 , h > 0 , s ∈ N , {(t,x)：tin(0,h),,xinmathbb{R}^{s}},quad h>0,,sinmathbb{N}, 为时间变量 x 1 , x 2 , ... , x s x_{1},x_{2},ldots,x_{s} 的二阶微分方程系统。我们提出了一种构建问题解的微分符号法，并确定了一类向量函数，在这类向量函数中，得到的解是唯一的。我们通过实例来说明层中 Dirichlet 问题的求解方法。
{"title":"The Dirichlet problem in an infinite layer for a system of differential equations with shifts","authors":"Zinovii Nytrebych, Roman Shevchuk, Ivan Savka","doi":"10.1515/gmj-2023-2104","DOIUrl":"https://doi.org/10.1515/gmj-2023-2104","url":null,"abstract":"In this paper, we study the problem with data on the boundary of the infinite layer <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo rspace=\"0.278em\" stretchy=\"false\">)</m:mo> </m:mrow> <m:mo rspace=\"0.278em\">:</m:mo> <m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>h</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo rspace=\"0.337em\">,</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi mathvariant=\"double-struck\">R</m:mi> <m:mi>s</m:mi> </m:msup> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> <m:mo rspace=\"1.167em\">,</m:mo> <m:mi>h</m:mi> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo rspace=\"0.337em\">,</m:mo> <m:mrow> <m:mi>s</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant=\"double-struck\">N</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2104_eq_9999.png\" /> <jats:tex-math>{(t,x):tin(0,h),,xinmathbb{R}^{s}},quad h>0,,sinmathbb{N},</jats:tex-math> </jats:alternatives> </jats:disp-formula> for the system of two differential equations of the second order in the time variable 𝑡 with shifts in the spatial variables <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>s</m:mi> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2104_ineq_0001.png\" /> <jats:tex-math>x_{1},x_{2},ldots,x_{s}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We propose a differential-symbol method of constructing a solution of the problem and identify a class of vector functions in which the obtained solution is unique. The method of solving the Dirichlet problem in the layer is illustrated by examples.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"14 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let 𝑅 be a commutative ring with identity <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2098_ineq_0001.png" /> <jats:tex-math>1neq 0</jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Z</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2098_ineq_0002.png" /> <jats:tex-math>Z(R)^{prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the set of all non-zero and non-unit elements of ring 𝑅. Further, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2098_ineq_0003.png" /> <jats:tex-math>Gamma^{prime}(R)</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Z</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2098_ineq_0002.png" /> <jats:tex-math>Z(R)^{prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>w</m:mi> <m:mo>∉</m:mo> <m:mrow> <m:mi>z</m:mi> <m:mo></m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2098_ineq_0005.png" /> <jats:tex-math>wnotin zR</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>z</m:mi> <m:mo>∉</m:mo> <m:mrow> <m:mi>w</m:mi> <m:mo></m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_gmj-2023-2098_ineq_0006.png" /> <jats:tex-math>znotin wR</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where <jats:
让𝑅 是一个交换环，其特征为 1≠0 1neq 0，让 Z ( R ) ′ Z(R)^{prime} 是环𝑅 中所有非零非单位元素的集合。此外，Γ ′ ( R ) Gamma^{prime}(R) 表示𝑅的零因子图，是一个无向图，其顶点集为 Z ( R ) ′ Z(R)^{prime} 、当且仅当两个不同的顶点 𝑤 和 𝑧 相邻时，w∉ z R w （notin zR）和 z ∉ w R z （notin wR），其中 q R qR 是元素 △ 在𝑅 中生成的理想。在本文中，我们将找到 n = p 1 N p 2 p 3 n=p_{1}^{N}p_{2}p_{3} 和 p 1 N p 2 M p 3 p_{1}^{N}p_{2}^{M}p_{3} 时，图 Γ ′ ( Z n ) 的无符号拉普拉奇特征值（Gamma^{prime}(mathbb{Z}_{n}）。 其中 p 1 , p 2 , p 3 p_{1},p_{2},p_{3} 是不同的素数，N , M N,M 是正整数。我们还证明了 cozero-divisor graph Γ ′ ( Z p 1 p 2 ) Gamma^{prime}(mathbb{Z}_{p_{1}p_{2}}) 是一个无符号的拉普拉斯积分。
{"title":"Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛","authors":"Mohd Rashid, Muzibur Rahman Mozumder, Mohd Anwar","doi":"10.1515/gmj-2023-2098","DOIUrl":"https://doi.org/10.1515/gmj-2023-2098","url":null,"abstract":"Let 𝑅 be a commutative ring with identity <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2098_ineq_0001.png\" /> <jats:tex-math>1neq 0</jats:tex-math> </jats:alternatives> </jats:inline-formula> and let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Z</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2098_ineq_0002.png\" /> <jats:tex-math>Z(R)^{prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the set of all non-zero and non-unit elements of ring 𝑅. Further, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi mathvariant=\"normal\">Γ</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2098_ineq_0003.png\" /> <jats:tex-math>Gamma^{prime}(R)</jats:tex-math> </jats:alternatives> </jats:inline-formula> denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Z</m:mi> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>R</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2098_ineq_0002.png\" /> <jats:tex-math>Z(R)^{prime}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>w</m:mi> <m:mo>∉</m:mo> <m:mrow> <m:mi>z</m:mi> <m:mo></m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2098_ineq_0005.png\" /> <jats:tex-math>wnotin zR</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>z</m:mi> <m:mo>∉</m:mo> <m:mrow> <m:mi>w</m:mi> <m:mo></m:mo> <m:mi>R</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2098_ineq_0006.png\" /> <jats:tex-math>znotin wR</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where <jats:","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138563869","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established.
{"title":"On a two-dimensional Dirichlet type problem for a linear hyperbolic equation of fourth order","authors":"Tariel Kiguradze, Reemah Alhuzally","doi":"10.1515/gmj-2023-2083","DOIUrl":"https://doi.org/10.1515/gmj-2023-2083","url":null,"abstract":"For a linear hyperbolic equation of fourth order, a Dirichlet type boundary problem in an orthogonally convex domain is investigated. Sharp sufficient conditions guaranteeing solvability and well-posedness of the problem under consideration are established.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"60 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mohammad W. Alomari, Mohammad Sababheh, Cristian Conde, Hamid Reza Moradi
In this paper, we introduce the f-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.
{"title":"Generalized Euclidean operator radius","authors":"Mohammad W. Alomari, Mohammad Sababheh, Cristian Conde, Hamid Reza Moradi","doi":"10.1515/gmj-2023-2079","DOIUrl":"https://doi.org/10.1515/gmj-2023-2079","url":null,"abstract":"In this paper, we introduce the <jats:italic>f</jats:italic>-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the <jats:italic>q</jats:italic>-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"20 6","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504242","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration.
{"title":"Multi-dimensional almost automorphic type sequences and applications","authors":"Marko Kostić, Halis Can Koyuncuoğlu","doi":"10.1515/gmj-2023-2092","DOIUrl":"https://doi.org/10.1515/gmj-2023-2092","url":null,"abstract":"In this paper, we investigate several new classes of multi-dimensional almost automorphic type sequences and focus on their applications to various difference equations involving Volterra difference equations. We provide many structural results, illustrative examples and open problems about the notion under consideration.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"176 3 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space X, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space X have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity capH(⋅){operatorname{cap}_{H}(,cdot,)} (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with capH(⋅){operatorname{cap}_{H}(,cdot,)} is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in X has mean curvature in L1(X,γ){L^{1}(X,gamma)}.
{"title":"BV capacity and perimeter in abstract Wiener spaces and applications","authors":"Guiyang Liu, He Wang, Yu Liu","doi":"10.1515/gmj-2023-2081","DOIUrl":"https://doi.org/10.1515/gmj-2023-2081","url":null,"abstract":"This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space <jats:italic>X</jats:italic>, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space <jats:italic>X</jats:italic> have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>cap</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo rspace=\"4.2pt\" stretchy=\"false\">(</m:mo> <m:mo rspace=\"4.2pt\">⋅</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2081_eq_0438.png\" /> <jats:tex-math>{operatorname{cap}_{H}(,cdot,)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>cap</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo rspace=\"4.2pt\" stretchy=\"false\">(</m:mo> <m:mo rspace=\"4.2pt\">⋅</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2081_eq_0438.png\" /> <jats:tex-math>{operatorname{cap}_{H}(,cdot,)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in <jats:italic>X</jats:italic> has mean curvature in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>γ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2081_eq_0347.png\" /> <jats:tex-math>{L^{1}(X,gamma)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"44 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented.
{"title":"On 〈s〉-generalized topologies","authors":"Jacek Hejduk, Mehmet Kucukaslan, Anna Loranty","doi":"10.1515/gmj-2023-2096","DOIUrl":"https://doi.org/10.1515/gmj-2023-2096","url":null,"abstract":"In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"21 2","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：