Georgian Mathematical Journal最新文献

{"title":"On the representation of solution for the perturbed quasi-linear controlled neutral functional-differential equation with the discontinuous initial condition","authors":"Abdeljalil Nachaoui, Tea Shavadze, Tamaz Tadumadze","doi":"10.1515/gmj-2023-2122","DOIUrl":"https://doi.org/10.1515/gmj-2023-2122","url":null,"abstract":"The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"53 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139422507","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}

{"title":"A note on b-generalized (α,β)-derivations in prime rings","authors":"Nripendu Bera, Basudeb Dhara","doi":"10.1515/gmj-2023-2121","DOIUrl":"https://doi.org/10.1515/gmj-2023-2121","url":null,"abstract":"Let <jats:italic>R</jats:italic> be a prime ring, let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>0</m:mn> <m:mo>≠</m:mo> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:mi>R</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0112.png\" /> <jats:tex-math>{0neq bin R}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and let α and β be two automorphisms of <jats:italic>R</jats:italic>. Suppose that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>F</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</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-2023-2121_eq_0147.png\" /> <jats:tex-math>{F:Rrightarrow R}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>F</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>:</m:mo> <m:mrow> <m:mi>R</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-2023-2121_eq_0155.png\" /> <jats:tex-math>{F_{1}:Rrightarrow R}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are two <jats:italic>b</jats:italic>-generalized <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0103.png\" /> <jats:tex-math>{(alpha,beta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivations of <jats:italic>R</jats:italic> associated with the same <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2121_eq_0103.png\" /> <jats:tex-math>{(alpha,beta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivation <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>d</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>R</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-2023-2121_eq_0098.png\" /> <jats:tex-math>d:Rrightarrow R</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>G</m:mi>","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"130 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139103439","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}

{"title":"Numerical approaches for solution of hyperbolic difference equations on circle","authors":"Allaberen Ashyralyev, Fatih Hezenci, Yasar Sozen","doi":"10.1515/gmj-2023-2103","DOIUrl":"https://doi.org/10.1515/gmj-2023-2103","url":null,"abstract":"The present paper considers nonlocal boundary value problems for hyperbolic equations on the circle <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"double-struck\">T</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2103_ineq_0001.png\" /> <jats:tex-math>mathbb{T}^{1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The first-order modified difference scheme for the numerical solution of nonlocal boundary value problems for hyperbolic equations on a circle is presented. The stability and coercivity estimates in various Hölder norms for solutions of the difference schemes are established. Moreover, numerical examples are provided.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"705 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139092302","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}

{"title":"Multiplicity result for a (p(x),q(x))-Laplacian-like system with indefinite weights","authors":"Khaled Kefi, Chaima Nefzi","doi":"10.1515/gmj-2023-2107","DOIUrl":"https://doi.org/10.1515/gmj-2023-2107","url":null,"abstract":"Under some suitable conditions, we show that at least three weak solutions exist for a system of differential equations involving the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2107_eq_0108.png\" /> <jats:tex-math>{(p(x),q(x))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Laplacian-like with indefinite weights. The proof is related to the Bonanno–Marano critical theorem (<jats:italic>Appl. Anal.</jats:italic> 89 (2010), 1–10).","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"124 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077826","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}

{"title":"Two presentations of a weak type inequality for geometric maximal operators","authors":"Paul Hagelstein, Giorgi Oniani, Alex Stokolos","doi":"10.1515/gmj-2023-2113","DOIUrl":"https://doi.org/10.1515/gmj-2023-2113","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;Φ&lt;/m:mi&gt; &lt;m:mo&gt;:&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;[&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;∞&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;→&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;[&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;∞&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0076.png\" /&gt; &lt;jats:tex-math&gt;{Phi:[0,infty)rightarrow[0,infty)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a Young’s function satisfying the &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0070.png\" /&gt; &lt;jats:tex-math&gt;{Delta_{2}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;-condition and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;M&lt;/m:mi&gt; &lt;m:mi mathvariant=\"script\"&gt;ℬ&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0063.png\" /&gt; &lt;jats:tex-math&gt;{M_{mathcal{B}}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be the geometric maximal operator associated to a homothecy invariant basis &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℬ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0098.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{B}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; acting on measurable functions on &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0090.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Let &lt;jats:italic&gt;Q&lt;/jats:italic&gt; be the unit cube in &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2113_eq_0090.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;L&lt;/m:mi&gt; &lt;m:mi mathvariant=\"normal\"&gt;Φ&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"25 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077895","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}

{"title":"Centralizing identities involving generalized derivations in prime rings","authors":"Vincenzo De Filippis, Pallavee Gupta, Shailesh Kumar Tiwari, Balchand Prajapati","doi":"10.1515/gmj-2023-2109","DOIUrl":"https://doi.org/10.1515/gmj-2023-2109","url":null,"abstract":"Let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℛ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{R}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be a prime ring of characteristic not equal to 2, let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒰&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0527.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{U}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be Utumi quotient ring of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℛ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{R}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;𝒞&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0481.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{C}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be the extended centroid of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℛ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{R}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Let Δ be a generalized derivation on &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"script\"&gt;ℛ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0519.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{R}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, and let &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;δ&lt;/m:mi&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0428.png\" /&gt; &lt;jats:tex-math&gt;{delta_{1}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;δ&lt;/m:mi&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2109_eq_0440.png\" /&gt; &lt;jats:tex-math&gt;{delta_{2}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be derivations on &lt;jats:inline-form","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"33 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077761","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}

{"title":"Almost measurable functions on probability spaces","authors":"Alexander Kharazishvili","doi":"10.1515/gmj-2023-2120","DOIUrl":"https://doi.org/10.1515/gmj-2023-2120","url":null,"abstract":"The notion of (real-valued) almost measurable functions on probability spaces is introduced and some of their properties are considered. It is shown that any almost measurable function may be treated as a quasi-random variable in the sense of [A. Kharazishvili, On some version of random variables, Trans. A. Razmadze Math. Inst. 177 2023, 1, 143–146].","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"21 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077822","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}

{"title":"On φ-u-S-flat modules and nonnil-u-S-injective modules","authors":"Hwankoo Kim, Najib Mahdou, El Houssaine Oubouhou","doi":"10.1515/gmj-2023-2117","DOIUrl":"https://doi.org/10.1515/gmj-2023-2117","url":null,"abstract":"This paper introduces and studies the ϕ-u-<jats:italic>S</jats:italic>-flat (resp., nonnil-u-<jats:italic>S</jats:italic>-injective) modules, which are a generalization of both ϕ-flat modules and u-<jats:italic>S</jats:italic>-flat modules (resp., both nonnil-injective modules and u-<jats:italic>S</jats:italic>-injective modules). We give the Cartan–Eilenberg–Bass theorem for nonnil-u-<jats:italic>S</jats:italic>-Noetherian rings. Finally, we offer some new characterizations of the ϕ-von Neumann regular ring.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077993","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}

{"title":"Some summation theorems and transformations for hypergeometric functions of Kampé de Fériet and Srivastava","authors":"Hari M. Srivastava, Bhawna Gupta, Mohammad Idris Qureshi, Mohd Shaid Baboo","doi":"10.1515/gmj-2023-2114","DOIUrl":"https://doi.org/10.1515/gmj-2023-2114","url":null,"abstract":"Owing to the remarkable success of the hypergeometric functions of one variable, the authors present a study of some families of hypergeometric functions of two or more variables. These functions include (for example) the Kampé de Fériet-type hypergeometric functions in two variables and Srivastava’s general hypergeometric function in three variables. The main aim of this paper is to provide several (presumably new) transformation and summation formulas for appropriately specified members of each of these families of hypergeometric functions in two and three variables. The methodology and techniques, which are used in this paper, are based upon the evaluation of some definite integrals involving logarithmic functions in terms of Riemann’s zeta function, Catalan’s constant, polylogarithm functions, and so on.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"173 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077828","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}

{"title":"Representations of a number in an arbitrary base with unbounded digits","authors":"ArtÅ«ras Dubickas","doi":"10.1515/gmj-2023-2118","DOIUrl":"https://doi.org/10.1515/gmj-2023-2118","url":null,"abstract":"In this paper, we prove that, for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mi&gt;ℂ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0137.png\" /&gt; &lt;jats:tex-math&gt;{betain{mathbb{C}}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, every &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mi&gt;ℂ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0122.png\" /&gt; &lt;jats:tex-math&gt;{alphain{mathbb{C}}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; has at most finitely many (possibly none at all) representations of the form &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;…&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0119.png\" /&gt; &lt;jats:tex-math&gt;{alpha=d_{n}beta^{n}+d_{n-1}beta^{n-1}+dots+d_{0}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; with nonnegative integers &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;n&lt;/m:mi&gt; &lt;m:mo&gt;-&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;…&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0231.png\" /&gt; &lt;jats:tex-math&gt;{n,d_{n},d_{n-1},dots,d_{0}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; if and only if β is a transcendental number or an algebraic number which has a conjugate over &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ℚ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0255.png\" /&gt; &lt;jats:tex-math&gt;{{mathbb{Q}}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; (possibly β itself) in the real interval &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"10 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077836","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}