Georgian Mathematical Journal最新文献

{"title":"Generalized derivations over amalgamated algebras along an ideal","authors":"Brahim Boudine, Mohammed Zerra","doi":"10.1515/gmj-2023-2108","DOIUrl":"https://doi.org/10.1515/gmj-2023-2108","url":null,"abstract":"Let <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> be two associative rings, let <jats:italic>I</jats:italic> be an ideal of <jats:italic>B</jats:italic> and let <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>Hom</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>A</m:mi> <m:mo>,</m:mo> <m:mi>B</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0167.png\" /> <jats:tex-math>{finmathrm{Hom}(A,B)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In this paper, we give a complete description of generalized derivations over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{Abowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Furthermore, when <jats:italic>A</jats:italic> is prime or semi-prime, we give several identities on generalized derivations which provide the commutativity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>A</m:mi> <m:msup> <m:mo>⋈</m:mo> <m:mi>f</m:mi> </m:msup> <m:mi>I</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2108_eq_0101.png\" /> <jats:tex-math>{Abowtie^{f}I}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"6 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077829","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":"Floquet theory and stability for a class of first order differential equations with delays","authors":"Alexander Domoshnitsky, Elnatan Berenson, Shai Levi, Elena Litsyn","doi":"10.1515/gmj-2023-2119","DOIUrl":"https://doi.org/10.1515/gmj-2023-2119","url":null,"abstract":"A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions.","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":"139077885","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":"Estimates for the commutators of Riesz transforms related to Schrödinger-type operators","authors":"Yanhui Wang, Kang Wang","doi":"10.1515/gmj-2023-2106","DOIUrl":"https://doi.org/10.1515/gmj-2023-2106","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:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;ℒ&lt;/m:mi&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&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:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msup&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;V&lt;/m:mi&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msup&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-2023-2106_eq_0396.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{L}_{2}=(-Delta)^{2}+V^{2}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be the Schrödinger-type operator 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-2023-2106_eq_0389.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{n}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; (&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:mn&gt;5&lt;/m:mn&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-2023-2106_eq_0468.png\" /&gt; &lt;jats:tex-math&gt;{ngeq 5}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;), 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:msubsup&gt; &lt;m:mi&gt;H&lt;/m:mi&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;ℒ&lt;/m:mi&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:msub&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:msubsup&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: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:mo stretchy=\"false\"&gt;)&lt;/m:mo&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-2023-2106_eq_0297.png\" /&gt; &lt;jats:tex-math&gt;{H^{1}_{mathcal{L}_{2}}(mathbb{R}^{n})}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be the Hardy space related to &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=\"script\"&gt;ℒ&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-2023-2106_eq_0397.png\" /&gt; &lt;jats:tex-math&gt;{mathcal{L}_{2}}&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:msub&gt; &lt;m:mi&gt;BMO&lt;/m:mi&gt; &lt;m:mi&gt;θ&lt;/m:mi&gt; &lt;/m:msub&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:mi&gt;ρ&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&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-2023-2106_eq_0406.png\" /&gt; &lt;jats:tex-math&gt;{mathrm{BMO}_{theta}(rho)}&lt;/jats:tex-math&gt; ","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"89 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139080261","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":"Essential norm of Riemann–Stieltjes operator on weighted Bergman spaces with doubling weights","authors":"Lian Hu, Songxiao Li, Rong Yang","doi":"10.1515/gmj-2023-2110","DOIUrl":"https://doi.org/10.1515/gmj-2023-2110","url":null,"abstract":"Let ω be a doubling weight and <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>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo>&lt;</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0161.png\" /> <jats:tex-math>{0&lt;pleq q&lt;infty}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The essential norm of Riemann–Stieltjes operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mi>g</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0210.png\" /> <jats:tex-math>{T_{g}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from the weighted Bergman space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>A</m:mi> <m:mi>ω</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0172.png\" /> <jats:tex-math>{A^{p}_{omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>A</m:mi> <m:mi>ω</m:mi> <m:mi>q</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0173.png\" /> <jats:tex-math>{A^{q}_{omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> was investigated in the unit ball of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℂ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2110_eq_0250.png\" /> <jats:tex-math>{mathbb{C}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","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":"139077938","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 radii of operator matrices in terms of certain complex combinations of operators","authors":"Cristian Conde, Fuad Kittaneh, Hamid Reza Moradi, Mohammad Sababheh","doi":"10.1515/gmj-2023-2112","DOIUrl":"https://doi.org/10.1515/gmj-2023-2112","url":null,"abstract":"Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"23 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077992","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 the comparison of translation invariant convex differentiation bases","authors":"Irakli Japaridze","doi":"10.1515/gmj-2023-2070","DOIUrl":"https://doi.org/10.1515/gmj-2023-2070","url":null,"abstract":"It is known that if <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are translation invariant convex density differentiation bases and the maximal operators associated to them locally majorize each other, then <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> differentiate the integrals of the same class of non-negative functions. We show that under the same conditions it is not possible to assert more about similarity of the differential properties of <jats:italic>B</jats:italic> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>B</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2070_eq_0070.png\" /> <jats:tex-math>{B^{prime}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in view of their positive equivalence.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"81 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077887","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 maximal estimate for an oscillatory operator","authors":"Jiawei Shen, Yali Pan","doi":"10.1515/gmj-2023-2115","DOIUrl":"https://doi.org/10.1515/gmj-2023-2115","url":null,"abstract":"We study the local maximal oscillatory integral operator &lt;jats:disp-formula-group&gt; &lt;jats:disp-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:mrow&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mi&gt;T&lt;/m:mi&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;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;∗&lt;/m:mo&gt; &lt;/m:msubsup&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:mi&gt;f&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:mi&gt;x&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:munder&gt; &lt;m:mo movablelimits=\"false\"&gt;sup&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;m:mo&gt;&lt;&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:munder&gt; &lt;m:mo&gt;⁡&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo maxsize=\"260%\" minsize=\"260%\"&gt;|&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mstyle displaystyle=\"true\"&gt; &lt;m:msub&gt; &lt;m:mo largeop=\"true\" symmetric=\"true\"&gt;∫&lt;/m:mo&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:msub&gt; &lt;/m:mstyle&gt; &lt;m:mrow&gt; &lt;m:mstyle displaystyle=\"true\"&gt; &lt;m:mfrac&gt; &lt;m:msup&gt; &lt;m:mi&gt;e&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;i&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mfrac&gt; &lt;/m:mstyle&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi mathvariant=\"normal\"&gt;Ψ&lt;/m:mi&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:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&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:mover accent=\"true\"&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;^&lt;/m:mo&gt; &lt;/m:mover&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:mi&gt;ξ&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:mpadded width=\"+1.7pt\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;e&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;π&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;i&lt;/m:mi&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:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;〉&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mpadded&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;𝑑&lt;/m:mo&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo maxsize=\"260%\" minsize=\"260%\"&gt;|&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2115_eq_0041.png\" /&gt; &lt;jats:tex-math&gt;displaystyle T_{alpha,beta}^{ast}(f)(x)=sup_{0&lt;t&lt;1}Bigg{|}int_{mathbb{% R}^{n}}frac{e^{i|txi|^{alpha}}}{|txi|^{beta}}Psi(|txi|)widehat{f}(xi)% e^{2pi ilangle x,xirangle},dxiBigg{|},&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt; &lt;/jats:disp-formula-group&gt; where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/M","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"86 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077888","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 statistical convergence of order α in partial metric spaces","authors":"Erdal Bayram, Çiğdem A. Bektaş, Yavuz Altın","doi":"10.1515/gmj-2023-2116","DOIUrl":"https://doi.org/10.1515/gmj-2023-2116","url":null,"abstract":"The present study introduces the notions of statistical convergence of order α and strong <jats:italic>p</jats:italic>-Cesàro summability of order α in partial metric spaces. Also, we examine the inclusion relations between these concepts. In addition, we introduce the notion of λ-statistical convergence of order α in partial metric spaces while providing relations linked to these sequence spaces.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077987","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":"Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping","authors":"Zayd Hajjej","doi":"10.1515/gmj-2023-2105","DOIUrl":"https://doi.org/10.1515/gmj-2023-2105","url":null,"abstract":"Abstract In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - ( ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ ( ∇ ⁡ z , ∇ ⁡ z t ) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ ( x ) ⁢ f ⁢ ( z t ) + g ⁢ ( z ) = 0 |z_{t}|^{alpha}z_{tt}-Delta z_{tt}-bigl{(}xi_{1}+xi_{2}|nabla z|^{2}+% sigma(nabla z,nabla z_{t})bigr{)}Delta z-Delta z_{t}+beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"9 3","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138943947","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 new fuzzy approach of vehicle routing problem for disaster-stricken zones","authors":"Gia Sirbiladze, Bezhan Ghvaberidze, Bidzina Midodashvili, Bidzina Matsaberidze, Irina Khutsishvili","doi":"10.1515/gmj-2023-2097","DOIUrl":"https://doi.org/10.1515/gmj-2023-2097","url":null,"abstract":"Route planning problems are among the activities that have the highest impact in emergency logistical planning, goods transportation and facility location-distribution because of their effects on efficiency in resource management, service levels and client satisfaction. In the extreme conditions, such as disaster-stricken zones, the difficulty of vehicle movement between nearest different affected areas (demand points) on planning routes cause the imprecision of time of movement and the uncertainty of feasibility of movement. In this paper, the imprecision is presented by triangular fuzzy numbers and the uncertainty is presented by a possibility measure. A new two-stage, fuzzy bi-criterion optimization approach for the vehicle routing problem (VRP) is considered. On the first stage, the sample of so-called “promising” closed routes are selected based on a “constructive” approach. On the second stage, triangular fuzzy valued Choquet aggregation (TFCA) operator is constructed for the selected closed routes. The evaluation of constructed routes, levels of failure and possibility of vehicle movement on the roads are aggregated by the TFCA operator by the new criterion – minimization of infeasibility of movement. The new criterion together with the classic criterion – minimization of the total distance traveled – creates a bi-criteria fuzzy VRP. The constructed VRP is reduced to the bi-criteria fuzzy partitioning problem, and an 𝜀-constraint approach is developed for solving it. For numerical experiments, a parallel algorithm is created on the basis of D. Knuth’s algorithm of Dancing Links (DLX). An example is presented with the results of our approach for the VRP, where all Pareto-optimal solutions are found from the set of promising routes. The optimal solutions tend to avoid roads that are problematic because of extreme situations.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"103 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138579260","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}