Georgian Mathematical Journal最新文献

{"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":null,"pages":null},"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>β</m:mi> <m:mo>∈</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0137.png\" /> <jats:tex-math>{betain{mathbb{C}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, every <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>∈</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0122.png\" /> <jats:tex-math>{alphain{mathbb{C}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> has at most finitely many (possibly none at all) representations of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>α</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msup> <m:mi>β</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>d</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:msup> <m:mi>β</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:mrow> <m:mo>+</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>+</m:mo> <m:msub> <m:mi>d</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0119.png\" /> <jats:tex-math>{alpha=d_{n}beta^{n}+d_{n-1}beta^{n-1}+dots+d_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with nonnegative integers <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>d</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>d</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>d</m:mi> <m:mn>0</m:mn> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0231.png\" /> <jats:tex-math>{n,d_{n},d_{n-1},dots,d_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> if and only if β is a transcendental number or an algebraic number which has a conjugate over <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℚ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2118_eq_0255.png\" /> <jats:tex-math>{{mathbb{Q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (possibly β itself) in the real interval <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"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}

{"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":null,"pages":null},"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":"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">ℒ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>V</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2106_eq_0396.png\" /> <jats:tex-math>{mathcal{L}_{2}=(-Delta)^{2}+V^{2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the Schrödinger-type operator on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</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-2023-2106_eq_0389.png\" /> <jats:tex-math>{mathbb{R}^{n}}</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:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>5</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2106_eq_0468.png\" /> <jats:tex-math>{ngeq 5}</jats:tex-math> </jats:alternatives> </jats:inline-formula>), let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msubsup> <m:mi>H</m:mi> <m:msub> <m:mi mathvariant=\"script\">ℒ</m:mi> <m:mn>2</m:mn> </m:msub> <m:mn>1</m:mn> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <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-2023-2106_eq_0297.png\" /> <jats:tex-math>{H^{1}_{mathcal{L}_{2}}(mathbb{R}^{n})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the Hardy space related to <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">ℒ</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2106_eq_0397.png\" /> <jats:tex-math>{mathcal{L}_{2}}</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:msub> <m:mi>BMO</m:mi> <m:mi>θ</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ρ</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-2023-2106_eq_0406.png\" /> <jats:tex-math>{mathrm{BMO}_{theta}(rho)}</jats:tex-math> ","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"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":"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":null,"pages":null},"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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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 <jats:disp-formula-group> <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mrow> <m:msubsup> <m:mi>T</m:mi> <m:mrow> <m:mi>α</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> <m:mo>∗</m:mo> </m:msubsup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>f</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <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:munder> <m:mo movablelimits=\"false\">sup</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>&lt;</m:mo> <m:mi>t</m:mi> <m:mo>&lt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:munder> <m:mo>⁡</m:mo> <m:mrow> <m:mo maxsize=\"260%\" minsize=\"260%\">|</m:mo> <m:mrow> <m:mstyle displaystyle=\"true\"> <m:msub> <m:mo largeop=\"true\" symmetric=\"true\">∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> </m:mstyle> <m:mrow> <m:mstyle displaystyle=\"true\"> <m:mfrac> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mi>α</m:mi> </m:msup> </m:mrow> </m:msup> <m:msup> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mi>β</m:mi> </m:msup> </m:mfrac> </m:mstyle> <m:mo>⁢</m:mo> <m:mi mathvariant=\"normal\">Ψ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:mi>ξ</m:mi> </m:mrow> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mover accent=\"true\"> <m:mi>f</m:mi> <m:mo>^</m:mo> </m:mover> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mpadded width=\"+1.7pt\"> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>π</m:mi> <m:mo>⁢</m:mo> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">〈</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo stretchy=\"false\">〉</m:mo> </m:mrow> </m:mrow> </m:msup> </m:mpadded> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>ξ</m:mi> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize=\"260%\" minsize=\"260%\">|</m:mo> </m:mrow> </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-2023-2115_eq_0041.png\" /> <jats:tex-math>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{|},</jats:tex-math> </jats:alternatives> </jats:disp-formula> </jats:disp-formula-group> where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/M","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"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":null,"pages":null},"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}