Georgian Mathematical Journal最新文献

{"title":"Approximation of functions in H ⃗p η1,η2 class by Fejér-type operators","authors":"Yogeshkumar K. Patel, Rajendra G. Vyas","doi":"10.1515/gmj-2024-2041","DOIUrl":"https://doi.org/10.1515/gmj-2024-2041","url":null,"abstract":"In the present work, we explore Fejér-type operators within the mixed Lebesgue space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mover accent=\"true\"> <m:mi>p</m:mi> <m:mo stretchy=\"false\">→</m:mo> </m:mover> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">[</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </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-2024-2041_eq_0165.png\"/> <jats:tex-math>{L_{vec{p}}[mathbb{R}^{2}]}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and establish the degree of approximation for functions belonging to the class <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi>H</m:mi> <m:mover accent=\"true\"> <m:mi>p</m:mi> <m:mo stretchy=\"false\">→</m:mo> </m:mover> <m:mrow> <m:msub> <m:mi>η</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>η</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2041_eq_0161.png\"/> <jats:tex-math>{H_{vec{p}}^{eta_{1},eta_{2}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> through the utilization of Fejér-type operators.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886861","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 minimal surfaces in ℍ2 × ℝ space","authors":"Bendehiba Senoussi","doi":"10.1515/gmj-2024-2038","DOIUrl":"https://doi.org/10.1515/gmj-2024-2038","url":null,"abstract":"A surface is minimal if the mean curvature <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">ℋ</m:mi> <m:mi>mean</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2038_eq_0115.png\"/> <jats:tex-math>{mathcal{H}_{rm mean}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> vanishes everywhere. In this paper, we study some surfaces in the product space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>ℍ</m:mi> <m:mn>2</m:mn> </m:msup> <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-2024-2038_eq_0102.png\"/> <jats:tex-math>{mathbb{H}^{2}timesmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. In particular, we completely classify minimal surfaces.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886866","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":"Ulyanov inequalities for the mixed moduli of smoothness in mixed metrics","authors":"Boris V. Simonov, Ainur A. Jumabayeva","doi":"10.1515/gmj-2024-2034","DOIUrl":"https://doi.org/10.1515/gmj-2024-2034","url":null,"abstract":"In this paper, mixed moduli of smoothness of functions of two variables are studied. We prove Ulyanov-type inequalities between mixed moduli of smoothness of positive orders in different metrics. Estimates for the mixed moduli of smoothness of the derivative of a function are also obtained in terms of the mixed moduli of smoothness of the function itself.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886863","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":"Marcinkiewicz spaces with variable exponents","authors":"Liuye Xia, Yingxiao Han, Mi Fang, Hongya Gao","doi":"10.1515/gmj-2024-2040","DOIUrl":"https://doi.org/10.1515/gmj-2024-2040","url":null,"abstract":"Marcinkiewicz spaces with variable exponents are defined and some basic properties are given.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886860","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":"Sobolev regularity for a class of local fractional new maximal operators","authors":"Rui Li, Shuangping Tao","doi":"10.1515/gmj-2024-2039","DOIUrl":"https://doi.org/10.1515/gmj-2024-2039","url":null,"abstract":"This paper is devoted to studying the regularity properties for the new maximal operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mi>φ</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2039_eq_0146.png\"/> <jats:tex-math>{M_{varphi}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the fractional new maximal operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2039_eq_0145.png\"/> <jats:tex-math>{M_{varphi,beta}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in the local case. Some new pointwise gradient estimates of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2039_eq_0143.png\"/> <jats:tex-math>{M_{varphi,Omega}}</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:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2039_eq_0144.png\"/> <jats:tex-math>{M_{varphi,beta,Omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are given. Moreover, the boundedness of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2039_eq_0143.png\"/> <jats:tex-math>{M_{varphi,Omega}}</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:msub> <m:mi>M</m:mi> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2039_eq_0144.png\"/> <jats:tex-math>{M_{varphi,beta,Omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on Sobolev space is established. As applications, we also obtain the bounds of the above operators on Sobolev space with zero boundary values.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886867","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":"Time-antiperiodic and space-periodic boundary value problem for one class of semilinear partial differential equations","authors":"Sergo Kharibegashvili, Bidzina Midodashvili","doi":"10.1515/gmj-2024-2046","DOIUrl":"https://doi.org/10.1515/gmj-2024-2046","url":null,"abstract":"In this work, a time-antiperiodic and space-periodic boundary value problem for one class of semilinear partial differential equations is studied. The theorems on existence, uniqueness and nonexistence of solutions of this problem are proved.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886862","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 class of nontrivial simple examples of a non-D-space","authors":"Yu-Lin Chou","doi":"10.1515/gmj-2024-2033","DOIUrl":"https://doi.org/10.1515/gmj-2024-2033","url":null,"abstract":"Given any regular <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0029.png\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (equivalently, regular <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0030.png\"/> <jats:tex-math>{T_{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>) space <jats:italic>X</jats:italic>, the question of whether <jats:italic>X</jats:italic> being Lindelöf implies <jats:italic>X</jats:italic> being a <jats:italic>D</jats:italic>-space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0029.png\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> space of uncountable cardinal, with at most countably many singletons being not closed, that is not a <jats:italic>D</jats:italic>-space. Also given is a class of handy examples of a second countable hyperconnected <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2033_eq_0029.png\"/> <jats:tex-math>{T_{0}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> space of uncountable cardinal, with at most countably many singletons being not closed, that is not a <jats:italic>D</jats:italic>-space.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504362","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 higher order Dirac operators in Clifford analysis","authors":"Daniel Alfonso Santiesteban","doi":"10.1515/gmj-2024-2024","DOIUrl":"https://doi.org/10.1515/gmj-2024-2024","url":null,"abstract":"In the framework of Clifford analysis, we study higher order Dirac operators constructed with <jats:italic>k</jats:italic>-vectors. We find a necessary and sufficient condition to determine whether a function cancels them.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526654","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":"Demicompact linear operator. Essential pseudospectra and perturbation","authors":"Aymen Ammar, Houcem Daoud, Aref Jeribi","doi":"10.1515/gmj-2024-2032","DOIUrl":"https://doi.org/10.1515/gmj-2024-2032","url":null,"abstract":"In this paper, we give new results on demicompact linear operators, study some properties and some results on Fredholm and upper semi-Fredholm relations involving demicompact operators. Our results are used to provide a fine description of the essential pseudospectra.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532164","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":"Wave propagation on hexagonal lattices","authors":"David Kapanadze, Ekaterina Pesetskaya","doi":"10.1515/gmj-2024-2035","DOIUrl":"https://doi.org/10.1515/gmj-2024-2035","url":null,"abstract":"We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msqrt> <m:mn>6</m:mn> </m:msqrt> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∖</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:msqrt> <m:mn>2</m:mn> </m:msqrt> <m:mo>,</m:mo> <m:msqrt> <m:mn>3</m:mn> </m:msqrt> <m:mo>,</m:mo> <m:mn>2</m:mn> <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-2024-2035_eq_0179.png\"/> <jats:tex-math>{kin(0,sqrt{6})setminus{sqrt{2},sqrt{3},2}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:msqrt> <m:mn>2</m:mn> </m:msqrt> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∪</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:msqrt> <m:mn>6</m:mn> </m:msqrt> <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-2024-2035_eq_0178.png\"/> <jats:tex-math>{kin(0,sqrt{2})cup(2,sqrt{6})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for the exterior Dirichlet problem.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526655","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}