In the present work, we explore Fejér-type operators within the mixed Lebesgue space Lp→[ℝ2]{L_{vec{p}}[mathbb{R}^{2}]} and establish the degree of approximation for functions belonging to the class Hp→η1,η2{H_{vec{p}}^{eta_{1},eta_{2}}} through the utilization of Fejér-type operators.
在本研究中，我们探讨了混合 Lebesgue 空间 L p → [ ℝ 2 ] {L_{vec{p}}[mathbb{R}^{2}]} 中的 Fejér 型算子，并通过利用 Fejér 型算子建立了属于 H p → η 1 , η 2 {H_{vec{p}}^{eta_{1},eta_{2}}} 类函数的逼近度。
{"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}
A surface is minimal if the mean curvature ℋmean{mathcal{H}_{rm mean}} vanishes everywhere. In this paper, we study some surfaces in the product space ℍ2×ℝ{mathbb{H}^{2}timesmathbb{R}}. In particular, we completely classify minimal surfaces.
{"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}
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.
{"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}
Marcinkiewicz spaces with variable exponents are defined and some basic properties are given.
定义了具有可变指数的 Marcinkiewicz 空间，并给出了一些基本性质。
{"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}
This paper is devoted to studying the regularity properties for the new maximal operator Mφ{M_{varphi}} and the fractional new maximal operator Mφ,β{M_{varphi,beta}} in the local case. Some new pointwise gradient estimates of Mφ,Ω{M_{varphi,Omega}} and Mφ,β,Ω{M_{varphi,beta,Omega}} are given. Moreover, the boundedness of Mφ,Ω{M_{varphi,Omega}} and Mφ,β,Ω{M_{varphi,beta,Omega}} on Sobolev space is established. As applications, we also obtain the bounds of the above operators on Sobolev space with zero boundary values.
本文致力于研究局部情况下新最大算子 M φ {M_{varphi} 和分数新最大算子 M φ , β {M_{varphi,beta} 的正则性。给出了 M φ , Ω {M_{varphi,Omega} 和 M φ , β , Ω {M_{varphi,beta,Omega} 的一些新的点梯度估计值。此外，我们还确定了 M φ , Ω {M_{varphi,Omega} 和 M φ , β , Ω {M_{varphi,beta,Omega} 在索波列夫空间上的有界性。作为应用，我们还得到了上述算子在边界值为零的索波列夫空间上的边界。
{"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}
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.
{"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}
Given any regular T0{T_{0}} (equivalently, regular T1{T_{1}}) space X, the question of whether X being Lindelöf implies X being a D-space is an active open problem. This article gives a class of handy examples of a second countable collectionwise normal collectionwise Hausdorff T0{T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D-space. Also given is a class of handy examples of a second countable hyperconnected T0{T_{0}} space of uncountable cardinal, with at most countably many singletons being not closed, that is not a D-space.
给定任何正则 T 0 {T_{0}} （等价于正则 T 1 {T_{1}} ）空间 X (等价地，正则 T 1 {T_{1}} ）空间 X，X 是林德洛夫是否意味着 X 是 D 空间是一个活跃的开放问题。本文给出了一类非 D 空间的第二可数集合正则集合 Hausdorff T 0 {T_{0}}空间的方便例子，该空间具有最多可数的单子不封闭。此外，还给出了一类非 D 空间的第二可数超连接 T 0 {T_{0}} 空间的方便示例，该空间具有最多可数个不封闭的单子。
{"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}
In the framework of Clifford analysis, we study higher order Dirac operators constructed with k-vectors. We find a necessary and sufficient condition to determine whether a function cancels them.
在克利福德分析框架内，我们研究了用 k 向量构造的高阶狄拉克算子。我们找到了确定函数是否抵消它们的必要条件和充分条件。
{"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}
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.
{"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}
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 k∈(0,6)∖{2,3,2}{kin(0,sqrt{6})setminus{sqrt{2},sqrt{3},2}} for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k∈(0,2)∪(2,6){kin(0,sqrt{2})cup(2,sqrt{6})} for the exterior Dirichlet problem.
{"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}
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