Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1415
Tarun Kumar Chauhan, Varun Jindal
We aim to study clopen linear subspaces and connectedness properties of the space C(X) of all real-valued continuous functions defined on a metric space (X,d) equipped with various topologies. In particular, we consider the topologies of strong Whitney and strong uniform convergence on bornology. We also examine when these topologies on C(X) are locally convex. While studying clopen subspaces, we give new characterizations for the notion of a shield and for a bornology to be shielded from closed sets.
{"title":"CLOPEN LINEAR SUBSPACES AND CONNECTEDNESS IN FUNCTION SPACES","authors":"Tarun Kumar Chauhan, Varun Jindal","doi":"10.1216/rmj.2023.53.1415","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1415","url":null,"abstract":"We aim to study clopen linear subspaces and connectedness properties of the space C(X) of all real-valued continuous functions defined on a metric space (X,d) equipped with various topologies. In particular, we consider the topologies of strong Whitney and strong uniform convergence on bornology. We also examine when these topologies on C(X) are locally convex. While studying clopen subspaces, we give new characterizations for the notion of a shield and for a bornology to be shielded from closed sets.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323337","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}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1459
Darshana Devi, Jayanta Borah
We consider a Caputo-type hybrid functional fractional differential equation of order 1
考虑一类具有非局部边界条件的1阶
{"title":"EXISTENCE OF SOLUTIONS FOR A NONLINEAR NONLOCAL HYBRID FUNCTIONAL FRACTIONAL DIFFERENTIAL EQUATION","authors":"Darshana Devi, Jayanta Borah","doi":"10.1216/rmj.2023.53.1459","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1459","url":null,"abstract":"We consider a Caputo-type hybrid functional fractional differential equation of order 1<q≤2 with nonlocal boundary conditions. By using the fixed-point theorem in Banach algebra due to Dhage (1988), we study the existence of the solutions.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323338","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}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1587
Keisuke Teramoto
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near nondegenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.
{"title":"FOCAL SURFACES OF FRONTS ASSOCIATED TO UNBOUNDED PRINCIPAL CURVATURES","authors":"Keisuke Teramoto","doi":"10.1216/rmj.2023.53.1587","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1587","url":null,"abstract":"We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near nondegenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of singularities and geometrical properties of initial fronts. Moreover, we investigate behavior of the Gaussian curvature of the focal surface.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323340","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}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1645
Yongming Wen, Huoxiong Wu
We obtain the quantitative weighted strong-type and weak-type estimates for variation operators associated with heat semigroups in the Schrödinger setting. In particular, we first established the quantitative endpoint bound for such operators in the Schrödinger setting, which is the main novelty of our results.
{"title":"QUANTITATIVE WEIGHTED BOUNDS FOR VARIATION OPERATORS ASSOCIATED WITH HEAT SEMIGROUPS IN THE SCHRÖDINGER SETTING","authors":"Yongming Wen, Huoxiong Wu","doi":"10.1216/rmj.2023.53.1645","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1645","url":null,"abstract":"We obtain the quantitative weighted strong-type and weak-type estimates for variation operators associated with heat semigroups in the Schrödinger setting. In particular, we first established the quantitative endpoint bound for such operators in the Schrödinger setting, which is the main novelty of our results.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323135","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}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1387
Maoli Chang, Liangliang Sun, Yuxin Wang
We study an inverse source problem for a multiterm time-fractional diffusion equation from a noisy final data in a general bounded domain. This problem is ill-posed. Uniqueness and a conditional stability for the inverse problem are derived based on an expression of the solution and some properties of the multinomial Mittag-Leffler function. Further we introduce the modified quasiboundary regularization method and the Landweber iterative regularization method to solve the inverse source problem. Convergence estimates between the regularization solution and the exact solution are given under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule, respectively. Finally, we use the finite difference method to solve the direct problem and the inverse source problem in the one-dimensional case, and apply the finite element method to solve them in the two-dimensional case. Numerical examples are provided to show the effectiveness of the proposed method in the one- and two-dimensional cases.
{"title":"TWO REGULARIZATION METHODS FOR IDENTIFYING THE UNKNOWN SOURCE IN A MULTITERM TIME-FRACTIONAL DIFFUSION EQUATION","authors":"Maoli Chang, Liangliang Sun, Yuxin Wang","doi":"10.1216/rmj.2023.53.1387","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1387","url":null,"abstract":"We study an inverse source problem for a multiterm time-fractional diffusion equation from a noisy final data in a general bounded domain. This problem is ill-posed. Uniqueness and a conditional stability for the inverse problem are derived based on an expression of the solution and some properties of the multinomial Mittag-Leffler function. Further we introduce the modified quasiboundary regularization method and the Landweber iterative regularization method to solve the inverse source problem. Convergence estimates between the regularization solution and the exact solution are given under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule, respectively. Finally, we use the finite difference method to solve the direct problem and the inverse source problem in the one-dimensional case, and apply the finite element method to solve them in the two-dimensional case. Numerical examples are provided to show the effectiveness of the proposed method in the one- and two-dimensional cases.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323331","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}
Pub Date : 2023-10-01DOI: 10.1216/rmj.2023.53.1489
Chengjun Guo, Baili Chen, Junming Liu, Ravi P. Agarwal
We study the existence of homoclinic orbits of the second order quasilinear Schrödinger equations u¨(t)−V(t)u(t)+2[u¨(t)u2(t)+u˙2(t)u(t)]+g(t,u(t+τ),u(t),u(t−τ))=h(t). containing both advance and retardation terms. By using critical point theory and variational approaches, we establish two different existence results. The first is based on g which does not satisfy the Ambrosetti–Rabinowitz growth condition. The second is based on g satisfying the Ambrosetti–Rabinowitz growth condition.
{"title":"EXISTENCE OF HOMOCLINIC ORBITS OF A CLASS OF SECOND-ORDER QUASILINEAR SCHRÖDINGER EQUATIONS WITH DELAY","authors":"Chengjun Guo, Baili Chen, Junming Liu, Ravi P. Agarwal","doi":"10.1216/rmj.2023.53.1489","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1489","url":null,"abstract":"We study the existence of homoclinic orbits of the second order quasilinear Schrödinger equations u¨(t)−V(t)u(t)+2[u¨(t)u2(t)+u˙2(t)u(t)]+g(t,u(t+τ),u(t),u(t−τ))=h(t). containing both advance and retardation terms. By using critical point theory and variational approaches, we establish two different existence results. The first is based on g which does not satisfy the Ambrosetti–Rabinowitz growth condition. The second is based on g satisfying the Ambrosetti–Rabinowitz growth condition.","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":"2013 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323332","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}
Pub Date : 2023-08-01DOI: 10.1216/rmj.2023.53.1129
Jiajie Hua
{"title":"STABILITY OF ROTATION RELATION OF TWO UNITARIES WITH ℤ3, ℤ4 AND ℤ6-ACTIONS IN C*-ALGEBRAS","authors":"Jiajie Hua","doi":"10.1216/rmj.2023.53.1129","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1129","url":null,"abstract":"","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41839746","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}
Pub Date : 2023-08-01DOI: 10.1216/rmj.2023.53.1087
A. Fošner, H. Ghahramani, F. Wei
{"title":"LIE CENTRALIZERS AND GENERALIZED LIE DERIVATIONS AT ZERO PRODUCTS","authors":"A. Fošner, H. Ghahramani, F. Wei","doi":"10.1216/rmj.2023.53.1087","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1087","url":null,"abstract":"","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45534804","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}
Pub Date : 2023-08-01DOI: 10.1216/rmj.2023.53.1327
Wenwan Yang, Cheng Yuan
{"title":"WOLFF’S IDEAL THEOREM ON ANALYTIC BESOV-TYPE SPACE AND ITS MÖBIUS INVARIANT SUBSPACE","authors":"Wenwan Yang, Cheng Yuan","doi":"10.1216/rmj.2023.53.1327","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1327","url":null,"abstract":"","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49437398","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}
Pub Date : 2023-08-01DOI: 10.1216/rmj.2023.53.1073
Moncef Elghribi
{"title":"A NEW CHARACTERIZATION OF HOMOGENEOUS FUNCTIONS AND APPLICATIONS","authors":"Moncef Elghribi","doi":"10.1216/rmj.2023.53.1073","DOIUrl":"https://doi.org/10.1216/rmj.2023.53.1073","url":null,"abstract":"","PeriodicalId":49591,"journal":{"name":"Rocky Mountain Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45048190","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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