. In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.
{"title":"Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition","authors":"Xu-Yang Fu, Jia-Yong Wu","doi":"10.2996/kmj/kmj45106","DOIUrl":"https://doi.org/10.2996/kmj/kmj45106","url":null,"abstract":". In this paper, we prove Souplet-Zhang type gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with the compact boundary under the Dirichlet boundary condition when the Bakry-Emery Ricci tensor and the weighted mean curvature are both bounded below. As an application, we obtain a new Liouville type result for some space-time functions on such smooth metric measure spaces. These results generalize previous linear equations to a nonlinear case.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47972727","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 compute the action of the mod p Steenrod operations on the modular invariants of the linear groups with p an odd prime number. Introduction Fix an odd prime p. Let Apn be the alternating group on p letters. Denote by Σpn,p a Sylow p-subgroup of Apn and E an elementary abelian p-group of rank n. Then we have the restriction homomorphisms Res(E, Σpn,p) : H (BΣpn,p) −→ H (BE), Res(E, Apn) : H (BApn ) −→ H (BE), induced by the regular permutation representation E ⊂ Σpn,p ⊂ Apn of E (see Mùi [4]). Here and throughout the paper, we assume that the coefficients are taken in the prime field Z/p. Using modular invariant theory of linear groups, Mùi proved in [3, 4] that ImRes(E, Σpn,p) = E(U1, . . . , Un) ⊗ P (V1, . . . , Vn), ImRes(E, Apn ) = E(M̃n,0, . . . , M̃n,n−1) ⊗ P (L̃n, Qn,1, . . . , Qn,n−1), Here and in what follows, E(., . . . , .) and P (., . . . , .) are the exterior and polynomial algebras over Z/p generated by the variables indicated. L̃n, Q,s are the Dickson invariants of dimensions p, 2(p − p), and M̃n,s, , Uk, Vk are the Mùi invariants of dimensions p − 2p, pk−1, 2pk−1 respectively (see Section 1). Let A be the mod p Steenrod algebra and let τs, ξi be the Milnor elements of dimensions 2p − 1, 2p − 2 respectively in the dual algebra A∗ of A. In [7], Milnor showed that, as an algebra A∗ = E(τ0, τ1, . . .) ⊗ P (ξ1, ξ2, . . .). Then A∗ has a basis consisting of all monomials τSξ = τs0 . . . τsk ξ r1 . . . ξm , with S = (s1, . . . , sk), 0 6 s1 < . . . < sk, R = (r1, . . . , rm), ri > 0. Let St ∈ A denote the dual of τSξ with respect to that basis. Then A has a basis consisting all operations St. For S = ∅, R = (r), St∅,(r) is nothing but the Steenrod operation P . Since H(BG), G = E, Σpn,p or Apn , is an A-module (see [13, Chap. VI]) and the restriction homomorphisms are A-linear, their images are A-submodules of H(BE). 2010 Mathematics Subject Classification. Primary 55S10; Secondary 55S05.
{"title":"Steenrod operations on the modular invariants","authors":"Nguyễn Sum","doi":"10.2996/kmj/1138040053","DOIUrl":"https://doi.org/10.2996/kmj/1138040053","url":null,"abstract":"In this paper, we compute the action of the mod p Steenrod operations on the modular invariants of the linear groups with p an odd prime number. Introduction Fix an odd prime p. Let Apn be the alternating group on p letters. Denote by Σpn,p a Sylow p-subgroup of Apn and E an elementary abelian p-group of rank n. Then we have the restriction homomorphisms Res(E, Σpn,p) : H (BΣpn,p) −→ H (BE), Res(E, Apn) : H (BApn ) −→ H (BE), induced by the regular permutation representation E ⊂ Σpn,p ⊂ Apn of E (see Mùi [4]). Here and throughout the paper, we assume that the coefficients are taken in the prime field Z/p. Using modular invariant theory of linear groups, Mùi proved in [3, 4] that ImRes(E, Σpn,p) = E(U1, . . . , Un) ⊗ P (V1, . . . , Vn), ImRes(E, Apn ) = E(M̃n,0, . . . , M̃n,n−1) ⊗ P (L̃n, Qn,1, . . . , Qn,n−1), Here and in what follows, E(., . . . , .) and P (., . . . , .) are the exterior and polynomial algebras over Z/p generated by the variables indicated. L̃n, Q,s are the Dickson invariants of dimensions p, 2(p − p), and M̃n,s, , Uk, Vk are the Mùi invariants of dimensions p − 2p, pk−1, 2pk−1 respectively (see Section 1). Let A be the mod p Steenrod algebra and let τs, ξi be the Milnor elements of dimensions 2p − 1, 2p − 2 respectively in the dual algebra A∗ of A. In [7], Milnor showed that, as an algebra A∗ = E(τ0, τ1, . . .) ⊗ P (ξ1, ξ2, . . .). Then A∗ has a basis consisting of all monomials τSξ = τs0 . . . τsk ξ r1 . . . ξm , with S = (s1, . . . , sk), 0 6 s1 < . . . < sk, R = (r1, . . . , rm), ri > 0. Let St ∈ A denote the dual of τSξ with respect to that basis. Then A has a basis consisting all operations St. For S = ∅, R = (r), St∅,(r) is nothing but the Steenrod operation P . Since H(BG), G = E, Σpn,p or Apn , is an A-module (see [13, Chap. VI]) and the restriction homomorphisms are A-linear, their images are A-submodules of H(BE). 2010 Mathematics Subject Classification. Primary 55S10; Secondary 55S05.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45262168","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 contraction of the principal series representations of $SL(2,mathbf{R})$","authors":"B. Cahen","doi":"10.2996/kmj/kmj44302","DOIUrl":"https://doi.org/10.2996/kmj/kmj44302","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43207394","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":"The growth of solutions of non-homogeneous linear differential equations","authors":"D. Kumar, Manisha Saini","doi":"10.2996/kmj/kmj44306","DOIUrl":"https://doi.org/10.2996/kmj/kmj44306","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49184992","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 prove a certain geometric version of the Grothendieck Conjecture for tautological curves over Hurwitz stacks. This result generalizes a similar result obtained by Hoshi and Mochizuki in the case of tautological curves over moduli stacks of pointed smooth curves. In the process of studying this version of the Grothendieck Conjecture, we also examine various fundamental geometric properties of “profiled log Hurwitz stacks”, i.e., log algebraic stacks that parametrize Hurwitz coverings for which the marked points are equipped with a certain ordering determined by combinatorial data which we refer to as a “profile”.
{"title":"Geometric version of the Grothendieck conjecture for universal curves over Hurwitz stacks","authors":"Shota Tsujimura","doi":"10.2996/kmj/kmj44305","DOIUrl":"https://doi.org/10.2996/kmj/kmj44305","url":null,"abstract":"In this paper, we prove a certain geometric version of the Grothendieck Conjecture for tautological curves over Hurwitz stacks. This result generalizes a similar result obtained by Hoshi and Mochizuki in the case of tautological curves over moduli stacks of pointed smooth curves. In the process of studying this version of the Grothendieck Conjecture, we also examine various fundamental geometric properties of “profiled log Hurwitz stacks”, i.e., log algebraic stacks that parametrize Hurwitz coverings for which the marked points are equipped with a certain ordering determined by combinatorial data which we refer to as a “profile”.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43910126","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}
Let $f^1,f^2,f^3$ are three holomorphic curves from a complex disc $Delta (R)$ into $mathbf{P}^n(mathbf{C}) (nge 2)$ with finite growth indexes $c_{f^1},c_{f^2},c_{f^3}$ and sharing $q (q ge 2n+2)$ hyperplanes in general position regardless of multiplicity. In this paper, we will show the above bounds for the sum $c_{f^1}+c_{f^2}+c_{f^3}$ to ensure that $f^1wedge f^2wedge f^3=0$ or there are two curves among ${f^1,f^2,f^3}$ coincide to each other. Our results are generalizations of the previous degeneracy and finiteness results for linearly non-degenerate meromorphic mappings from $mathbf{C}^m$ into $mathbf{P}^n(mathbf{C})$ sharing $(2n+2)$ hyperplanes regardless of multiplicities.
{"title":"Degeneracy and finiteness problems for holomorphic curves from a disc into $mathbf{P}^n(C)$ with finite growth index","authors":"Duc Quang Si","doi":"10.2996/kmj44209","DOIUrl":"https://doi.org/10.2996/kmj44209","url":null,"abstract":"Let $f^1,f^2,f^3$ are three holomorphic curves from a complex disc $Delta (R)$ into $mathbf{P}^n(mathbf{C}) (nge 2)$ with finite growth indexes $c_{f^1},c_{f^2},c_{f^3}$ and sharing $q (q ge 2n+2)$ hyperplanes in general position regardless of multiplicity. In this paper, we will show the above bounds for the sum $c_{f^1}+c_{f^2}+c_{f^3}$ to ensure that $f^1wedge f^2wedge f^3=0$ or there are two curves among ${f^1,f^2,f^3}$ coincide to each other. Our results are generalizations of the previous degeneracy and finiteness results for linearly non-degenerate meromorphic mappings from $mathbf{C}^m$ into $mathbf{P}^n(mathbf{C})$ sharing $(2n+2)$ hyperplanes regardless of multiplicities.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41953073","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":"Correction to the paper \"On complex analytic properties of limit sets and julia sets\"","authors":"H. Shiga","doi":"10.2996/kmj44210","DOIUrl":"https://doi.org/10.2996/kmj44210","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47629678","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 study the univalency and quasiconformal extension of sense-preserving harmonic mappings $f$ in the unit disk. For $f$, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of $f$, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernandez and Martin [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmuller mappings.
{"title":"Criteria for univalency and quasiconformal extension for harmonic mappings","authors":"Zhenyong Hu, Jinhua Fan","doi":"10.2996/kmj44203","DOIUrl":"https://doi.org/10.2996/kmj44203","url":null,"abstract":"In this paper, we study the univalency and quasiconformal extension of sense-preserving harmonic mappings $f$ in the unit disk. For $f$, we introduce a quantity similar to Ahlfors's criteria and obtain a criterion of univalency and quasiconformal extension of $f$, which can be regarded as generalizations of the results obtained by Ahlfors [Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974), 23-29], Hernandez and Martin [Quasiconformal extensions of harmonic mappings in the plane, Ann. Acad. Sci. Fenn. Math. 38 (2013), 617-630], and Chen and Que [Quasiconformal extension of harmonic mappings with a complex parameter, J. Aust. Math. Soc. 102 (2017), 307-315]. By Schwarzian derivatives of harmonic mappings, we also obtain a criterion for univalency and quasiconformal extension for harmonic Techmuller mappings.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42605282","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 characterize conformally flat Riemannian manifolds with constant scalar curvature by the standard contact metric structure of their unit tangent sphere bundles.
本文利用单位切球丛的标准接触度量结构刻画了常标曲率保形平坦黎曼流形。
{"title":"Unit tangent sphere bundles of conformally flat manifolds","authors":"Jong Taek Cho, S. Chun","doi":"10.2996/kmj44205","DOIUrl":"https://doi.org/10.2996/kmj44205","url":null,"abstract":"In this paper, we characterize conformally flat Riemannian manifolds with constant scalar curvature by the standard contact metric structure of their unit tangent sphere bundles.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44936810","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 investigate which plane curves admit rational families of quasi-toric relations. This extends previous results of Takahashi and Tokunaga in the positive case and of the author in the negative case.
{"title":"Curves with rational families of quasi-toric relations","authors":"R. Kloosterman","doi":"10.2996/kmj45203","DOIUrl":"https://doi.org/10.2996/kmj45203","url":null,"abstract":"We investigate which plane curves admit rational families of quasi-toric relations. This extends previous results of Takahashi and Tokunaga in the positive case and of the author in the negative case.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45389782","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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