Let p be a prime integer, let be a commutative complete local Noetherian ring with an algebraically closed residue field k of charateristic p and letG be a finite group. Let P be a p-subgroup of G and let X be an indecomposable P-module with vertex P. Let Λ(G, P, X) denote a set of representatives for the isomorphism classes of indecomposable G-modules with vertex-source pair (P, X) (so that Λ(G, P, X) is a finite set by the Green correspondence). As mentioned in [5, Notes on Section 26], L. Puig asserted that a defect multiplicity module determined by (P, X) can be used to obtain an extended parameterization of Λ(G, P, X). In [5, Proposition 26.3], J. Thévenaz completed this program under the hypotheses that X is -free. Here we use the methods of proof of [5, Theorem 26.3] to show that the -free hypothesis on X is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let B be a block of G. Then we obtain a corresponding paramaterization of the (G)B-modules in Λ(G, P, X).
{"title":"A Block Refinement of the Green-Puig Parameterization of the Isomorphism Types of Indecomposable Modules","authors":"M. E. Harris","doi":"10.18910/72315","DOIUrl":"https://doi.org/10.18910/72315","url":null,"abstract":"Let p be a prime integer, let be a commutative complete local Noetherian ring with an algebraically closed residue field k of charateristic p and letG be a finite group. Let P be a p-subgroup of G and let X be an indecomposable P-module with vertex P. Let Λ(G, P, X) denote a set of representatives for the isomorphism classes of indecomposable G-modules with vertex-source pair (P, X) (so that Λ(G, P, X) is a finite set by the Green correspondence). As mentioned in [5, Notes on Section 26], L. Puig asserted that a defect multiplicity module determined by (P, X) can be used to obtain an extended parameterization of Λ(G, P, X). In [5, Proposition 26.3], J. Thévenaz completed this program under the hypotheses that X is -free. Here we use the methods of proof of [5, Theorem 26.3] to show that the -free hypothesis on X is superfluous. (M. Linckelmann has also proved this, cf. [3]). Let B be a block of G. Then we obtain a corresponding paramaterization of the (G)B-modules in Λ(G, P, X).","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"229-236"},"PeriodicalIF":0.4,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45339670","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":"STRONG-VISCOSITY SOLUTIONS: CLASSICAL AND PATH-DEPENDENT PDEs","authors":"Andrea Cosso, F. Russo","doi":"10.18910/72322","DOIUrl":"https://doi.org/10.18910/72322","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"323-373"},"PeriodicalIF":0.4,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44748041","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 will introduce the concepts of symmetric twisted partial Hopf coactions, and discuss under which conditions a given symmetric twisted partial Hopf coaction is globalizable. Then we will introduce the notion of partial cleft coextensions which are dual to partial cleft extensions introduced by M. M. S. Alves et.al., and discuss its relation with partial crossed coproducts introduced by the first author of this paper, which covers the classical results in classical Hopf algebra theory.
本文引入了对称扭曲部分Hopf协同的概念，并讨论了给定对称扭曲部分Hopf协同在哪些条件下是可全局的。然后我们将引入部分裂共延的概念，它是M. M. S. Alves等人引入的部分裂共延的对偶。，并讨论了它与第一作者所介绍的部分交叉余积的关系，其中涵盖了经典Hopf代数理论中的经典结果。
{"title":"Cleft Coextension for symmetric twisted partial coactions on coalgebras","authors":"Q.-G. Chen, B. Yang","doi":"10.18910/72321","DOIUrl":"https://doi.org/10.18910/72321","url":null,"abstract":"In this paper, we will introduce the concepts of symmetric twisted partial Hopf coactions, and discuss under which conditions a given symmetric twisted partial Hopf coaction is globalizable. Then we will introduce the notion of partial cleft coextensions which are dual to partial cleft extensions introduced by M. M. S. Alves et.al., and discuss its relation with partial crossed coproducts introduced by the first author of this paper, which covers the classical results in classical Hopf algebra theory.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"301-322"},"PeriodicalIF":0.4,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42827896","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 a class of Rauzy fractals Ra given by the polynomial x − ax + x− 1 where a ≥ 2 is an integer. In particular, we give explicitly an automaton that generates the boundary of Ra and using an unusual numeration system we prove that Ra is homeomorphic to a topological disk.
{"title":"Parametrization for a class of Rauzy Fractals","authors":"J. Bastos, T. R. Souza","doi":"10.18910/71138","DOIUrl":"https://doi.org/10.18910/71138","url":null,"abstract":"In this paper, we study a class of Rauzy fractals Ra given by the polynomial x − ax + x− 1 where a ≥ 2 is an integer. In particular, we give explicitly an automaton that generates the boundary of Ra and using an unusual numeration system we prove that Ra is homeomorphic to a topological disk.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"101-122"},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67906816","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 a su ffi cient condition for a (weakly reducible) Heegaard splitting to be unstabilized and uncritical. We also give a su ffi cient condition for a Heegaard splitting to be critical.
{"title":"Unstabilized weakly reducible Heegaard splittings","authors":"K. Du","doi":"10.18910/71141","DOIUrl":"https://doi.org/10.18910/71141","url":null,"abstract":"In this paper, we give a su ffi cient condition for a (weakly reducible) Heegaard splitting to be unstabilized and uncritical. We also give a su ffi cient condition for a Heegaard splitting to be critical.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"165-172"},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67906995","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 show that n-dimensional perfect fluid spacetimes with divergence-free conformal curvature tensor and constant scalar curvature are generalized Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid Yang pure space is a GRW spacetime. We also prove that perfect fluid spacetimes with harmonic generalized curvature tensor are, under certain conditions, GRW spacetimes. As particular cases, perfect fluids with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensor are GRW spacetimes. Finally, we explore some physical consequences of such results.
{"title":"Perfect fluid spacetimes with harmonic generalized curvature tensor","authors":"C. Mantica, U. De, Y. Suh, L. Molinari","doi":"10.18910/71142","DOIUrl":"https://doi.org/10.18910/71142","url":null,"abstract":"We show that n-dimensional perfect fluid spacetimes with divergence-free conformal curvature tensor and constant scalar curvature are generalized Robertson Walker (GRW) spacetimes; as a consequence a perfect fluid Yang pure space is a GRW spacetime. We also prove that perfect fluid spacetimes with harmonic generalized curvature tensor are, under certain conditions, GRW spacetimes. As particular cases, perfect fluids with divergence-free projective, concircular, conharmonic or quasi-conformal curvature tensor are GRW spacetimes. Finally, we explore some physical consequences of such results.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"173-182"},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67906920","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 R be a left Noetherian ring, S a right Noetherian ring and RU a Gorenstein module with S = End(RU). If the injective dimensions of RU and US are finite, then the last term in the minimal injective resolution of RU has an essential socle.
{"title":"The Socle of the Last Term in the Minimal Injective Resolution of a Gorenstein Module","authors":"Weiling Song, Xiaojin Zhang, Zhaoyong Huang","doi":"10.18910/71139","DOIUrl":"https://doi.org/10.18910/71139","url":null,"abstract":"Let R be a left Noetherian ring, S a right Noetherian ring and RU a Gorenstein module with S = End(RU). If the injective dimensions of RU and US are finite, then the last term in the minimal injective resolution of RU has an essential socle.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"56 1","pages":"123-132"},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67906874","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 the permutation module kP↑GLn(p f ), where P is a parabolic group in the general linear group GLn(p f ) and k is an algebraically closed field of prime characteristic p. The vertices of the components of these modules have been calculated in [9] by Tinberg, who studied these modules for all groups with split BN-pairs in characteristic p. In this paper we show that the idea of suitability is strong enough to find all p-groups that are vertex of some component of kP↑GLn(p f ). Furthermore using a result of Burry and Carlson we show that all components have a different vertex.
{"title":"THE VERTICES OF THE COMPONENTS OF THE PERMUTATION MODULE INDUCED FROM PARABOLIC GROUPS","authors":"Lars Pforte","doi":"10.18910/70821","DOIUrl":"https://doi.org/10.18910/70821","url":null,"abstract":"We consider the permutation module kP↑GLn(p f ), where P is a parabolic group in the general linear group GLn(p f ) and k is an algebraically closed field of prime characteristic p. The vertices of the components of these modules have been calculated in [9] by Tinberg, who studied these modules for all groups with split BN-pairs in characteristic p. In this paper we show that the idea of suitability is strong enough to find all p-groups that are vertex of some component of kP↑GLn(p f ). Furthermore using a result of Burry and Carlson we show that all components have a different vertex.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"55 1","pages":"769-775"},"PeriodicalIF":0.4,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47837974","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 studies the zero noise limit for the solution of a class of one-dimensional stochastic differential equations involving local time with irregular drift. These solutions are expected to approach one of the solutions to the ordinary differential equation formally obtained by cutting off the noise term. By determining the limit, we reveal that the presence of the local time really affects the asymptotic behavior, while it is observed only when intensity of the drift term is close to symmetric around the irregular point. Related with this problem, we also establish the Wentzel-Freidlin type large deviation principle.
{"title":"Zero noise limit of a stochastic differential equation involving a local time","authors":"Kazumasa Kuwada, Taroujirou Matsumura","doi":"10.18910/70822","DOIUrl":"https://doi.org/10.18910/70822","url":null,"abstract":"This paper studies the zero noise limit for the solution of a class of one-dimensional stochastic differential equations involving local time with irregular drift. These solutions are expected to approach one of the solutions to the ordinary differential equation formally obtained by cutting off the noise term. By determining the limit, we reveal that the presence of the local time really affects the asymptotic behavior, while it is observed only when intensity of the drift term is close to symmetric around the irregular point. Related with this problem, we also establish the Wentzel-Freidlin type large deviation principle.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"55 1","pages":"777-794"},"PeriodicalIF":0.4,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42597425","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":"Initial-boundary value problem for the degenerate hyperbolic equation of a hanging string","authors":"M. Takayama","doi":"10.18910/70087","DOIUrl":"https://doi.org/10.18910/70087","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"55 1","pages":"547-565"},"PeriodicalIF":0.4,"publicationDate":"2018-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45049294","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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