In this article we study a three dimensional contact metric manifold M 3 with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with Qx 1⁄4 rx, where r is a smooth function on M constant along Reeb vector field x and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with x or is a gradient vector field. Moreover, if r is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: SUð2Þ or SOð3Þ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field x. Secondly, it is proved that a ðk; m; nÞ-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that M is Sasakian, flat, a contact metric ð0; 4Þ-space or a contact metric ðk; 0Þ-space with k < 1 and k0 0. For the potential vector field being orthogonal to x, if n is constant we prove that M is either Sasakian, or a ðk; mÞ-contact metric space.
{"title":"Three dimensional contact metric manifolds with Cotton solitons","authors":"Xiaomin Chen","doi":"10.32917/h2020064","DOIUrl":"https://doi.org/10.32917/h2020064","url":null,"abstract":"In this article we study a three dimensional contact metric manifold M 3 with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with Qx 1⁄4 rx, where r is a smooth function on M constant along Reeb vector field x and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with x or is a gradient vector field. Moreover, if r is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: SUð2Þ or SOð3Þ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field x. Secondly, it is proved that a ðk; m; nÞ-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that M is Sasakian, flat, a contact metric ð0; 4Þ-space or a contact metric ðk; 0Þ-space with k < 1 and k0 0. For the potential vector field being orthogonal to x, if n is constant we prove that M is either Sasakian, or a ðk; mÞ-contact metric space.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48106976","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 problem of minimal volume vector fields on a given Riemann surface, specialising on the case of $M^star$, that is, the arbitrary radius 2-sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle $(T^1M^star,partial T^1M^star)$ in relation with calibrations and a certain minimal volume equation. A particular family $X_{mathrm{m},k},:kinmathbb{N}$, of minimal vector fields on $M^star$ is found in an original fashion. The family has unbounded volume, $lim_kmathrm{vol}({X_{mathrm{m},k}}_{|Omega})=+infty$, on any given open subset $Omega$ of $M^star$ and indeed satisfies the necessary differential equation for minimality. Another vector field $X_ell$ is discovered on a region $Omega_1subsetmathbb{S}^2$, with volume smaller than any other known textit{optimal} vector field restricted to $Omega_1$.
{"title":"Vector fields with big and small volume on the 2-sphere","authors":"R. Albuquerque","doi":"10.32917/h2022009","DOIUrl":"https://doi.org/10.32917/h2022009","url":null,"abstract":"We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on the case of $M^star$, that is, the arbitrary radius 2-sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle $(T^1M^star,partial T^1M^star)$ in relation with calibrations and a certain minimal volume equation. A particular family $X_{mathrm{m},k},:kinmathbb{N}$, of minimal vector fields on $M^star$ is found in an original fashion. The family has unbounded volume, $lim_kmathrm{vol}({X_{mathrm{m},k}}_{|Omega})=+infty$, on any given open subset $Omega$ of $M^star$ and indeed satisfies the necessary differential equation for minimality. Another vector field $X_ell$ is discovered on a region $Omega_1subsetmathbb{S}^2$, with volume smaller than any other known textit{optimal} vector field restricted to $Omega_1$.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49108640","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}
Fairness and centredness of ideals in commutative rings, i.e., the relations between assassins and weak assassins of a module, its small or large torsion submodule, and the corresponding quotients, are studied. General criteria as well as more specific results about idempotent or nil ideals are given, and several examples are presented.
{"title":"Assassins and torsion functors II","authors":"F. Rohrer","doi":"10.32917/h2020095","DOIUrl":"https://doi.org/10.32917/h2020095","url":null,"abstract":"Fairness and centredness of ideals in commutative rings, i.e., the relations between assassins and weak assassins of a module, its small or large torsion submodule, and the corresponding quotients, are studied. General criteria as well as more specific results about idempotent or nil ideals are given, and several examples are presented.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43017595","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 introduce the concept CFA modules and their applications in investigation the coassociated primes of local homology modules. The main result of this paper says that if $M$ is a CFA linearly compact $R$-module and $t$ is a non-negative integer such that H i I ( M ) is CFA for all $i < t$, then R / I ⊗ R H t I ( M ) is CFA. Hence, the set $mathrm{Coass}_R$ H t I ( M ) is finite.
我们介绍了CFA模的概念及其在研究局部同源模的共缔合素数中的应用。本文的主要结果表明，如果$M$是CFA线性紧致$R$-模，$t$是一个非负整数，使得H i i（M）是所有$i＜t$的CFA，那么R/i⊗R H t i（M）是CFA。因此，集合$mathrm{Coass}_R$HtI（M）是有限的。
{"title":"CFA modules and the finiteness of coassociated primes of local homology modules","authors":"N. Tri","doi":"10.32917/H2020073","DOIUrl":"https://doi.org/10.32917/H2020073","url":null,"abstract":"We introduce the concept CFA modules and their applications in investigation the coassociated primes of local homology modules. The main result of this paper says that if $M$ is a CFA linearly compact $R$-module and $t$ is a non-negative integer such that H i I ( M ) is CFA for all $i < t$, then R / I ⊗ R H t I ( M ) is CFA. Hence, the set $mathrm{Coass}_R$ H t I ( M ) is finite.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44778026","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 multivariate generalized ridge (MGR) regression provides a shrinkage estimator of the multivariate linear regression by multiple ridge parameters. Since the ridge parameters which adjust the amount of shrinkage of the estimator are unknown, their optimization is an important task to obtain a better estimator. For the univariate case, a fast algorithm has been proposed for optimizing ridge parameters based on minimizing a model selection criterion (MSC) and the algorithm can be applied to various MSCs. In this paper, we extend this algorithm to MGR regression. We also describe the relationship between the MGR estimator which is not sparse and a multivariate adaptive group Lasso estimator which is sparse, under orthogonal explanatory variables.
{"title":"Ridge parameters optimization based on minimizing model selection criterion in multivariate generalized ridge regression","authors":"M. Ohishi","doi":"10.32917/H2020104","DOIUrl":"https://doi.org/10.32917/H2020104","url":null,"abstract":"A multivariate generalized ridge (MGR) regression provides a shrinkage estimator of the multivariate linear regression by multiple ridge parameters. Since the ridge parameters which adjust the amount of shrinkage of the estimator are unknown, their optimization is an important task to obtain a better estimator. For the univariate case, a fast algorithm has been proposed for optimizing ridge parameters based on minimizing a model selection criterion (MSC) and the algorithm can be applied to various MSCs. In this paper, we extend this algorithm to MGR regression. We also describe the relationship between the MGR estimator which is not sparse and a multivariate adaptive group Lasso estimator which is sparse, under orthogonal explanatory variables.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47814894","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 if three meromorphic functions share three two-point sets CM, then there exist two of the meromorphic functions such that one of them is a Mobius transform of the other.
{"title":"On meromorphic functions sharing three two-point sets CM","authors":"M. Shirosaki","doi":"10.32917/H2020058","DOIUrl":"https://doi.org/10.32917/H2020058","url":null,"abstract":"We show that if three meromorphic functions share three two-point sets CM, then there exist two of the meromorphic functions such that one of them is a Mobius transform of the other.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47510559","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, several necessary and su‰cient conditions are presented for a module M to be cohomologically complete intersection module with respect to I , i.e. H i I ðMÞ 1⁄4 0 for all i0 c 1⁄4 gradeðI ;MÞ. This notion is a generalization of cohomologically complete intersection ideals.
在本文中，给出了模M是关于I的上同调完全交模的几个必要和充分条件，即对于所有i0 c 1⁄4等级的H I I？MÞ1⁄40；MÞ。这个概念是上同调完全交集理想的推广。
{"title":"On cohomologically complete intersection\u0000 modules","authors":"Waqas Mahmood","doi":"10.32917/H2019073","DOIUrl":"https://doi.org/10.32917/H2019073","url":null,"abstract":"In this paper, several necessary and su‰cient conditions are presented for a module M to be cohomologically complete intersection module with respect to I , i.e. H i I ðMÞ 1⁄4 0 for all i0 c 1⁄4 gradeðI ;MÞ. This notion is a generalization of cohomologically complete intersection ideals.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"51 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41452989","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 a planar Jordan domain G with rectifiable boundary, it is well known that smooth functions on the closure of G do not always admit smooth extensions to C. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to f A CðG;CÞ. In this note we show that for Vitushkin sets K with K 1⁄4 K it is always possible to uniformly approximate on K the smooth function f A C ðK ;CÞ by smooth functions fn in C so that also qfn converges uniformly to qf on K. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in G merely admit continuous extensions to its boundary.
给定具有可直边界的平面Jordan域G，众所周知，G的闭包上的光滑函数并不总是允许C的光滑扩展。边界上的进一步条件是保证这种扩展所必需的。另一方面，Weierstrass的近似定理产生了一致收敛到fAC？G的多项式；CÞ。在这个注记中，我们证明了对于K为1⁄4K的Vitushkin集K，总是可以在K上一致逼近光滑函数f A CğK；通过C中的光滑函数fn，使得qfn也一致收敛于K上的qf。作为副产品，我们从其“邻域中的光滑版本”中推导出G中偏导数仅允许其边界连续扩展的函数的一般高斯积分定理。
{"title":"A note on simultaneous approximation on Vitushkin\u0000 sets","authors":"R. Mortini, R. Rupp","doi":"10.32917/H2020009","DOIUrl":"https://doi.org/10.32917/H2020009","url":null,"abstract":"Given a planar Jordan domain G with rectifiable boundary, it is well known that smooth functions on the closure of G do not always admit smooth extensions to C. Further conditions on the boundary are necessary to guarantee such extensions. On the other hand, Weierstrass’ approximation theorem yields polynomials converging uniformly to f A CðG;CÞ. In this note we show that for Vitushkin sets K with K 1⁄4 K it is always possible to uniformly approximate on K the smooth function f A C ðK ;CÞ by smooth functions fn in C so that also qfn converges uniformly to qf on K. As a byproduct we deduce from its ‘‘smooth in a neighborhood version’’ the general Gauss integral theorem for functions whose partial derivatives in G merely admit continuous extensions to its boundary.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45254757","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}
Our aim in this paper is to deal with the boundedness of the HardyLittlewood maximal operator and the Hardy operator on non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces and to establish a generalization of Sobolev’s inequalities for Riesz potentials of functions in such spaces.
{"title":"Boundedness of maximal operator, Hardy operator and\u0000 Sobolev’s inequalities on non-homogeneous central Herz-Morrey-Musielak-Orlicz\u0000 spaces","authors":"F. Maeda, Y. Mizuta, T. Ohno, T. Shimomura","doi":"10.32917/H2019141","DOIUrl":"https://doi.org/10.32917/H2019141","url":null,"abstract":"Our aim in this paper is to deal with the boundedness of the HardyLittlewood maximal operator and the Hardy operator on non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces and to establish a generalization of Sobolev’s inequalities for Riesz potentials of functions in such spaces.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45794570","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 the ring of modular forms with characters for the even unimodular lattice of signature (2,18) is obtained from the invariant ring of $mathrm{Sym}(mathrm{Sym}^8(V) oplus mathrm{Sym}^{12}(V))$ with respect to the action of $mathrm{SL}(V)$ by adding a Borcherds product of weight 132 with one relation of weight 264, where $V$ is a 2-dimensional $mathbb{C}$-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.
{"title":"The ring of modular forms for the even unimodular lattice of signature (2,18)","authors":"Atsuhira Nagano, K. Ueda","doi":"10.32917/h2021012","DOIUrl":"https://doi.org/10.32917/h2021012","url":null,"abstract":"We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,18) is obtained from the invariant ring of $mathrm{Sym}(mathrm{Sym}^8(V) oplus mathrm{Sym}^{12}(V))$ with respect to the action of $mathrm{SL}(V)$ by adding a Borcherds product of weight 132 with one relation of weight 264, where $V$ is a 2-dimensional $mathbb{C}$-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44370190","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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