A bstract . Firstly, we obtain conditions for stable extendibility and extendibility of complex vector bundles over the ð 2 n þ 1 Þ -dimensional standard lens space L n ð p Þ mod p , where p is a prime. Secondly, we prove that the complexification c ð t n ð p ÞÞ of the tangent bundle t n ð p Þ ð¼ t ð L n ð p ÞÞÞ of L n ð p Þ is extendible to L 2 n þ 1 ð p Þ if p is a prime, and is not stably extendible to L 2 n þ 2 ð p Þ if p is an odd prime and n b 2 p (cid:1) 2. Thirdly, we show, for some odd prime p and positive integers n and m with m > n , that t ð L n ð p ÞÞ is stably extendible to L m ð p Þ but is not extendible to L m ð p Þ .
一个混蛋。Firstly,我们得到条件为马厩extendibility》和bundle情结向量extendibility完毕《ð2 nþ1Þ-dimensional标准版的太空L n pðÞmod p, p是a prime在哪里。Secondly,我们证明那个《complexification c p t nððÞÞ相切之穿t p nðÞð¼t p L nððÞÞÞn pðÞ是extendible到我的2个nþ1ðpÞ如果p是a prime, and is not stably extendible to L 2 nþðpÞ如果p是一个古怪的擎天柱和n b p (cid): 1) 2。Thirdly,我们的节目,因为一些奇怪的擎天柱积极integers n和m和p p > n,那t L nððÞÞ是stably extendible to L mðpÞ但是extendible to L mðpÞ音符。
{"title":"Stable extendibility and extendibility of vector bundles over lens spaces","authors":"M. Imaoka, Teiichi Kobayashi","doi":"10.32917/hmj/1520478023","DOIUrl":"https://doi.org/10.32917/hmj/1520478023","url":null,"abstract":"A bstract . Firstly, we obtain conditions for stable extendibility and extendibility of complex vector bundles over the ð 2 n þ 1 Þ -dimensional standard lens space L n ð p Þ mod p , where p is a prime. Secondly, we prove that the complexification c ð t n ð p ÞÞ of the tangent bundle t n ð p Þ ð¼ t ð L n ð p ÞÞÞ of L n ð p Þ is extendible to L 2 n þ 1 ð p Þ if p is a prime, and is not stably extendible to L 2 n þ 2 ð p Þ if p is an odd prime and n b 2 p (cid:1) 2. Thirdly, we show, for some odd prime p and positive integers n and m with m > n , that t ð L n ð p ÞÞ is stably extendible to L m ð p Þ but is not extendible to L m ð p Þ .","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"48 1","pages":"57-66"},"PeriodicalIF":0.2,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41977156","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 bstract . In this paper we study the LCM-stability property and other related concepts, and their universality in the case of polynomial and formal power series extensions.
{"title":"LCM-stability and formal power series","authors":"Walid Maaref, A. Benhissi","doi":"10.32917/HMJ/1520478022","DOIUrl":"https://doi.org/10.32917/HMJ/1520478022","url":null,"abstract":"A bstract . In this paper we study the LCM-stability property and other related concepts, and their universality in the case of polynomial and formal power series extensions.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"48 1","pages":"39-55"},"PeriodicalIF":0.2,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49272763","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 define a new class of Riemannian submanifolds which we call arid submanifolds. A Riemannian submanifold is called an arid submanifold if no nonzero normal vectors are invariant under the full slice representation. We see that arid submanifolds are a generalization of weakly reflective submanifolds, and arid submanifolds are minimal submanifolds. We also introduce an application of arid submanifolds to the study of left-invariant metrics on Lie groups. We give a sufficient condition for a left-invariant metric on an arbitrary Lie group to be a Ricci soliton.
{"title":"On a Riemannian submanifold whose slice representation has no nonzero fixed points","authors":"Y. Taketomi","doi":"10.32917/HMJ/1520478020","DOIUrl":"https://doi.org/10.32917/HMJ/1520478020","url":null,"abstract":"In this paper, we define a new class of Riemannian submanifolds which we call arid submanifolds. A Riemannian submanifold is called an arid submanifold if no nonzero normal vectors are invariant under the full slice representation. We see that arid submanifolds are a generalization of weakly reflective submanifolds, and arid submanifolds are minimal submanifolds. We also introduce an application of arid submanifolds to the study of left-invariant metrics on Lie groups. We give a sufficient condition for a left-invariant metric on an arbitrary Lie group to be a Ricci soliton.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"48 1","pages":"1-20"},"PeriodicalIF":0.2,"publicationDate":"2018-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44722868","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}
For a finite collection $mathbf A=(A_i)_{iin I}$ of locally closed sets in $mathbb R^n$, $ngeqslant3$, with the sign $pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $alpha$-Riesz kernel $|x-y|^{alpha-n}$, $alphain(0,2]$, over positive vector Radon measures $boldsymbolmu=(mu^i)_{iin I}$ such that each $mu^i$, $iin I$, is carried by $A_i$ and normalized by $mu^i(A_i)=a_iin(0,infty)$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $boldsymbollambda^{boldsymbolxi}_{mathbf A}=(lambda^i_{mathbf A})_{iin I}$ (also in the presence of an external field) if we restrict ourselves to $boldsymbolmu$ with $mu^ileqslantxi^i$, $iin I$, where the constraint $boldsymbolxi=(xi^i)_{iin I}$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector $alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $lambda^i_{mathbf A}$, $iin I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $alpha$-Riesz energy on a set of vector measures associated with $mathbf A$, as well as on the establishment of an intimate relationship between the constrained minimum $alpha$-Riesz energy problem and a constrained minimum $alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.
对于$mathbb R^n$，$ngeqslant3$中局部闭集的有限集合$mathbf a=（a_i）_｛iin i｝$，符号为$pm1$，使得带相反电荷的板相互不相交，我们考虑相对于$alpha$-Riesz核$|x-y|^｛alpha-n｝$的最小能量问题，$alphain（0,2]$，在正向量Radon上测量$boldsymbol mu=（mu^i）_｛iin i｝$，使得每个$mu^i$，$iin i$由$A_i$携带，并由$mu^ i（A_i）=A_iin（0，infty）$归一化。我们证明，即使在一组非零容量中，带相反电荷的板的闭包也可能相互交叉，但如果我们将自己限制为$boldsymbolmu$与$mu^ileqslantnenenebb xi ^i$，$iin i$，其中约束$boldsymbolneneneba xi=（nenenebb xi ^i）_｛iin i｝$被正确选择。我们建立了由此获得的可解性的充分条件的尖锐性，给出了解的加权向量$alpha$-Riesz势的描述，指出了它们的特征性质，并分析了i$中$lambda^i_｛mathbf A｝$，$i的支持。我们的方法基于在与$mathbf a$相关的一组向量测度上同时使用模糊拓扑和根据$alpha$-Reesz能量定义的适当的半度量结构，以及在约束最小$alph$-Riesz能量问题和约束最小$alpha$-Green能量问题之间建立密切关系，适当配制。通过实例说明了结果。
{"title":"Constrained minimum Riesz and Green energy problems for vector measures associated with a generalized condenser","authors":"B. Fuglede, N. Zorii","doi":"10.32917/hmj/1573787036","DOIUrl":"https://doi.org/10.32917/hmj/1573787036","url":null,"abstract":"For a finite collection $mathbf A=(A_i)_{iin I}$ of locally closed sets in $mathbb R^n$, $ngeqslant3$, with the sign $pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $alpha$-Riesz kernel $|x-y|^{alpha-n}$, $alphain(0,2]$, over positive vector Radon measures $boldsymbolmu=(mu^i)_{iin I}$ such that each $mu^i$, $iin I$, is carried by $A_i$ and normalized by $mu^i(A_i)=a_iin(0,infty)$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this problem has a solution $boldsymbollambda^{boldsymbolxi}_{mathbf A}=(lambda^i_{mathbf A})_{iin I}$ (also in the presence of an external field) if we restrict ourselves to $boldsymbolmu$ with $mu^ileqslantxi^i$, $iin I$, where the constraint $boldsymbolxi=(xi^i)_{iin I}$ is properly chosen. We establish the sharpness of the sufficient conditions on the solvability thus obtained, provide descriptions of the weighted vector $alpha$-Riesz potentials of the solutions, single out their characteristic properties, and analyze the supports of the $lambda^i_{mathbf A}$, $iin I$. Our approach is based on the simultaneous use of the vague topology and an appropriate semimetric structure defined in terms of the $alpha$-Riesz energy on a set of vector measures associated with $mathbf A$, as well as on the establishment of an intimate relationship between the constrained minimum $alpha$-Riesz energy problem and a constrained minimum $alpha$-Green energy problem, suitably formulated. The results are illustrated by examples.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2018-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47614082","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 derive the asymptotic distributions of the characteristic roots in multivariate linear models when the dimension p and the sample size n are large. The results are given for the case that the population characteristic roots have multiplicities greater than unity, and their orders are O(np) or O(n). Next, similar results are given for the asymptotic distributions of the canonical correlations when one of the dimensions and the sample size are large, assuming that the order of the population canonical correlations is O( √ p) or O(1). AMS 2000 Subject Classification: primary 62H10; secondary 62E20
{"title":"High-dimensional asymptotic distributions of characteristic roots in multivariate linear models and canonical correlation analysis","authors":"Y. Fujikoshi","doi":"10.32917/HMJ/1509674447","DOIUrl":"https://doi.org/10.32917/HMJ/1509674447","url":null,"abstract":"In this paper, we derive the asymptotic distributions of the characteristic roots in multivariate linear models when the dimension p and the sample size n are large. The results are given for the case that the population characteristic roots have multiplicities greater than unity, and their orders are O(np) or O(n). Next, similar results are given for the asymptotic distributions of the canonical correlations when one of the dimensions and the sample size are large, assuming that the order of the population canonical correlations is O( √ p) or O(1). AMS 2000 Subject Classification: primary 62H10; secondary 62E20","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"47 1","pages":"249-271"},"PeriodicalIF":0.2,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48198832","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 skew growth functions for the monoid of type $mathrm{B_{ii}}$ and others","authors":"Tadashi Ishibe","doi":"10.32917/HMJ/1509674449","DOIUrl":"https://doi.org/10.32917/HMJ/1509674449","url":null,"abstract":"","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"47 1","pages":"289-317"},"PeriodicalIF":0.2,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48081244","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":"Asymptotic cut-off point in linear discriminant rule to adjust the misclassification probability for large dimensions","authors":"Takayuki Yamada, Tetsuto Himeno, Tetsuro Sakurai","doi":"10.32917/hmj/1509674450","DOIUrl":"https://doi.org/10.32917/hmj/1509674450","url":null,"abstract":"","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"47 1","pages":"319-348"},"PeriodicalIF":0.2,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69464716","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 two-sample test for high-dimension, low-sample-size data under the strongly spiked eigenvalue model","authors":"Aki Ishii","doi":"10.32917/HMJ/1509674448","DOIUrl":"https://doi.org/10.32917/HMJ/1509674448","url":null,"abstract":"","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"47 1","pages":"273-288"},"PeriodicalIF":0.2,"publicationDate":"2017-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44376852","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}
For a finite subgroup $G$ of $operatorname{GL}(2, mathbb C)$, we consider the moduli space ${mathcal M}_theta$ of $G$-constellations. It depends on the stability parameter $theta$ and if $theta$ is generic it is a resolution of singularities of $mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $mathbb C^2/G$ is isomorphic to ${mathcal M}_theta$ for some generic $theta$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.
{"title":"$G$-constellations and the maximal resolution of a quotient surface singularity","authors":"A. Ishii","doi":"10.32917/hmj/1607396494","DOIUrl":"https://doi.org/10.32917/hmj/1607396494","url":null,"abstract":"For a finite subgroup $G$ of $operatorname{GL}(2, mathbb C)$, we consider the moduli space ${mathcal M}_theta$ of $G$-constellations. It depends on the stability parameter $theta$ and if $theta$ is generic it is a resolution of singularities of $mathbb C^2/G$. In this paper, we show that a resolution $Y$ of $mathbb C^2/G$ is isomorphic to ${mathcal M}_theta$ for some generic $theta$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43235517","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}
Atsufumi Honda, K. Naokawa, M. Umehara, Kotaro Yamada
Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called "Kossowski metrics". A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points(the definition of "non-parabolic singular point" is stated in the introduction here) can be realized as wave front germs (Kossowski's realization theorem). On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of "coherent tangent bundle". Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle. In this paper, we shall explain Kossowski's realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.
{"title":"Isometric deformations of wave fronts at non-degenerate singular points","authors":"Atsufumi Honda, K. Naokawa, M. Umehara, Kotaro Yamada","doi":"10.32917/hmj/1607396490","DOIUrl":"https://doi.org/10.32917/hmj/1607396490","url":null,"abstract":"Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean $3$-space. Their first fundamental forms belong to a class of positive semi-definite metrics called \"Kossowski metrics\". A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points(the definition of \"non-parabolic singular point\" is stated in the introduction here) can be realized as wave front germs (Kossowski's realization theorem). \u0000On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of \"coherent tangent bundle\". Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle. \u0000In this paper, we shall explain Kossowski's realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2017-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43082895","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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