The non-existence of CR submanifolds of maximal CR dimension with umbilical shape operator in holomorphic statistical manifolds is proven. Our results are a generalization of the known results in the theory of CR submanifolds in complex space forms. Statistical manifolds in this paper are considered as manifolds consisting of certain probability density functions. In this setting we have two shape operators in the distinguished normal vector field direction with respect to the affine connection of the ambient space, and the one with respect to the dual connection. After obtaining the fundamental equations for CR submanifolds in holomorphic statistical manifolds, we examine umbilical dual shape operators.
{"title":"CR STATISTICAL SUBMANIFOLDS","authors":"M. Milijević","doi":"10.2206/kyushujm.73.89","DOIUrl":"https://doi.org/10.2206/kyushujm.73.89","url":null,"abstract":"The non-existence of CR submanifolds of maximal CR dimension with umbilical shape operator in holomorphic statistical manifolds is proven. Our results are a generalization of the known results in the theory of CR submanifolds in complex space forms. Statistical manifolds in this paper are considered as manifolds consisting of certain probability density functions. In this setting we have two shape operators in the distinguished normal vector field direction with respect to the affine connection of the ambient space, and the one with respect to the dual connection. After obtaining the fundamental equations for CR submanifolds in holomorphic statistical manifolds, we examine umbilical dual shape operators.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556917","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 classical congruence of Eisenstein and Lerch for the Fermat quotient with base 2 is generalized by Skula and Dobson. We give an alternative proof for the general congruence using a ‘Fermat quotient’ associated to a unit of an abelian number field.
{"title":"NOTE ON A CONGRUENCE FOR THE FERMAT QUOTIENT WITH BASE 2","authors":"H. Ichimura","doi":"10.2206/kyushujm.73.115","DOIUrl":"https://doi.org/10.2206/kyushujm.73.115","url":null,"abstract":"A classical congruence of Eisenstein and Lerch for the Fermat quotient with base 2 is generalized by Skula and Dobson. We give an alternative proof for the general congruence using a ‘Fermat quotient’ associated to a unit of an abelian number field.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/kyushujm.73.115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555533","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 define two parabolic flows on compact almost Hermitian manifolds, which coincide with the pluriclosed flow and the Hermitian curvature flow, respectively, on compact Hermitian manifolds with pluriclosed metric. We study the relation between these two parabolic evolution equations.
{"title":"PARABOLIC FLOWS ON ALMOST HERMITIAN MANIFOLDS","authors":"Masaya Kawamura","doi":"10.2206/kyushujm.73.69","DOIUrl":"https://doi.org/10.2206/kyushujm.73.69","url":null,"abstract":"We define two parabolic flows on compact almost Hermitian manifolds, which coincide with the pluriclosed flow and the Hermitian curvature flow, respectively, on compact Hermitian manifolds with pluriclosed metric. We study the relation between these two parabolic evolution equations.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556800","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 the present paper, we discuss the hyperbolic ordinariness of hyperelliptic curves in characteristic three. In particular, we prove that every hyperelliptic projective hyperbolic curve of genus less than or equal to five in characteristic three is hyperbolically ordinary.
{"title":"HYPERBOLIC ORDINARINESS OF HYPERELLIPTIC CURVES OF LOWER GENUS IN CHARACTERISTIC THREE","authors":"Yuichiro Hoshi","doi":"10.2206/kyushujm.73.317","DOIUrl":"https://doi.org/10.2206/kyushujm.73.317","url":null,"abstract":"— In the present paper, we discuss the hyperbolic ordinariness of hyperelliptic curves in characteristic three. In particular, we prove that every hyperelliptic projective hyperbolic curve of genus less than or equal to five in characteristic three is hyperbolically ordinary.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556662","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 shall introduce the notion of a KSM-manifold, which has the structure of a fiber bundle over an Einstein–K ¨ ahler Fano manifold whose fiber is a toric Fano manifold, and prove that every KSM-manifold admits a K ¨ ahler–Ricci soliton.
{"title":"KÄHLER-RICCI SOLITONS ON CERTAIN TORIC BUNDLES","authors":"Y. Nakagawa","doi":"10.2206/kyushujm.73.379","DOIUrl":"https://doi.org/10.2206/kyushujm.73.379","url":null,"abstract":". In this paper, we shall introduce the notion of a KSM-manifold, which has the structure of a fiber bundle over an Einstein–K ¨ ahler Fano manifold whose fiber is a toric Fano manifold, and prove that every KSM-manifold admits a K ¨ ahler–Ricci soliton.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556765","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}
– Asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by Shirai and Takahashi in terms of the Fredholm determinant. In the subclass, the obtained leading orders are same with the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point field associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of the states are also investigated in the case where the single site potentials have compact supports. The same method also applies to another well studied example, the zeros of a Gaussian random analytic function.
{"title":"ASYMPTOTIC BEHAVIOR OF THE INTEGRATED DENSITY OF STATES FOR RANDOM POINT FIELDS ASSOCIATED WITH CERTAIN FREDHOLM DETERMINANTS","authors":"N. Ueki","doi":"10.2206/kyushujm.73.43","DOIUrl":"https://doi.org/10.2206/kyushujm.73.43","url":null,"abstract":"– Asymptotic behavior of the integrated density of states of a Schrödinger operator with positive potentials located around all sample points of some random point field at the infimum of the spectrum is investigated. The random point field is taken from a subclass of the class given by Shirai and Takahashi in terms of the Fredholm determinant. In the subclass, the obtained leading orders are same with the well known results for the Poisson point fields, and the character of the random field appears in the leading constants. The random point field associated with the sine kernel and the Ginibre random point field are well studied examples not included in the above subclass, though they are included in the class by Shirai and Takahashi. By applying the results on asymptotics of the hole probability for these random fields, the corresponding asymptotic behaviors of the densities of the states are also investigated in the case where the single site potentials have compact supports. The same method also applies to another well studied example, the zeros of a Gaussian random analytic function.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/kyushujm.73.43","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556722","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}
The loxodromic Eisenstein series is defined for a loxodromic element of cofinite Kleinian groups. It is the analogue of the ordinary Eisenstein series associated to cusps. We study the asymptotic behavior of the loxodromic Eisenstein series for degenerating sequences of three-dimensional hyperbolic manifolds of finite volume. In particular, we prove that if the loxodromic element corresponds to the degenerating geodesic, then the associated loxodromic Eisenstein series converges to the ordinary Eisenstein series associated to the newly developing cusp on the limit manifold.
{"title":"LOXODROMIC EISENSTEIN SERIES THROUGH DEGENERATION","authors":"Y. Irie","doi":"10.2206/kyushujm.73.357","DOIUrl":"https://doi.org/10.2206/kyushujm.73.357","url":null,"abstract":"The loxodromic Eisenstein series is defined for a loxodromic element of cofinite Kleinian groups. It is the analogue of the ordinary Eisenstein series associated to cusps. We study the asymptotic behavior of the loxodromic Eisenstein series for degenerating sequences of three-dimensional hyperbolic manifolds of finite volume. In particular, we prove that if the loxodromic element corresponds to the degenerating geodesic, then the associated loxodromic Eisenstein series converges to the ordinary Eisenstein series associated to the newly developing cusp on the limit manifold.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556693","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}
Fuminori Kawamoto, Y. Kishi, H. Suzuki, Koshi Tomita
For a non-square positive integer d with 4 d , put ω(d) := (1+ √ d)/2 if d is congruent to 1 modulo 4 and ω(d) := √ d otherwise. Let a1, a2, . . . , a`−1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a1, a2, . . . , a[`/2] is the primary symmetric part of the simple continued fraction expansion of ω(d). A notion of ‘ELE type’ for a finite sequence was introduced in Kawamoto et al (Comment. Math. Univ. St. Pauli 64(2) (2015), 131–155). The aims of this paper are to introduce a notion of ‘pre-ELE type’ for a finite sequence and to give a way of constructing primary symmetric parts of ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields with period ` of minimal type for each even `≥ 6.
对于具有4d的非平方正整数d，如果d等于1模4，则设ω(d):=(1+√d)/2，否则设ω(d):=√d。设a1 a2。， a '−1是ω(d)的简单连分式展开式的对称部分。我们说序列a1, a2，…， a[' /2]是ω(d)的简单连分式展开式的初等对称部分。Kawamoto等人(评论)引入了有限序列的“ELE型”概念。数学。圣保利大学学报，64(2)(2015)，131-155。本文的目的是为有限序列引入“前ELE型”的概念，并给出构造ELE型的初级对称部分的一种方法。作为副产物，我们证明了存在无穷多个周期为最小型的实二次域，对于每一个偶数≥6。
{"title":"REAL QUADRATIC FIELDS, CONTINUED FRACTIONS, AND A CONSTRUCTION OF PRIMARY SYMMETRIC PARTS OF ELE TYPE","authors":"Fuminori Kawamoto, Y. Kishi, H. Suzuki, Koshi Tomita","doi":"10.2206/kyushujm.73.165","DOIUrl":"https://doi.org/10.2206/kyushujm.73.165","url":null,"abstract":"For a non-square positive integer d with 4 d , put ω(d) := (1+ √ d)/2 if d is congruent to 1 modulo 4 and ω(d) := √ d otherwise. Let a1, a2, . . . , a`−1 be the symmetric part of the simple continued fraction expansion of ω(d). We say that the sequence a1, a2, . . . , a[`/2] is the primary symmetric part of the simple continued fraction expansion of ω(d). A notion of ‘ELE type’ for a finite sequence was introduced in Kawamoto et al (Comment. Math. Univ. St. Pauli 64(2) (2015), 131–155). The aims of this paper are to introduce a notion of ‘pre-ELE type’ for a finite sequence and to give a way of constructing primary symmetric parts of ELE type. As a byproduct, we show that there exist infinitely many real quadratic fields with period ` of minimal type for each even `≥ 6.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2206/kyushujm.73.165","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555956","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 prove that there are only finitely many perfect powers in any linear recurrence sequence of integers of order at least two and whose characteristic polynomial is irreducible and has a dominant root.
证明了在特征多项式不可约且有一个优势根的至少2阶整数线性递推序列中，只有有限多个完全幂。
{"title":"ON PERFECT POWERS IN LINEAR RECURRENCE SEQUENCES OF INTEGERS","authors":"Y. Bugeaud, H. Kaneko","doi":"10.2206/kyushujm.73.221","DOIUrl":"https://doi.org/10.2206/kyushujm.73.221","url":null,"abstract":"We prove that there are only finitely many perfect powers in any linear recurrence sequence of integers of order at least two and whose characteristic polynomial is irreducible and has a dominant root.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556049","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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