The word type number of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a fixed genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number T of such isomorphism classes are called type number or G-type number , where G is the group of quaternion hermitian similitudes. We express T in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I) from the principal genus to general lattices. We also apply our result to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over F p such that the polarizations are in a ”fixed genus of lattices”. This is a generalization of [8] and has an application to the number of components in the supersingular locus which are defined over F p .
{"title":"Type numbers of quaternion hermitian forms and supersingular abelian varieties","authors":"T. Ibukiyama","doi":"10.18910/68357","DOIUrl":"https://doi.org/10.18910/68357","url":null,"abstract":"The word type number of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a fixed genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number T of such isomorphism classes are called type number or G-type number , where G is the group of quaternion hermitian similitudes. We express T in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I) from the principal genus to general lattices. We also apply our result to the number of isomorphism classes of any polarized superspecial abelian varieties which have a model over F p such that the polarizations are in a ”fixed genus of lattices”. This is a generalization of [8] and has an application to the number of components in the supersingular locus which are defined over F p .","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41705769","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}
Inspired by the work of Kuniba-Okado-Yamada, we study some tensor product representations of quantized coordinate algebras of symmetrizable Kac-Moody Lie algebras in terms of quantized enveloping algebras. As a consequence, we describe structures and properties of certain reducible representations of quantized coordinate algebras. This paper includes alternative proofs of Soibelman’s tensor product theorem and Kuniba-Okado-Yamada’s common structure theorem based on our direct calculation method using global bases.
{"title":"Representations of quantized coordinate algebras via PBW-type elements","authors":"Hironori Oya","doi":"10.18910/67750","DOIUrl":"https://doi.org/10.18910/67750","url":null,"abstract":"Inspired by the work of Kuniba-Okado-Yamada, we study some tensor product representations of quantized coordinate algebras of symmetrizable Kac-Moody Lie algebras in terms of quantized enveloping algebras. As a consequence, we describe structures and properties of certain reducible representations of quantized coordinate algebras. This paper includes alternative proofs of Soibelman’s tensor product theorem and Kuniba-Okado-Yamada’s common structure theorem based on our direct calculation method using global bases.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67899654","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 study the global existence of solutions to an $n$-dimensional parabolic-parabolic system for chemotaxis with logistic-type growth. We introduce superlinear production of a chemoattractant. We then show the global existence of solutions in $L_p$ space $( p > n )$ under certain relations between the degradation and production orders.
{"title":"GLOBAL EXISTENCE OF SOLUTIONS TO AN n-DIMENSIONAL PARABOLIC-PARABOLIC SYSTEM FOR CHEMOTAXIS WITH LOGISTIC-TYPE GROWTH AND SUPERLINEAR PRODUCTION","authors":"E. Nakaguchi, Koichi Osaki","doi":"10.18910/67749","DOIUrl":"https://doi.org/10.18910/67749","url":null,"abstract":"We study the global existence of solutions to an $n$-dimensional parabolic-parabolic system for chemotaxis with logistic-type growth. We introduce superlinear production of a chemoattractant. We then show the global existence of solutions in $L_p$ space $( p > n )$ under certain relations between the degradation and production orders.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67899604","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 behavior of the solutions for the Laplace equation with a large spectral parameter and the inhomogeneous Robin type conditions","authors":"Masaru Ikehata, M. Kawashita","doi":"10.18910/67751","DOIUrl":"https://doi.org/10.18910/67751","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67899718","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 study holomorphic maps induced from orthogonal direct sums of holomorphic line bundles over a compact simply connected homogeneous K¨ahler manifold into a complex Grassmannian. Then we show if such maps are equivariant, then they are unique up to complex isometry.
{"title":"A rigidity of equivariant holomorphic maps into a complex Grassmannian induced from orthogonal direct sums of holomorphic line bundles","authors":"Isami Koga","doi":"10.18910/67752","DOIUrl":"https://doi.org/10.18910/67752","url":null,"abstract":"In the present paper, we study holomorphic maps induced from orthogonal direct sums of holomorphic line bundles over a compact simply connected homogeneous K¨ahler manifold into a complex Grassmannian. Then we show if such maps are equivariant, then they are unique up to complex isometry.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67899886","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 extend some results about Föllmer’s pathwise Itô calculus that have only been derived for continuous paths to càdlàg paths with quadratic variation. We study some fundamental properties of pathwise Itô integrals with respect to càdlàg integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise Itô integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of Föllmer’s calculus.
{"title":"REMARKS ON FÖLLMER’S PATHWISE ITÔ CALCULUS","authors":"Y. Hirai","doi":"10.18910/73360","DOIUrl":"https://doi.org/10.18910/73360","url":null,"abstract":"We extend some results about Föllmer’s pathwise Itô calculus that have only been derived for continuous paths to càdlàg paths with quadratic variation. We study some fundamental properties of pathwise Itô integrals with respect to càdlàg integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise Itô integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of Föllmer’s calculus.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42430171","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}
Dedicated to Prof. Taizo Kanenobu, Makoto Sakuma, Yasutaka Nakanishi on their 60-th birthday
在金信泰三、佐久间诚、中西康孝教授60岁生日之际向他们致敬
{"title":"Commensurability of link complements","authors":"H. Yoshida","doi":"10.18910/67004","DOIUrl":"https://doi.org/10.18910/67004","url":null,"abstract":"Dedicated to Prof. Taizo Kanenobu, Makoto Sakuma, Yasutaka Nakanishi on their 60-th birthday","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46383988","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 p be an odd prime number, W an absolutely unramified p-adically complete discrete valuation ring with algebraically closed residue field, and X a curve of genus at least two over the field of fractions K of W. In the present paper, we study, under the assumption that X has stable reduction over W, torsion points on X, i.e., torsion points of the Jacobian variety J of X which lie on the image of the Albanese embedding X ↪→ J with respect to a K-rational point of X. A consequence of the main result of the present paper is that if, moreover, J has good reduction over W, then every torsion point on X is K-rational after multiplying p. This result is closely related to a conjecture of R. Coleman concerning the ramification of torsion points. For instance, this result leads us to a solution of the conjecture in the case where a given curve is hyperelliptic and of genus at least p.
{"title":"On ramified torsion points on a curve with stable reduction over an absolutely unramified base","authors":"Yuichiro Hoshi","doi":"10.18910/67013","DOIUrl":"https://doi.org/10.18910/67013","url":null,"abstract":"Let p be an odd prime number, W an absolutely unramified p-adically complete discrete valuation ring with algebraically closed residue field, and X a curve of genus at least two over the field of fractions K of W. In the present paper, we study, under the assumption that X has stable reduction over W, torsion points on X, i.e., torsion points of the Jacobian variety J of X which lie on the image of the Albanese embedding X ↪→ J with respect to a K-rational point of X. A consequence of the main result of the present paper is that if, moreover, J has good reduction over W, then every torsion point on X is K-rational after multiplying p. This result is closely related to a conjecture of R. Coleman concerning the ramification of torsion points. For instance, this result leads us to a solution of the conjecture in the case where a given curve is hyperelliptic and of genus at least p.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47448049","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 $mathbf{PU}(2,1)$ denote the holomorphic isometry group of the $2$-dimensional complex hyperbolic space $mathbf{H}_{mathbb{C}}^{2}$, and the group $mathbf{SU}(2,1)$ is a 3-fold covering of $mathbf{PU}(2,1)$: $mathbf{PU}(2,1)=mathbf{SU}(2,1)/{omega I:omega^{3}=1}$. We study how to decompose a given pair of isometries $(A,B)in mathbf{SU}(2,1)^{2}$ under the form $A=I_{1}I_{2}$ and $B=I_{3}I_{2},$ where the $I_{k}$'s are complex symmetries about complex lines. If $(A,B)$ can be written as above, we call it is $mathbb{C}$-decomposable. The main results are decomposability criteria, which improve and supplement the result of [17].
{"title":"Decomposition of complex hyperbolic isometries by two complex symmetries","authors":"Xue-Jing Ren, Baohua Xie, Yue-Ping Jiang","doi":"10.18910/67006","DOIUrl":"https://doi.org/10.18910/67006","url":null,"abstract":"Let $mathbf{PU}(2,1)$ denote the holomorphic isometry group of the $2$-dimensional complex hyperbolic space $mathbf{H}_{mathbb{C}}^{2}$, and the group $mathbf{SU}(2,1)$ is a 3-fold covering of $mathbf{PU}(2,1)$: $mathbf{PU}(2,1)=mathbf{SU}(2,1)/{omega I:omega^{3}=1}$. We study how to decompose a given pair of isometries $(A,B)in mathbf{SU}(2,1)^{2}$ under the form $A=I_{1}I_{2}$ and $B=I_{3}I_{2},$ where the $I_{k}$'s are complex symmetries about complex lines. If $(A,B)$ can be written as above, we call it is $mathbb{C}$-decomposable. The main results are decomposability criteria, which improve and supplement the result of [17].","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42894245","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":"Perturbation of irregular Weyl-Heisenberg wave packet frames in $L^2(mathbb{R})$","authors":"Raj Kumar, A. Sah","doi":"10.18910/67014","DOIUrl":"https://doi.org/10.18910/67014","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42254125","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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