{"title":"Indices of Some Meromorphic Functions of Degree 3 on Tori","authors":"Sarenhu","doi":"10.3836/tjm/1502179375","DOIUrl":"https://doi.org/10.3836/tjm/1502179375","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47969579","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":"Curvature Invariants of Equivariant Isometric Minimal Immersions into Grassmannian Manifolds","authors":"Shota Morii","doi":"10.3836/tjm/1502179381","DOIUrl":"https://doi.org/10.3836/tjm/1502179381","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44356431","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":"Algebraic Independence of the Values of Power Series and Their Derivatives Generated by Linear Recurrence","authors":"Haruki Ide, Taka-aki Tanaka, Kento Toyama","doi":"10.3836/tjm/1502179362","DOIUrl":"https://doi.org/10.3836/tjm/1502179362","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43818745","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 an elliptic curve E over Q , putting K = Q ( E [ p ]) which is the p -th division field of E for an odd prime p , we study the ideal class group Cl K of K as a Gal( K/ Q ) -module. More precisely, for any j with 1 6 j 6 p − 2 , we give a condition that Cl K ⊗ F p has the symmetric power Sym j E [ p ] of E [ p ] as its quotient Gal( K/ Q ) -module, in terms of Bloch-Kato’s Tate-Shafarevich group of Sym j V p E . Here V p E denotes the rational p -adic Tate module of E . This is a partial generalization of a result of Prasad and Shekhar for the case j = 1 .
． 对于椭圆曲线E / Q，设K = Q (E [p])为E对奇素数p的p -分域，研究了K的理想类群Cl K作为Gal(K/ Q) -模。更确切地说，对于任意具有1 6 j 6 p−2的j，我们给出了一个条件，即Cl K⊗F p具有E [p]的对称幂Sym j E [p]作为它的商Gal(K/ Q)模，即Sym j V p E的Bloch-Kato的ate- shafarevich群。这里vpe表示E的有理p进的Tate模。这是Prasad和Shekhar对j = 1的结果的部分推广。
{"title":"Ideal Class Groups of Number Fields and Bloch-Kato's Tate-Shafarevich Groups for Symmetric Powers of Elliptic Curves","authors":"Naoto Dainobu","doi":"10.3836/tjm/1502179361","DOIUrl":"https://doi.org/10.3836/tjm/1502179361","url":null,"abstract":". For an elliptic curve E over Q , putting K = Q ( E [ p ]) which is the p -th division field of E for an odd prime p , we study the ideal class group Cl K of K as a Gal( K/ Q ) -module. More precisely, for any j with 1 6 j 6 p − 2 , we give a condition that Cl K ⊗ F p has the symmetric power Sym j E [ p ] of E [ p ] as its quotient Gal( K/ Q ) -module, in terms of Bloch-Kato’s Tate-Shafarevich group of Sym j V p E . Here V p E denotes the rational p -adic Tate module of E . This is a partial generalization of a result of Prasad and Shekhar for the case j = 1 .","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47349621","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":"Moduli of Stable Sheaves on a K3 Surface of Picard Number 1","authors":"Akira Mori, K. Yoshioka","doi":"10.3836/tjm/1502179369","DOIUrl":"https://doi.org/10.3836/tjm/1502179369","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43226416","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":"Classifications of Prime Ideals and Simple Modules of the Weyl Algebra $A_1$ in Prime Characteristic","authors":"V. Bavula","doi":"10.3836/tjm/1502179377","DOIUrl":"https://doi.org/10.3836/tjm/1502179377","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42799759","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}
Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in 4-space. A finite type invariant theory for ribbon knotted surfaces was developped by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to wk-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to wk-equivalence. All results have direct corollaries for ribbon knotted surfaces.
{"title":"On Finite Type Invariants of Welded String Links and Ribbon Tubes","authors":"Adrien Casejuane, Jean-Baptiste Meilhan","doi":"10.3836/tjm/1502179380","DOIUrl":"https://doi.org/10.3836/tjm/1502179380","url":null,"abstract":"Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in 4-space. A finite type invariant theory for ribbon knotted surfaces was developped by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to wk-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to wk-equivalence. All results have direct corollaries for ribbon knotted surfaces.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45537971","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 study is based on the paper [1]. We give the formula to extract the position and the shape of the defect D generated in the object (conductor) Ω from the observation data on the boundary ∂Ω for the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [2]. We show a reconstruction formula of the convex hull of the defect D from the observed data, assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary ∂D in the two and three dimensional case. Let Ω ⊂ R(n = 2, 3) be a bounded domain where the boundary ∂Ω is C and let D be an open set satisfying D ⊂ Ω and Ω D is connected. The defect D consists of the union of disjoint bounded domains {Dj}j=1, where the boundary of D is Lipschitz continuous. First, we define the DN map for the magnetic Schrödinger equation with no defect D in Ω. Here, let D Au := ∑n j=1 DA,j(DA,ju), where DA,j := 1 i ∂j +Aj and A = (A1, A2, · · · , An). Definition 1. Suppose q ∈ L∞(Ω), q ≥ 0, A ∈ C(Ω, R). For a given f ∈ H(∂Ω), we say u ∈ H(Ω) is a weak solution to the following boundary value problem for the magnetic Schrödinger equation { D Au+ qu = 0 in Ω, u = f on ∂Ω, (1.1)
{"title":"Reconstruction of the Defect by the Enclosure Method for Inverse Problems of the Magnetic Schrödinger Operator","authors":"K. Kurata, Ryusei Yamashita","doi":"10.3836/tjm/1502179363","DOIUrl":"https://doi.org/10.3836/tjm/1502179363","url":null,"abstract":"This study is based on the paper [1]. We give the formula to extract the position and the shape of the defect D generated in the object (conductor) Ω from the observation data on the boundary ∂Ω for the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [2]. We show a reconstruction formula of the convex hull of the defect D from the observed data, assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary ∂D in the two and three dimensional case. Let Ω ⊂ R(n = 2, 3) be a bounded domain where the boundary ∂Ω is C and let D be an open set satisfying D ⊂ Ω and Ω D is connected. The defect D consists of the union of disjoint bounded domains {Dj}j=1, where the boundary of D is Lipschitz continuous. First, we define the DN map for the magnetic Schrödinger equation with no defect D in Ω. Here, let D Au := ∑n j=1 DA,j(DA,ju), where DA,j := 1 i ∂j +Aj and A = (A1, A2, · · · , An). Definition 1. Suppose q ∈ L∞(Ω), q ≥ 0, A ∈ C(Ω, R). For a given f ∈ H(∂Ω), we say u ∈ H(Ω) is a weak solution to the following boundary value problem for the magnetic Schrödinger equation { D Au+ qu = 0 in Ω, u = f on ∂Ω, (1.1)","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46564749","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 discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+ 1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equation. 2010 Mathematics Subject Classification. Primary 58A15; Secondary 58A17.
{"title":"On a Generalization of Monge–Ampère Equations and Monge–Ampère Systems","authors":"M. Kawamata, K. Shibuya","doi":"10.3836/tjm/1502179374","DOIUrl":"https://doi.org/10.3836/tjm/1502179374","url":null,"abstract":"We discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+ 1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equation. 2010 Mathematics Subject Classification. Primary 58A15; Secondary 58A17.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47527732","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":"Virtual Knots with Properties of Kishino's Knot","authors":"Y. Ohyama, Migiwa Sakurai","doi":"10.3836/tjm/1502179365","DOIUrl":"https://doi.org/10.3836/tjm/1502179365","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43419315","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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