We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $ggeq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.
{"title":"Torsion generators of the twist subgroup","authors":"Tulin Altunoz, Mehmetcik Pamuk, O. Yildiz","doi":"10.2748/tmj.20210407","DOIUrl":"https://doi.org/10.2748/tmj.20210407","url":null,"abstract":"We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $ggeq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45998347","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}
Newton’s law of universal gravitation states that any particle in the universe attracts every other particle with a force (pointing along the line intersecting both particles) which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
{"title":"Newton’s Law of Universal Gravitation","authors":"A. E. Kossovsky","doi":"10.1090/clrm/050/16","DOIUrl":"https://doi.org/10.1090/clrm/050/16","url":null,"abstract":"Newton’s law of universal gravitation states that any particle in the universe attracts every other particle with a force (pointing along the line intersecting both particles) which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60550179","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 $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite total curvature where $n geq 2$. Note here that our radial curvatures can change signs wildly. We then show that $lim_{ttoinfty} mathrm{vol} B_t(p) / t^n$ exists where $mathrm{vol} B_t(p)$ denotes the volume of the open metric ball $B_t(p)$ with center $p$ and radius $t$. Moreover we show that in addition if the limit above is positive, then $M$ has finite topological type and there is therefore a finitely upper bound on the number of ends of $M$.
设$M$是一个连通的完全非紧$n$维黎曼流形，其基点$p在M$中，其在$p$处的径向截面曲率从下界于非紧旋转表面的曲率，该非紧旋转曲面允许有限的总曲率，其中$ngeq2$。请注意，我们的径向曲率可以剧烈地改变符号。然后我们证明了$lim_｛t to infty｝mathrm｛vol｝B_t（p）/t^n$存在，其中$mathrm{vol}B_t（p）$表示中心为$p$、半径为$t$的开度量球$B_t（p$的体积。此外，我们还证明了如果上面的极限是正的，那么$M$具有有限拓扑类型，因此在$M$的端数上存在有限上界。
{"title":"On sufficient conditions to extend Huber's finite connectivity theorem to higher dimensions","authors":"K. Kondo, Yusuke Shinoda","doi":"10.2748/tmj.20200701","DOIUrl":"https://doi.org/10.2748/tmj.20200701","url":null,"abstract":"Let $M$ be a connected complete noncompact $n$-dimensional Riemannian manifold with a base point $p in M$ whose radial sectional curvature at $p$ is bounded from below by that of a noncompact surface of revolution which admits a finite total curvature where $n geq 2$. Note here that our radial curvatures can change signs wildly. We then show that $lim_{ttoinfty} mathrm{vol} B_t(p) / t^n$ exists where $mathrm{vol} B_t(p)$ denotes the volume of the open metric ball $B_t(p)$ with center $p$ and radius $t$. Moreover we show that in addition if the limit above is positive, then $M$ has finite topological type and there is therefore a finitely upper bound on the number of ends of $M$.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43805526","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 Schwarz's map for Appell's second system $cF_2$ of hypergeometric differential equations in two variables with parameters $a=c_1=c_2=frac{1}{2}$, $b_1=b_2=frac{1}{4}$. By using theta functions with characteristics, we give a defining equation of an analytic set in $C^2times H$ of its image, and express its inverse.
{"title":"Schwarz's map for Appell's second hypergeometric system with quarter integer parameters","authors":"Keiji Matsumoto, Shohei Osafune, T. Terasoma","doi":"10.2748/tmj.20201207","DOIUrl":"https://doi.org/10.2748/tmj.20201207","url":null,"abstract":"We study Schwarz's map for Appell's second system $cF_2$ of hypergeometric differential equations in two variables with parameters $a=c_1=c_2=frac{1}{2}$, $b_1=b_2=frac{1}{4}$. By using theta functions with characteristics, we give a defining equation of an analytic set in $C^2times H$ of its image, and express its inverse.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49438663","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 Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.
{"title":"Ricci flat Calabi's metric is not projectively\u0000 induced","authors":"A. Loi, Michela Zedda, F. Zuddas","doi":"10.2748/TMJ.20191211","DOIUrl":"https://doi.org/10.2748/TMJ.20191211","url":null,"abstract":"We show that the Ricci flat Calabi's metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [arXiv:1705.03908v2 [math.DG]] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46089181","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 the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah. The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as $ttoinfty$, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.
{"title":"Global existence for a system of multiple-speed wave equations violating the null condition","authors":"K. Hidano, K. Yokoyama, Dongbing Zha","doi":"10.2748/tmj.20210826","DOIUrl":"https://doi.org/10.2748/tmj.20210826","url":null,"abstract":"We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah. \u0000The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as $ttoinfty$, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47063353","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}
Choe et. al. have recently characterized compact double differences formed by four composition operators acting on the standard weighted Bergman spaces over the disk of the complex plane. In this paper, we extend such a result to the ball setting. Our characterization is obtained under a suitable restriction on inducing maps, which is automatically satisfied in the case of the disk. We exhibit concrete examples, for the first time even for single composition operators, which shows that such a restriction is essential in the case of the ball.
{"title":"Compact double differences of composition operators on the Bergman spaces over the ball","authors":"B. Choe, H. Koo, Jongho Yang","doi":"10.2748/tmj/1576724796","DOIUrl":"https://doi.org/10.2748/tmj/1576724796","url":null,"abstract":"Choe et. al. have recently characterized compact double differences formed by four composition operators acting on the standard weighted Bergman spaces over the disk of the complex plane. In this paper, we extend such a result to the ball setting. Our characterization is obtained under a suitable restriction on inducing maps, which is automatically satisfied in the case of the disk. We exhibit concrete examples, for the first time even for single composition operators, which shows that such a restriction is essential in the case of the ball.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":"71 1","pages":"609-637"},"PeriodicalIF":0.5,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42954776","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}
Summary: We investigate the automorphism group of a plane curve, introducing the notion of a quasi-Galois point. We show that the automorphism group of several curves, for example, Klein quartic, Wiman sextic and Fermat curves, is generated by the groups associated with quasi-Galois points.
{"title":"Quasi-Galois points, I: Automorphism groups of plane curves","authors":"Satoru Fukasawa, Kei Miura, Takeshi Takahashi","doi":"10.2748/tmj/1576724789","DOIUrl":"https://doi.org/10.2748/tmj/1576724789","url":null,"abstract":"Summary: We investigate the automorphism group of a plane curve, introducing the notion of a quasi-Galois point. We show that the automorphism group of several curves, for example, Klein quartic, Wiman sextic and Fermat curves, is generated by the groups associated with quasi-Galois points.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43419837","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 singular corank 1 surfaces in $mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact properties of the surface with respect to the plane orthogonal to the axial vector and show how they are related to the axial curvature. Finally, for certain fold type singularities, we relate the axial curvature with the Gaussian curvature of an appropriate blow up.
{"title":"The axial curvature for corank 1 singular surfaces","authors":"R. O. Sinha, K. Saji","doi":"10.2748/tmj.20210322","DOIUrl":"https://doi.org/10.2748/tmj.20210322","url":null,"abstract":"For singular corank 1 surfaces in $mathbb R^3$ we introduce a distinguished normal vector called the axial vector. Using this vector and the curvature parabola we define a new type of curvature called the axial curvature, which generalizes the singular curvature for frontal type singularities. We then study contact properties of the surface with respect to the plane orthogonal to the axial vector and show how they are related to the axial curvature. Finally, for certain fold type singularities, we relate the axial curvature with the Gaussian curvature of an appropriate blow up.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42397115","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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