It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.
{"title":"A classification of left-invariant Lorentzian metrics on some nilpotent Lie groups","authors":"Yuji Kondo, H. Tamaru","doi":"10.2748/tmj.20211122","DOIUrl":"https://doi.org/10.2748/tmj.20211122","url":null,"abstract":"It has been known that there exist exactly three left-invariant Lorentzian metrics up to scaling and automorphisms on the three dimensional Heisenberg group. In this paper, we classify left-invariant Lorentzian metrics on the direct product of three dimensional Heisenberg group and the Euclidean space of dimension $n-3$ with $n geq 4$, and prove that there exist exactly six such metrics on this Lie group up to scaling and automorphisms. Moreover we show that only one of them is flat, and the other five metrics are Ricci solitons but not Einstein. We also characterize this flat metric as the unique closed orbit, where the equivalence class of each left-invariant metric can be identified with an orbit of a certain group action on some symmetric space.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44354754","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 sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.
{"title":"Generating functions for sums of polynomial multiple zeta values","authors":"M. Hirose, H. Murahara, Shingo Saito","doi":"10.2748/tmj.20210409","DOIUrl":"https://doi.org/10.2748/tmj.20210409","url":null,"abstract":"The sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values bear a striking resemblance. We explain the resemblance in a rather straightforward manner using an identity that involves the Schur multiple zeta values. We also obtain the sum formula for polynomial multiple zeta(-star) values in terms of generating functions, simultaneously generalizing the sum formulas for multiple zeta(-star) values and symmetric multiple zeta(-star) values.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44102261","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 give a new proof of the fact that Milnor-Witt K-theory has geometric transfers. The proof yields to a simplification of Morel's conjecture about transfers on contracted homotopy sheaves.
给出了Milnor-Witt k -理论具有几何迁移的一个新的证明。该证明是对Morel关于收缩同伦轴上转移的猜想的简化。
{"title":"Transfers on Milnor-Witt K-theory","authors":"Niels Feld","doi":"10.2748/tmj.20211005","DOIUrl":"https://doi.org/10.2748/tmj.20211005","url":null,"abstract":"We give a new proof of the fact that Milnor-Witt K-theory has geometric transfers. The proof yields to a simplification of Morel's conjecture about transfers on contracted homotopy sheaves.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48256048","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":"Erratum: Stability of stationary solution for the Lugiato-Lefever equation","authors":"T. Miyaji, I. Ohnishi, Y. Tsutsumi","doi":"10.2748/tmj/1601085626","DOIUrl":"https://doi.org/10.2748/tmj/1601085626","url":null,"abstract":"","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":"72 1","pages":"487-492"},"PeriodicalIF":0.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41701080","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":"Boundedness of maximal operators on Herz spaces with radial variable exponent","authors":"Y. Mizuta, T. Shimomura","doi":"10.2748/tmj/1601085619","DOIUrl":"https://doi.org/10.2748/tmj/1601085619","url":null,"abstract":"","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":"72 1","pages":"335-348"},"PeriodicalIF":0.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43853402","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 consider the following nonlinear Schr"{o}dinger equation with a potential in $mathbb{R}^N$. We studied the existence of an initial value with critical mass for which the corresponding solution blows up. A previous study demonstrated the existence of an initial value for which the corresponding solution blows up when $N=1$ or $2$. In this work, without any restrictions on the number of dimensions $N$, we construct a critical-mass initial value for which the corresponding solution blows up in finite time and derive its blow-up rate.
{"title":"Minimal mass blow-up solutions for nonlinear Schrödinger equations with a potential","authors":"Naoki Matsui","doi":"10.2748/tmj.20211216","DOIUrl":"https://doi.org/10.2748/tmj.20211216","url":null,"abstract":"We consider the following nonlinear Schr\"{o}dinger equation with a potential in $mathbb{R}^N$. We studied the existence of an initial value with critical mass for which the corresponding solution blows up. A previous study demonstrated the existence of an initial value for which the corresponding solution blows up when $N=1$ or $2$. In this work, without any restrictions on the number of dimensions $N$, we construct a critical-mass initial value for which the corresponding solution blows up in finite time and derive its blow-up rate.","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46591976","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":"On a construction of harmonic function for recurrent relativistic $alpha$-stable processes","authors":"K. Tsuchida","doi":"10.2748/tmj/1593136823","DOIUrl":"https://doi.org/10.2748/tmj/1593136823","url":null,"abstract":"","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":"72 1","pages":"299-315"},"PeriodicalIF":0.5,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42992239","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":"Invariant Einstein metrics on $mathrm{SU}(n)$ and complex Stiefel manifolds","authors":"A. Arvanitoyeorgos, Y. Sakane, Marina Statha","doi":"10.2748/tmj/1593136818","DOIUrl":"https://doi.org/10.2748/tmj/1593136818","url":null,"abstract":"","PeriodicalId":54427,"journal":{"name":"Tohoku Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43730007","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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