The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of connected sums, which was recently invented by Seki and the author. Interestingly, the connected sum involves the hypergeometric function in its connector. Moreover, we introduce two kinds of $q$-analogues of the one-variable multiple poylogarithms and generalize the duality to them.
{"title":"Duality of One-variable Multiple Polylogarithms and Their $q$-analogues","authors":"Shuji Yamamoto","doi":"10.3836/tjm/1502179378","DOIUrl":"https://doi.org/10.3836/tjm/1502179378","url":null,"abstract":"The duality relation of one-variable multiple polylogarithms was proved by Hirose, Iwaki, Sato and Tasaka by means of iterated integrals. In this paper, we give a new proof using the method of connected sums, which was recently invented by Seki and the author. Interestingly, the connected sum involves the hypergeometric function in its connector. Moreover, we introduce two kinds of $q$-analogues of the one-variable multiple poylogarithms and generalize the duality to them.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41767843","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 consider a class of singular nonlinear first order partial differential equations $t(partial u/partial t)=F(t,x,u, partial u/partial x)$ with $(t,x) in mathbb{R} times mathbb{C}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.
{"title":"On a Class of Singular Nonlinear First Order Partial Differential Equations","authors":"H. Tahara","doi":"10.3836/tjm/1502179352","DOIUrl":"https://doi.org/10.3836/tjm/1502179352","url":null,"abstract":"In this paper, we consider a class of singular nonlinear first order partial differential equations $t(partial u/partial t)=F(t,x,u, partial u/partial x)$ with $(t,x) in mathbb{R} times mathbb{C}$ under the assumption that $F(t,x,z_1,z_2)$ is a function which is continuous in $t$ and holomorphic in the other variables. Under suitable conditions, we determine all the solutions of this equation in a neighborhood of the origin.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47090105","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 2-adic Lie Iterated Extensions of Number Fields Arising from a Joukowski Map","authors":"Yasushi Mizusawa, Kota Yamamoto","doi":"10.3836/tjm/1502179321","DOIUrl":"https://doi.org/10.3836/tjm/1502179321","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44719688","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":"Mean Square of Double Zeta-function","authors":"D. Banerjee, T. Minamide, Y. Tanigawa","doi":"10.3836/tjm/1502179322","DOIUrl":"https://doi.org/10.3836/tjm/1502179322","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45898474","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 equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.
{"title":"Fitting Ideals in Two-variable Equivariant Iwasawa Theory and an Application to CM Elliptic Curves","authors":"T. Kataoka","doi":"10.3836/tjm/1502179347","DOIUrl":"https://doi.org/10.3836/tjm/1502179347","url":null,"abstract":"We study equivariant Iwasawa theory for two-variable abelian extensions of an imaginary quadratic field. One of the main goals of this paper is to describe the Fitting ideals of Iwasawa modules using $p$-adic $L$-functions. We also provide an application to Selmer groups of elliptic curves with complex multiplication.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41480160","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 consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $varepsilon$ in a space of analytic functions which have a holomorphic extension in a strip of size $sigma$, then the solution exists up to a time of size $C/varepsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $sigma - C'varepsilon t$.
{"title":"Cauchy Theory for the Water Waves System in an Analytic Framework","authors":"T. Alazard, N. Burq, C. Zuily","doi":"10.3836/tjm/1502179355","DOIUrl":"https://doi.org/10.3836/tjm/1502179355","url":null,"abstract":"In this paper we consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $varepsilon$ in a space of analytic functions which have a holomorphic extension in a strip of size $sigma$, then the solution exists up to a time of size $C/varepsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $sigma - C'varepsilon t$.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48845686","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":"Constructions of Smooth $SU(p,q)$-actions on the Projective Space ${mathbf P}^{p+q-1}_{mathbb C}$ with $m$ Closed and $m+1$ Open Orbits","authors":"Kazuo Mukōyama, T. Ono, K. Takei","doi":"10.3836/TJM/1502179299","DOIUrl":"https://doi.org/10.3836/TJM/1502179299","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48142226","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 Some Families of Certain Divisible Polynomials and Their Zeta Functions","authors":"K. Chinen","doi":"10.3836/tjm/1502179317","DOIUrl":"https://doi.org/10.3836/tjm/1502179317","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46504741","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":"An Application of the Arithmetic of Elliptic Curves to the Class Number Problem for Quadratic Fields","authors":"Yoshichika Iizuka, Yutaka Konomi, S. Nakano","doi":"10.3836/tjm/1502179314","DOIUrl":"https://doi.org/10.3836/tjm/1502179314","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43826998","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 introduce a straightforward method to analyze the blow-up of solutions to systems of ordinary differential inequalities, and apply it to study the blow-up of solutions to a weakly coupled system of semilinear heat equations. In particular, we give upper and lower estimates of the lifespan of the solution in the subcritical case.
{"title":"Lifespan of Solutions for a Weakly Coupled System of Semilinear Heat Equations","authors":"K. Fujiwara, M. Ikeda, Yuta Wakasugi","doi":"10.3836/TJM/1502179300","DOIUrl":"https://doi.org/10.3836/TJM/1502179300","url":null,"abstract":"We introduce a straightforward method to analyze the blow-up of solutions to systems of ordinary differential inequalities, and apply it to study the blow-up of solutions to a weakly coupled system of semilinear heat equations. In particular, we give upper and lower estimates of the lifespan of the solution in the subcritical case.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41545986","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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