We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.
我们用Kashiwara晶体给出了第二个Rogers-Ramanujan恒等式的另一个证明。
{"title":"A proof of the second Rogers-Ramanujan identity via Kleshchev multipartitions","authors":"Shunsuke Tsuchioka","doi":"10.3792/pjaa.99.005","DOIUrl":"https://doi.org/10.3792/pjaa.99.005","url":null,"abstract":"We give another proof of the second Rogers-Ramanujan identity by Kashiwara crystals.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"36 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90457222","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":"Proving dualities for $q$MZVs with connected sums","authors":"Benjamin Brindle","doi":"10.3792/pjaa.98.006","DOIUrl":"https://doi.org/10.3792/pjaa.98.006","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"15 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90383690","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":"Concordant pairs in ratios with rank at least two and the distribution of $theta$-congruent numbers","authors":"J. T. Dimabayao","doi":"10.3792/pjaa.98.005","DOIUrl":"https://doi.org/10.3792/pjaa.98.005","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"168 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76188736","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 authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan’s τ -function has a bias to being positive. This phenomenon is an analogue of Chebyshev’s bias.
{"title":"Chebyshev’s bias for Ramanujan’s $tau$-function via the Deep Riemann Hypothesis","authors":"S. Koyama, N. Kurokawa","doi":"10.3792/pjaa.98.007","DOIUrl":"https://doi.org/10.3792/pjaa.98.007","url":null,"abstract":"The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan’s τ -function has a bias to being positive. This phenomenon is an analogue of Chebyshev’s bias.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82218969","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 geometric criterion for the bounded multiplicity property of ‘‘small’’ infinite-dimensional representations of real reductive Lie groups in both induction and restrictions. In particular, for a reductive symmetric pair ð G; H Þ , we determine the reductive subgroups G 0 having the property that any irreducible H -distinguished admissible representations of G are of bounded multiplicity when restricted to G 0 .
{"title":"Multiplicity in restricting small representations","authors":"Toshiyuki Kobayashi","doi":"10.3792/pjaa.98.004","DOIUrl":"https://doi.org/10.3792/pjaa.98.004","url":null,"abstract":": We give a geometric criterion for the bounded multiplicity property of ‘‘small’’ infinite-dimensional representations of real reductive Lie groups in both induction and restrictions. In particular, for a reductive symmetric pair ð G; H Þ , we determine the reductive subgroups G 0 having the property that any irreducible H -distinguished admissible representations of G are of bounded multiplicity when restricted to G 0 .","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84064799","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":"Some remarks on finiteness of extremal rays of divisorial type","authors":"Yoshio Fujimoto","doi":"10.3792/pjaa.98.002","DOIUrl":"https://doi.org/10.3792/pjaa.98.002","url":null,"abstract":"","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"22 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74919483","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 [Cho09], Choi studied congruences of coefficients (modulo T q − T ) for Drinfeld modular forms of level Γ 0 ( T ), trivial type and the linear relations between the initial coefficients of those. In this article, we generalize these results for level Γ 0 ( T ), arbitrary type.
{"title":"On congruences and linear relations for Drinfeld modular forms of level $Gamma_{0}(T)$, arbitrary type","authors":"Tarun Dalal, Narasimha Kumar","doi":"10.3792/pjaa.99.003","DOIUrl":"https://doi.org/10.3792/pjaa.99.003","url":null,"abstract":". In [Cho09], Choi studied congruences of coefficients (modulo T q − T ) for Drinfeld modular forms of level Γ 0 ( T ), trivial type and the linear relations between the initial coefficients of those. In this article, we generalize these results for level Γ 0 ( T ), arbitrary type.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"16 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75035386","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 give a necessary and sufficient condition under which a woven frame can be preserved when applying an operator to it, which provides a correction to one existing corresponding result due to Rahimi et al. We also present some new results on the stability of woven frames under perturbation of two different operators.
{"title":"A note on operator perturbation of woven frames","authors":"Zhong-Qi Xiang, Xiang-Chun Xiao","doi":"10.3792/pjaa.97.017","DOIUrl":"https://doi.org/10.3792/pjaa.97.017","url":null,"abstract":": In this paper, we give a necessary and sufficient condition under which a woven frame can be preserved when applying an operator to it, which provides a correction to one existing corresponding result due to Rahimi et al. We also present some new results on the stability of woven frames under perturbation of two different operators.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"137 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80520257","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 (cid:2) be a discontinuous group for the 3-dimensional anti-de Sitter space AdS 3 : ¼ SO 0 ð 2 ; 2 Þ = SO 0 ð 2 ; 1 Þ . In this article, we discuss a growth rate of the counting of (cid:2) -orbits at infinity and the discrete spectrum of the hyperbolic Laplacian of the complete anti-de Sitter manifold (cid:2) n AdS 3 .
{"title":"A quantitative study of orbit counting and discrete spectrum for anti-de Sitter 3-manifolds","authors":"K. Kannaka","doi":"10.3792/pjaa.97.018","DOIUrl":"https://doi.org/10.3792/pjaa.97.018","url":null,"abstract":": Let (cid:2) be a discontinuous group for the 3-dimensional anti-de Sitter space AdS 3 : ¼ SO 0 ð 2 ; 2 Þ = SO 0 ð 2 ; 1 Þ . In this article, we discuss a growth rate of the counting of (cid:2) -orbits at infinity and the discrete spectrum of the hyperbolic Laplacian of the complete anti-de Sitter manifold (cid:2) n AdS 3 .","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"46 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76797263","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 every pair of an analytic family $f=f_{t}$ of endomorphisms of degree $>1$ of the Berkovich projective line $mathbb{P}^{1,mathrm{an}}$ over an algebraically closed and complete non-trivially valued field $K$ and an analytically marked point $a=a(t)$ in $mathbb{P}^{1,mathrm{an}}$ both parametrized by a domain $V$ in the Berkovich analytification of a smooth projective algebraic curve $C/K$, we establish the equidistribution of the averaged pullbacks of any value in $mathbb{P}^{1,mathrm{an}}$ but a subset of logarithmic capacity 0 under the sequence of the morphisms $a_{n}=a_{n}(t)=f_{t}^{n}(a(t)):Vtomathbb{P}^{1,mathrm{an}}$, towards the activity measure $mu_{(f,a)}$ on $V$ associated with $f$ and $a$.
{"title":"Equidistribution in non-archimedean parameter curves towards the activity measures","authors":"Reimi Irokawa, Y. Okuyama","doi":"10.3792/pjaa.97.011","DOIUrl":"https://doi.org/10.3792/pjaa.97.011","url":null,"abstract":"For every pair of an analytic family $f=f_{t}$ of endomorphisms of degree $>1$ of the Berkovich projective line $mathbb{P}^{1,mathrm{an}}$ over an algebraically closed and complete non-trivially valued field $K$ and an analytically marked point $a=a(t)$ in $mathbb{P}^{1,mathrm{an}}$ both parametrized by a domain $V$ in the Berkovich analytification of a smooth projective algebraic curve $C/K$, we establish the equidistribution of the averaged pullbacks of any value in $mathbb{P}^{1,mathrm{an}}$ but a subset of logarithmic capacity 0 under the sequence of the morphisms $a_{n}=a_{n}(t)=f_{t}^{n}(a(t)):Vtomathbb{P}^{1,mathrm{an}}$, towards the activity measure $mu_{(f,a)}$ on $V$ associated with $f$ and $a$.","PeriodicalId":49668,"journal":{"name":"Proceedings of the Japan Academy Series A-Mathematical Sciences","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72887307","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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