Pub Date : 2009-06-10DOI: 10.1215/0023608X-2009-001
Masaki Izumi, R. Srinivasan
Toeplitz CAR flows are a class of E_0-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate E_0-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.
Toeplitz CAR流是一类e_0 -半群，包括R. T. Powers构造的第一个III型例子。我们证明了Toeplitz CAR流包含不可数的互非循环共轭e_0半群。我们还推广了power使用的Toeplitz CAR流的III型准则(后来由W. Arveson改进)，并表明Toeplitz CAR流总是I型或III型。
{"title":"Toeplitz CAR flows and type I factorizations","authors":"Masaki Izumi, R. Srinivasan","doi":"10.1215/0023608X-2009-001","DOIUrl":"https://doi.org/10.1215/0023608X-2009-001","url":null,"abstract":"Toeplitz CAR flows are a class of E_0-semigroups including the first type III example constructed by R. T. Powers. We show that the Toeplitz CAR flows contain uncountably many mutually non cocycle conjugate E_0-semigroups of type III. We also generalize the type III criterion for Toeplitz CAR flows employed by Powers (and later refined by W. Arveson), and show that Toeplitz CAR flows are always either of type I or type III.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"50 1","pages":"1-32"},"PeriodicalIF":0.0,"publicationDate":"2009-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/0023608X-2009-001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66040714","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2009-04-10DOI: 10.1215/21562261-1424875
B. Feigin, A. Tsymbaliuk
In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].
{"title":"Equivariant K-theory of Hilbert schemes via shuffle algebra","authors":"B. Feigin, A. Tsymbaliuk","doi":"10.1215/21562261-1424875","DOIUrl":"https://doi.org/10.1215/21562261-1424875","url":null,"abstract":"In this paper we construct the action of Ding-Iohara and shuffle algebras in the sum of localized equivariant K-groups of Hilbert schemes of points on C^2. We show that commutative elements K_i of shuffle algebra act through vertex operators over positive part {h_i}_{i>0} of the Heisenberg algebra in these K-groups. Hence we get the action of Heisenberg algebra itself. Finally, we normalize the basis of the structure sheaves of fixed points in such a way that it corresponds to the basis of Macdonald polynomials in the Fock space k[h_1,h_2,...].","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"51 1","pages":"831-854"},"PeriodicalIF":0.0,"publicationDate":"2009-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/21562261-1424875","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66024579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main purpose of this article is to establish the existence result concerning to the problem −Δu(x )+ c(x)u(x )= a(x)f (u(x)), x ∈ R N , N> 2, u(x) → 0a s|x |→∞ . Similary problems have been also studied. The proofs of the existence are based on the maximum principle and sub and super solutions method.
本文的主要目的是建立关于−Δu(x)+ c(x)u(x)= a(x)f (u(x))， x∈rn, N >2, u(x)→0a s|x |→∞问题的存在性结果。类似的问题也被研究过。存在性的证明是基于极大值原理和上、下解方法。
{"title":"A Lane-Emden-Fowler type problem with singular nonlinearity","authors":"D. Covei","doi":"10.1215/KJM/1256219159","DOIUrl":"https://doi.org/10.1215/KJM/1256219159","url":null,"abstract":"The main purpose of this article is to establish the existence result concerning to the problem −Δu(x )+ c(x)u(x )= a(x)f (u(x)), x ∈ R N , N> 2, u(x) → 0a s|x |→∞ . Similary problems have been also studied. The proofs of the existence are based on the maximum principle and sub and super solutions method.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"325-338"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66113650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain the tail estimation of the quadratic variation of a local martingale with no assumption with respect to positive jumps. Moreover, applying it, we also discuss a tail property of the first-passage times of stochastic integrals.
{"title":"The quadratic variations of local martingales and the first-passage times of stochastic integrals","authors":"S. Kaji","doi":"10.1215/KJM/1260975037","DOIUrl":"https://doi.org/10.1215/KJM/1260975037","url":null,"abstract":"We obtain the tail estimation of the quadratic variation of a local martingale with no assumption with respect to positive jumps. Moreover, applying it, we also discuss a tail property of the first-passage times of stochastic integrals.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"491-502"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1260975037","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66113952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An asymptotic estimate for the hitting time of a half-line by two-dimensional Brownian motion","authors":"Y. Isozaki","doi":"10.1215/KJM/1260975041","DOIUrl":"https://doi.org/10.1215/KJM/1260975041","url":null,"abstract":"","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"603-617"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1260975041","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66114075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Decaying solution of a Navier-Stokes flow without surface tension","authors":"Yasushi Hataya","doi":"10.1215/KJM/1265899478","DOIUrl":"https://doi.org/10.1215/KJM/1265899478","url":null,"abstract":"","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"691-717"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1265899478","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66114197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a class of linear higher order hyperbolic equations with a single degenerate point. We give sufficient conditions in order for the Cauchy problem to be well-posed in Gevrey classes and in the $C^infty$ class.
{"title":"Levi conditions to the Gevrey well-posedness for hyperbolic operators of higher order","authors":"H. Ishida","doi":"10.1215/KJM/1248983035","DOIUrl":"https://doi.org/10.1215/KJM/1248983035","url":null,"abstract":"We consider a class of linear higher order hyperbolic equations with a single degenerate point. We give sufficient conditions in order for the Cauchy problem to be well-posed in Gevrey classes and in the $C^infty$ class.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"173-191"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66087650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dolgopyat [4] showed that a class of Axiom A flows has exponential decay of correlations for smooth observables, and Baladi-Vallée [2] gave a nice interpretation of it on suspension semiflows of one-dimensional expanding countable Markov maps. Avila-Gouëzel-Yoccoz [1] extends the result of Baladi-Vallée to higher dimensional systems. In this paper we show that a class of non-Markov semiflows also has exponential decay of correlations. We prove that such exponential decay can be shown on an open dense condition for the suspensions of piecewise expanding maps.
{"title":"Exponential decay of correlations for surface semiflows with an expanding direction","authors":"I. Obayashi","doi":"10.1215/KJM/1256219166","DOIUrl":"https://doi.org/10.1215/KJM/1256219166","url":null,"abstract":"Dolgopyat [4] showed that a class of Axiom A flows has exponential decay of correlations for smooth observables, and Baladi-Vallée [2] gave a nice interpretation of it on suspension semiflows of one-dimensional expanding countable Markov maps. Avila-Gouëzel-Yoccoz [1] extends the result of Baladi-Vallée to higher dimensional systems. In this paper we show that a class of non-Markov semiflows also has exponential decay of correlations. We prove that such exponential decay can be shown on an open dense condition for the suspensions of piecewise expanding maps.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"427-440"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1256219166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66113888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A variational representation for random functionals on abstract Wiener spaces","authors":"Xicheng Zhang","doi":"10.1215/KJM/1260975036","DOIUrl":"https://doi.org/10.1215/KJM/1260975036","url":null,"abstract":"","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"49 1","pages":"475-490"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1215/KJM/1260975036","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66113942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We determine the composition factors of the polynomial representation of DAHA, conjectured by M. Kasatani in [Kasa, Conjecture 6.4.]. He constructed an increasing sequence of subrepresentations in the polynomial representation of DAHA using the “multi-wheel condition”, and conjectured that it is a composition series. On the other hand, DAHA has two degenerate versions called the “degenerate DAHA” and the “rational DAHA”. The category O of modules over these three algebras and the category of modules over the v -Schur algebra are closely related. By using this relationship, we reduce the determination of composition factors of polynomial representations of DAHA to the determination of the composition factors of the Weyl module W ( n ) v for the v -Schur algebra. By using the LLT-Ariki type theorem of v -Schur algebra proved by Varagnolo-Vasserot, we determine the composition factors of W ( n ) v by calculating the upper global basis and crystal basis of Fock space of U q ( b sl (cid:2) ) when v is a primitive (cid:2) -th root of unity. This result gives a different way from the determination of decomposition number of W ( n ) v by H. Miyachi or B. Ackermann via the modular representation theory of the general linear groups.
{"title":"Composition factors of polynomial representation of DAHA and q-decomposition numbers","authors":"Naoya Enomoto","doi":"10.1215/KJM/1260975035","DOIUrl":"https://doi.org/10.1215/KJM/1260975035","url":null,"abstract":"We determine the composition factors of the polynomial representation of DAHA, conjectured by M. Kasatani in [Kasa, Conjecture 6.4.]. He constructed an increasing sequence of subrepresentations in the polynomial representation of DAHA using the “multi-wheel condition”, and conjectured that it is a composition series. On the other hand, DAHA has two degenerate versions called the “degenerate DAHA” and the “rational DAHA”. The category O of modules over these three algebras and the category of modules over the v -Schur algebra are closely related. By using this relationship, we reduce the determination of composition factors of polynomial representations of DAHA to the determination of the composition factors of the Weyl module W ( n ) v for the v -Schur algebra. By using the LLT-Ariki type theorem of v -Schur algebra proved by Varagnolo-Vasserot, we determine the composition factors of W ( n ) v by calculating the upper global basis and crystal basis of Fock space of U q ( b sl (cid:2) ) when v is a primitive (cid:2) -th root of unity. This result gives a different way from the determination of decomposition number of W ( n ) v by H. Miyachi or B. Ackermann via the modular representation theory of the general linear groups.","PeriodicalId":50142,"journal":{"name":"Journal of Mathematics of Kyoto University","volume":"41 1","pages":"441-473"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66113905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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