Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S 1 -equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphisms of 4-manifolds. Our invariants and the Bauer-Furuta classes are directly comparable for 4-manifolds with b1 =0 ;they are equivalent when b1 =0 and b+ > 1, but are finer in the case b1 =0 ,b+ =1 (they detect the wall-crossing phenomena). We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants.
{"title":"Cohomotopy invariants and the universal cohomotopy invariant jump formula","authors":"C. Okonek, A. Teleman","doi":"10.5167/UZH-21451","DOIUrl":"https://doi.org/10.5167/UZH-21451","url":null,"abstract":"Starting from ideas of Furuta, we develop a general formalism for the construction of cohomotopy invariants associated with a certain class of S 1 -equivariant non-linear maps between Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of cohomotopy Seiberg-Witten invariants which have clear functorial properties with respect to diffeomorphisms of 4-manifolds. Our invariants and the Bauer-Furuta classes are directly comparable for 4-manifolds with b1 =0 ;they are equivalent when b1 =0 and b+ > 1, but are finer in the case b1 =0 ,b+ =1 (they detect the wall-crossing phenomena). We study fundamental properties of the new invariants in a very general framework. In particular we prove a universal cohomotopy invariant jump formula and a multiplicative property. The formalism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromov-Witten invariants.","PeriodicalId":50143,"journal":{"name":"Journal of Mathematical Sciences-The University of Tokyo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80169491","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 : 2005-06-01DOI: 10.14492/hokmj/1285766230
A. Shimomura
We study the scattering theory for the coupled Klein- Gordon-Schrodinger equation with the Yukawa type interaction in two space dimensions.The scattering problem for this equation belongs to the borderline between the short range case and the long range one. We show the existence of the wave operators to this equation without any size restriction on the Klein-Gordon component of the final state. 2 . (KGS) Here u and v are complex and real valued unknown functions of (t, x) ∈ R × R 2 , respectively.In the present paper, we prove the existence of the wave operators to the equation (KGS) without any size restriction on the Klein-Gordon component of the final state. A large amount of work has been devoted to the asymptotic behavior of solutions for the nonlinear Schrodinger equation and for the nonlinear Klein- Gordon equation.We consider the scattering theory for systems centering on the Schrodinger equation, in particular, the Klein-Gordon-Schrodinger, the Wave-Schrodinger and the Maxwell-Schrodinger equations.In the scat- tering theory for the linear Schrodinger equation, (ordinary) wave operators
{"title":"Scattering Theory for the Coupled Klein-Gordon-Schrödinger Equations in Two Space Dimensions","authors":"A. Shimomura","doi":"10.14492/hokmj/1285766230","DOIUrl":"https://doi.org/10.14492/hokmj/1285766230","url":null,"abstract":"We study the scattering theory for the coupled Klein- Gordon-Schrodinger equation with the Yukawa type interaction in two space dimensions.The scattering problem for this equation belongs to the borderline between the short range case and the long range one. We show the existence of the wave operators to this equation without any size restriction on the Klein-Gordon component of the final state. 2 . (KGS) Here u and v are complex and real valued unknown functions of (t, x) ∈ R × R 2 , respectively.In the present paper, we prove the existence of the wave operators to the equation (KGS) without any size restriction on the Klein-Gordon component of the final state. A large amount of work has been devoted to the asymptotic behavior of solutions for the nonlinear Schrodinger equation and for the nonlinear Klein- Gordon equation.We consider the scattering theory for systems centering on the Schrodinger equation, in particular, the Klein-Gordon-Schrodinger, the Wave-Schrodinger and the Maxwell-Schrodinger equations.In the scat- tering theory for the linear Schrodinger equation, (ordinary) wave operators","PeriodicalId":50143,"journal":{"name":"Journal of Mathematical Sciences-The University of Tokyo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2005-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91389088","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}
In this paper, we present a filtering model on a default risk related to mathematical finance. We regard as the time when a default occurs the first hitting time at zero of a one dimensional process which starts at some positive number and is not directly observed. We discuss the conditional law of the hitting time under imperfect information. We use the reference measure change technique and a new formula on a kind of conditional expectation to obtain a so-called hazard rate process. It is also discussed what the relation between the hazard rate process and the conditional law of the hitting time is like.
{"title":"A Filtering Model on Default Risk","authors":"H. Nakagawa","doi":"10.5687/sss.2001.231","DOIUrl":"https://doi.org/10.5687/sss.2001.231","url":null,"abstract":"In this paper, we present a filtering model on a default risk related to mathematical finance. We regard as the time when a default occurs the first hitting time at zero of a one dimensional process which starts at some positive number and is not directly observed. We discuss the conditional law of the hitting time under imperfect information. We use the reference measure change technique and a new formula on a kind of conditional expectation to obtain a so-called hazard rate process. It is also discussed what the relation between the hazard rate process and the conditional law of the hitting time is like.","PeriodicalId":50143,"journal":{"name":"Journal of Mathematical Sciences-The University of Tokyo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2001-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73051670","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 : 1997-01-01DOI: 10.1007/978-4-431-68413-8_6
Keisuke Uchikoshi
{"title":"Stokes Operators for Microhyperbolic Equations","authors":"Keisuke Uchikoshi","doi":"10.1007/978-4-431-68413-8_6","DOIUrl":"https://doi.org/10.1007/978-4-431-68413-8_6","url":null,"abstract":"","PeriodicalId":50143,"journal":{"name":"Journal of Mathematical Sciences-The University of Tokyo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1997-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80469526","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 : 1997-01-01DOI: 10.1515/9783112319185-027
Akira Kaneko
Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary Abstract. We give a sufficient condition for the removability of thin singularities of Gevrey class solutions of linear partial differential equations. In §1we give a sufficient condition for the removability in the case of equations with constant coefficients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients. In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear par- tial differential equations. Some of the results given here are easily derived from Grushin's pioneering works on continuation of C ∞ solutions and from the author's former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be ob- vious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-differentiable functions and ultradistributions. Here is a brief plan of the present article. The first two sections treat equations with constant coefficients. In §1we give a sufficient condition for
{"title":"On Continuation of Gevrey Class Solutions of Linear Partial Differential Equations","authors":"Akira Kaneko","doi":"10.1515/9783112319185-027","DOIUrl":"https://doi.org/10.1515/9783112319185-027","url":null,"abstract":"Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary Abstract. We give a sufficient condition for the removability of thin singularities of Gevrey class solutions of linear partial differential equations. In §1we give a sufficient condition for the removability in the case of equations with constant coefficients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients. In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear par- tial differential equations. Some of the results given here are easily derived from Grushin's pioneering works on continuation of C ∞ solutions and from the author's former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be ob- vious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-differentiable functions and ultradistributions. Here is a brief plan of the present article. The first two sections treat equations with constant coefficients. In §1we give a sufficient condition for","PeriodicalId":50143,"journal":{"name":"Journal of Mathematical Sciences-The University of Tokyo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1997-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90149164","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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