{"title":"Cardinal invariants for $kappa$-box products: weight, density character and Suslin number","authors":"W. Comfort, I. Gotchev","doi":"10.4064/DM748-2-2016","DOIUrl":"https://doi.org/10.4064/DM748-2-2016","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157871","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a Hidden Markov Model (HMM) such that both the state space and the observation space are complete, separable, metric spaces and for which both the transition probability function (tr.pr.f.) determining the hidden Markov chain of the HMM and the tr.pr.f. determining the observation sequence of the HMM have densities. Such HMMs are called fully dominated. In this paper we consider a subclass of fully dominated HMMs which we call regular. A fully dominated, regular HMM induces a tr.pr.f. on the set of probability density functions on the state space which we call the filter kernel induced by the HMM and which can be interpreted as the Markov kernel associated to the sequence of conditional state distributions. We show that if the underlying hidden Markov chain of the fully dominated, regular HMM is strongly ergodic and a certain coupling condition is fulfilled, then, in the limit, the distribution of the conditional distribution becomes independent of the initial distribution of the hidden Markov chain and, if also the hidden Markov chain is uniformly ergodic, then the distributions tend towards a limit distribution. In the last part of the paper, we present some more explicit conditions, implying that the coupling condition mentioned above is satisfied.
{"title":"On convergence in distribution of the Markov chain generated by the filter kernel induced by a fully dominated Hidden Markov Model","authors":"Thomas Kaijser","doi":"10.4064/dm739-9-2015","DOIUrl":"https://doi.org/10.4064/dm739-9-2015","url":null,"abstract":"Consider a Hidden Markov Model (HMM) such that both the state space and the observation space are complete, separable, metric spaces and for which both the transition probability function (tr.pr.f.) determining the hidden Markov chain of the HMM and the tr.pr.f. determining the observation sequence of the HMM have densities. Such HMMs are called fully dominated. In this paper we consider a subclass of fully dominated HMMs which we call regular. A fully dominated, regular HMM induces a tr.pr.f. on the set of probability density functions on the state space which we call the filter kernel induced by the HMM and which can be interpreted as the Markov kernel associated to the sequence of conditional state distributions. We show that if the underlying hidden Markov chain of the fully dominated, regular HMM is strongly ergodic and a certain coupling condition is fulfilled, then, in the limit, the distribution of the conditional distribution becomes independent of the initial distribution of the hidden Markov chain and, if also the hidden Markov chain is uniformly ergodic, then the distributions tend towards a limit distribution. In the last part of the paper, we present some more explicit conditions, implying that the coupling condition mentioned above is satisfied.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sur des inégalités intégrales et applications à la stabilité de quelques systèmes distribués non dissipatifs","authors":"A. Guesmia","doi":"10.4064/DM506-0-1","DOIUrl":"https://doi.org/10.4064/DM506-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157071","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local free boundary problem for incompressible magnetohydrodynamics","authors":"P. Kacprzyk","doi":"10.4064/DM509-0-1","DOIUrl":"https://doi.org/10.4064/DM509-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Breaking the curse of dimensionality","authors":"Markus Weimar","doi":"10.4064/DM505-0-1","DOIUrl":"https://doi.org/10.4064/DM505-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Boundary value problems and singular integral equations on Banach function spaces","authors":"E. Rojas","doi":"10.4064/DM742-12-2015","DOIUrl":"https://doi.org/10.4064/DM742-12-2015","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Free algebras and noncommutative power series in the analysis of nonlinear control systems: an application to approximation problems","authors":"G. Sklyar, S. Ignatovich","doi":"10.4064/DM504-0-1","DOIUrl":"https://doi.org/10.4064/DM504-0-1","url":null,"abstract":"","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70156991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
L. Grafakos, Liguang Liu, Diego Maldonado, Dachun Yang
The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel–Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel–Lizorkin spaces. Acknowledgements. Loukas Grafakos is supported by grant DMS 0900946 of the National Science Foundation of the USA. Liguang Liu is supported by the National Natural Science Foundation of China (grant No. 11101425). Diego Maldonado is supported by grant DMS 0901587 of the National Science Foundation of the USA. Dachun Yang (the corresponding author) is supported by the National Natural Science Foundation of China (grant nos. 11171027 & 11361020) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant No. 20120003110003). All authors would like to thank the copy editor, Jerzy Trzeciak, for his valuable remarks which made this article more readable. 2010 Mathematics Subject Classification: Primary 42B20, 42B25, 42B35; Secondary 35S50, 42C15, 47G30, 30L99.
{"title":"Multilinear analysis on metric spaces","authors":"L. Grafakos, Liguang Liu, Diego Maldonado, Dachun Yang","doi":"10.4064/DM497-0-1","DOIUrl":"https://doi.org/10.4064/DM497-0-1","url":null,"abstract":"The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel–Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel–Lizorkin spaces. Acknowledgements. Loukas Grafakos is supported by grant DMS 0900946 of the National Science Foundation of the USA. Liguang Liu is supported by the National Natural Science Foundation of China (grant No. 11101425). Diego Maldonado is supported by grant DMS 0901587 of the National Science Foundation of the USA. Dachun Yang (the corresponding author) is supported by the National Natural Science Foundation of China (grant nos. 11171027 & 11361020) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (grant No. 20120003110003). All authors would like to thank the copy editor, Jerzy Trzeciak, for his valuable remarks which made this article more readable. 2010 Mathematics Subject Classification: Primary 42B20, 42B25, 42B35; Secondary 35S50, 42C15, 47G30, 30L99.","PeriodicalId":51016,"journal":{"name":"Dissertationes Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4064/DM497-0-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70157222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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