In this paper,we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore,hence all focal submanifolds of isoparametric hypersurfaces in th...In this paper,we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore,hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.展开更多
In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probabil...In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.展开更多
基金Supported by National Natural Science Foundation of China(Grant Nos.11401151,11326071)
文摘In this paper,we show that both focal submanifolds of each isoparametric hypersurface in the sphere with six distinct principal curvatures are Willmore,hence all focal submanifolds of isoparametric hypersurfaces in the sphere are Willmore.
基金Hunan Provincial Natural Science Foundation of China (06JJ50004)the Construct Program of the Key Discipline in Hunan Province
文摘In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.
