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Wei Sun - One of the best experts on this subject based on the ideXlab platform.
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Strong Continuity of generalized feynman kac semigroups necessary and sufficient conditions
Journal of Functional Analysis, 2006Co-Authors: Chuan-zhong Chen, Wei SunAbstract:Let (E,D(E)) be a Strongly local, quasi-regular symmetric Dirichlet form on L2(E;m) and ((Xt)t⩾0,(Px)x∈E) the diffusion process associated with (E,D(E)). For u∈D(E)e, u has a quasi-continuous version u˜ and u˜(Xt) has Fukushima's decomposition: u˜(Xt)−u˜(X0)=Mtu+Ntu, where Mtu is the martingale part and Ntu is the zero energy part. In this paper, we study the Strong Continuity of the generalized Feynman–Kac semigroup defined by Ptuf(x)=Ex[eNtuf(Xt)], t⩾0. Two necessary and sufficient conditions for (Ptu)t⩾0 to be Strongly continuous are obtained by considering the quadratic form (Qu,D(E)b), where Qu(f,f):=E(f,f)+E(u,f2) for f∈D(E)b, and the energy measure μ〈u〉 of u, respectively. An example is also given to show that (Ptu)t⩾0 is Strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant δμ〈u〉(E)⩽12 (cf. Definition 4.5).
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Strong Continuity of generalized Feynman–Kac semigroups: Necessary and sufficient conditions
Journal of Functional Analysis, 2006Co-Authors: Chuan-zhong Chen, Wei SunAbstract:Let (E,D(E)) be a Strongly local, quasi-regular symmetric Dirichlet form on L2(E;m) and ((Xt)t⩾0,(Px)x∈E) the diffusion process associated with (E,D(E)). For u∈D(E)e, u has a quasi-continuous version u˜ and u˜(Xt) has Fukushima's decomposition: u˜(Xt)−u˜(X0)=Mtu+Ntu, where Mtu is the martingale part and Ntu is the zero energy part. In this paper, we study the Strong Continuity of the generalized Feynman–Kac semigroup defined by Ptuf(x)=Ex[eNtuf(Xt)], t⩾0. Two necessary and sufficient conditions for (Ptu)t⩾0 to be Strongly continuous are obtained by considering the quadratic form (Qu,D(E)b), where Qu(f,f):=E(f,f)+E(u,f2) for f∈D(E)b, and the energy measure μ〈u〉 of u, respectively. An example is also given to show that (Ptu)t⩾0 is Strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant δμ〈u〉(E)⩽12 (cf. Definition 4.5).
Tatsuji Kawai - One of the best experts on this subject based on the ideXlab platform.
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a Continuity principle equivalent to the monotone pi 0 _ 1 π 1 0 fan theorem
Archive for Mathematical Logic, 2019Co-Authors: Tatsuji KawaiAbstract:The Strong Continuity principle reads “every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone $$\varPi ^{0}_{1}$$ bars. We work in the context of constructive reverse mathematics.
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A Continuity principle equivalent to the monotone $$\Pi ^{0}_{1}$$ Π 1 0 fan theorem
Archive for Mathematical Logic, 2018Co-Authors: Tatsuji KawaiAbstract:The Strong Continuity principle reads “every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone $$\varPi ^{0}_{1}$$ bars. We work in the context of constructive reverse mathematics.
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A Continuity principle equivalent to the monotone $\Pi^{0}_{1}$ fan theorem
arXiv: Logic, 2018Co-Authors: Tatsuji KawaiAbstract:The Strong Continuity principle reads "every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan theorem for monotone $\Pi^{0}_{1}$ bars. We work in the context of constructive reverse mathematics.
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a Continuity principle equivalent to the monotone pi 0 _ 1 fan theorem
arXiv: Logic, 2018Co-Authors: Tatsuji KawaiAbstract:The Strong Continuity principle reads "every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan theorem for monotone $\Pi^{0}_{1}$ bars. We work in the context of constructive reverse mathematics.
Chuan-zhong Chen - One of the best experts on this subject based on the ideXlab platform.
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Strong Continuity of generalized feynman kac semigroups necessary and sufficient conditions
Journal of Functional Analysis, 2006Co-Authors: Chuan-zhong Chen, Wei SunAbstract:Let (E,D(E)) be a Strongly local, quasi-regular symmetric Dirichlet form on L2(E;m) and ((Xt)t⩾0,(Px)x∈E) the diffusion process associated with (E,D(E)). For u∈D(E)e, u has a quasi-continuous version u˜ and u˜(Xt) has Fukushima's decomposition: u˜(Xt)−u˜(X0)=Mtu+Ntu, where Mtu is the martingale part and Ntu is the zero energy part. In this paper, we study the Strong Continuity of the generalized Feynman–Kac semigroup defined by Ptuf(x)=Ex[eNtuf(Xt)], t⩾0. Two necessary and sufficient conditions for (Ptu)t⩾0 to be Strongly continuous are obtained by considering the quadratic form (Qu,D(E)b), where Qu(f,f):=E(f,f)+E(u,f2) for f∈D(E)b, and the energy measure μ〈u〉 of u, respectively. An example is also given to show that (Ptu)t⩾0 is Strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant δμ〈u〉(E)⩽12 (cf. Definition 4.5).
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Strong Continuity of generalized Feynman–Kac semigroups: Necessary and sufficient conditions
Journal of Functional Analysis, 2006Co-Authors: Chuan-zhong Chen, Wei SunAbstract:Let (E,D(E)) be a Strongly local, quasi-regular symmetric Dirichlet form on L2(E;m) and ((Xt)t⩾0,(Px)x∈E) the diffusion process associated with (E,D(E)). For u∈D(E)e, u has a quasi-continuous version u˜ and u˜(Xt) has Fukushima's decomposition: u˜(Xt)−u˜(X0)=Mtu+Ntu, where Mtu is the martingale part and Ntu is the zero energy part. In this paper, we study the Strong Continuity of the generalized Feynman–Kac semigroup defined by Ptuf(x)=Ex[eNtuf(Xt)], t⩾0. Two necessary and sufficient conditions for (Ptu)t⩾0 to be Strongly continuous are obtained by considering the quadratic form (Qu,D(E)b), where Qu(f,f):=E(f,f)+E(u,f2) for f∈D(E)b, and the energy measure μ〈u〉 of u, respectively. An example is also given to show that (Ptu)t⩾0 is Strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant δμ〈u〉(E)⩽12 (cf. Definition 4.5).
Terence Jegaraj - One of the best experts on this subject based on the ideXlab platform.
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Large Deviations and Transitions Between Equilibria for Stochastic Landau–Lifshitz–Gilbert Equation
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Zdzisław Brzeźniak, Ben Goldys, Terence JegarajAbstract:We study a stochastic Landau–Lifshitz equation on a bounded interval and with finite dimensional noise. We first show that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Next, we prove the large deviations principle for the small noise asymptotic of solutions using the weak convergence method. An essential ingredient of the proof is the compactness, or weak to Strong Continuity, of the solution map for a deterministic Landau–Lifschitz equation when considered as a transformation of external fields. We then apply this large deviations principle to show that small noise can cause magnetisation reversal. We also show the importance of the shape anisotropy parameter for reducing the disturbance of the solution caused by small noise. The problem is motivated by applications from ferromagnetic nanowires to the fabrication of magnetic memories.
D. Singh - One of the best experts on this subject based on the ideXlab platform.
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Between Strong Continuity and almost Continuity
Applied General Topology, 2010Co-Authors: J. K. Kohli, D. SinghAbstract:As embodied in the title of the paper Strong and weak variants of Continuity that lie strictly between Strong Continuity of Levine and almost Continuity due to Singal and Singal are considered. Basic properties of almost completely continuous functions (≡ R-maps) and δ-continuous functions are studied. Direct and inverse transfer of topological properties under almost completely continuous functions and δ-continuous functions are investigated and their place in the hier- archy of variants of Continuity that already exist in the literature is out- lined. The class of almost completely continuous functions lies strictly between the class of completely continuous functions studied by Arya and Gupta (Kyungpook Math. J. 14 (1974), 131-143) and δ-continuous functions defined by Noiri (J. Korean Math. Soc. 16, (1980), 161-166). The class of almost completely continuous functions properly contains each of the classes of (1) completely continuous functions, and (2) al- most perfectly continuous (≡ regular set connected) functions defined by Dontchev, Ganster and Reilly (Indian J. Math. 41 (1999), 139-146) and further studied by Singh (Quaestiones Mathematicae 33(2)(2010), 1–11) which in turn include all δ-perfectly continuous functions initi- ated by Kohli and Singh (Demonstratio Math. 42(1), (2009), 221-231) and so include all perfectly continuous functions introduced by Noiri (Indian J. Pure Appl. Math. 15(3) (1984), 241-250).
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Function Spaces and Strong Variants of Continuity
Applied General Topology, 2008Co-Authors: J. K. Kohli, D. SinghAbstract:It is shown that if domain is a sum connected space and range is a T0-space, then the notions of Strong Continuity, perfect Continuity and cl-superContinuity coincide. Further, it is proved that if X is a sum connected space and Y is Hausdorff, then the set of all Strongly continuous (perfectly continuous, cl-supercontinuous) functions is closed in Y X in the topology of pointwise convergence. The results obtained in the process strengthen and extend certain results of Levine and Naimpally.
