This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference be...This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space ( E,ℰ,μ ) such that the N.B.A. is isomorphic to ( ℰ ˜ , μ ˜ ) resulting from the factorisation of initial σ-algebra by the ideal of negligible sets. We focus on topological properties ( ℰ ˜ , μ ˜ ) in general setting when μ can be an infinite measure. In case when μ is infinite, we also consider properties of ℰ ˜ fin ⊆ ℰ ˜ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on ( E,ℰ,μ ) are derived for separability and compactness of ℰ ˜ and ℰ ˜ fin .展开更多
The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topol...The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum function is Borel from the space of bounded operators into the Polish space of compact subsets of the closed unit disc. Alternative results are given when other topologies are used.展开更多
This paper studies the limit average variance criterion for continuous-time Markov decision processes in Polish spaces. Based on two approaches, this paper proves not only the existence of solutions to the variance mi...This paper studies the limit average variance criterion for continuous-time Markov decision processes in Polish spaces. Based on two approaches, this paper proves not only the existence of solutions to the variance minimization optimality equation and the existence of a variance minimal policy that is canonical, but also the existence of solutions to the two variance minimization optimality inequalities and the existence of a variance minimal policy which may not be canonical. An example is given to illustrate all of our conditions.展开更多
In this paper we study the average sample-path cost （ASPC） problem for continuous-time Markov decision processes in Polish spaces. To the best of our knowledge, this paper is a first attempt to study the ASPC criter...In this paper we study the average sample-path cost （ASPC） problem for continuous-time Markov decision processes in Polish spaces. To the best of our knowledge, this paper is a first attempt to study the ASPC criterion on continuous-time MDPs with Polish state and action spaces. The corresponding transition rates are allowed to be unbounded, and the cost rates may have neither upper nor lower bounds. Under some mild hypotheses, we prove the existence of （ε〉 0）-ASPC optimal stationary policies based on two different approaches： one is the ＂optimality equation＂ approach and the other is the ＂two optimality inequalities＂ approach.展开更多
This paper studies the strong n(n =—1,0)-discount and finite horizon criteria for continuoustime Markov decision processes in Polish spaces.The corresponding transition rates are allowed to be unbounded,and the rewar...This paper studies the strong n(n =—1,0)-discount and finite horizon criteria for continuoustime Markov decision processes in Polish spaces.The corresponding transition rates are allowed to be unbounded,and the reward rates may have neither upper nor lower bounds.Under mild conditions,the authors prove the existence of strong n(n =—1,0)-discount optimal stationary policies by developing two equivalence relations:One is between the standard expected average reward and strong—1-discount optimality,and the other is between the bias and strong 0-discount optimality.The authors also prove the existence of an optimal policy for a finite horizon control problem by developing an interesting characterization of a canonical triplet.展开更多
We constructed a class of self-similar sets and proved the convergence in this paper.Besides these,the upper bound and lower bound of Hausdorff measures of them were given too.
Let LSC(X) denote the set of all proper lower semicontinuous functions on X with the epi-topology. In this paper we give characterizations of the separation axioms. Baire properties and metrizability of LSC(X). We sho...Let LSC(X) denote the set of all proper lower semicontinuous functions on X with the epi-topology. In this paper we give characterizations of the separation axioms. Baire properties and metrizability of LSC(X). We show also that the continuous function space C(X) with the epi-topology is of first category when N is first countable.展开更多
文摘This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space ( E,ℰ,μ ) such that the N.B.A. is isomorphic to ( ℰ ˜ , μ ˜ ) resulting from the factorisation of initial σ-algebra by the ideal of negligible sets. We focus on topological properties ( ℰ ˜ , μ ˜ ) in general setting when μ can be an infinite measure. In case when μ is infinite, we also consider properties of ℰ ˜ fin ⊆ ℰ ˜ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on ( E,ℰ,μ ) are derived for separability and compactness of ℰ ˜ and ℰ ˜ fin .
文摘The paper investigates the variation of the spectrum of operators in infinite dimensional Banach spaces. Consider the space of bounded operators on a separable Banach space when equipped with the strong operator topology, and the Polish space of compact subsets of the closed unit disc of the complex plane when equipped with the Hausdorff topology. Then, it is shown that the unit spectrum function is Borel from the space of bounded operators into the Polish space of compact subsets of the closed unit disc. Alternative results are given when other topologies are used.
基金supported by the National Natural Science Foundation of China(10801056)the Natural Science Foundation of Ningbo(2010A610094)
文摘This paper studies the limit average variance criterion for continuous-time Markov decision processes in Polish spaces. Based on two approaches, this paper proves not only the existence of solutions to the variance minimization optimality equation and the existence of a variance minimal policy that is canonical, but also the existence of solutions to the two variance minimization optimality inequalities and the existence of a variance minimal policy which may not be canonical. An example is given to illustrate all of our conditions.
基金Supported by the National Natural Science Foundation of China (No.10801056)the Natural Science Foundation of Ningbo (No. 2010A610094)K.C. Wong Magna Fund in Ningbo University
文摘In this paper we study the average sample-path cost （ASPC） problem for continuous-time Markov decision processes in Polish spaces. To the best of our knowledge, this paper is a first attempt to study the ASPC criterion on continuous-time MDPs with Polish state and action spaces. The corresponding transition rates are allowed to be unbounded, and the cost rates may have neither upper nor lower bounds. Under some mild hypotheses, we prove the existence of （ε〉 0）-ASPC optimal stationary policies based on two different approaches： one is the ＂optimality equation＂ approach and the other is the ＂two optimality inequalities＂ approach.
基金supported by the National Natural Science Foundation of China under Grant Nos.61374080 and 61374067the Natural Science Foundation of Zhejiang Province under Grant No.LY12F03010+1 种基金the Natural Science Foundation of Ningbo under Grant No.2012A610032Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions
文摘This paper studies the strong n(n =—1,0)-discount and finite horizon criteria for continuoustime Markov decision processes in Polish spaces.The corresponding transition rates are allowed to be unbounded,and the reward rates may have neither upper nor lower bounds.Under mild conditions,the authors prove the existence of strong n(n =—1,0)-discount optimal stationary policies by developing two equivalence relations:One is between the standard expected average reward and strong—1-discount optimality,and the other is between the bias and strong 0-discount optimality.The authors also prove the existence of an optimal policy for a finite horizon control problem by developing an interesting characterization of a canonical triplet.
文摘We constructed a class of self-similar sets and proved the convergence in this paper.Besides these,the upper bound and lower bound of Hausdorff measures of them were given too.
文摘Let LSC(X) denote the set of all proper lower semicontinuous functions on X with the epi-topology. In this paper we give characterizations of the separation axioms. Baire properties and metrizability of LSC(X). We show also that the continuous function space C(X) with the epi-topology is of first category when N is first countable.
