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Central limit theorem for the occupation time of catalyticsuper-Brownian motion

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Chinese Science Bulletin 1998年第24期43卷 2035-2040页

作者： hong Wenming Department of mathematics,{2., beijing 100875, china Corresponding address: Institute of mathematics, Shantou university, Shantou 515063, china Department of mathematics Beijing Normal University Beijing China

The catalytic super_Brownian motion has been considered. If both the catalytic medium process Q and CSBM started with Lebesgue measure λ, the central limit theorem for occupation time of CSBM has been obtained in dim... 详细信息

关键词： branching rate functional Brownian collision local time catalytic super_Brownian motion occupation time central limit theorem.

Multiplicity of Periodic Solutions of Duffing Equations with Jumping Nonlinearities

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Acta Mathematicae Applicatae Sinica 2002年第3期18卷 513-522页

作者： Zai-hong WangDepartment of mathematics, Capital {2. university, beijing 100037, china Department of mathematics Capital Normal University Beijing China

We provide sufficient conditions for the existence and multiplicity of subharrnonic solutions for Duffing's equations with jumping nonlinearities.

关键词： Periodic solution. Jumping nonlinearity PS-condition

Three industries and wastewater and COD discharge of beijing

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Journal of Environmental Sciences 2000年第4期12卷 474-477页

作者： WANG Yan 1, WANG hong rui 2. NING Da tong 2.(1. beijing Municipal Research Institute of Environmental Protection, beijing 100037, china 2. Institute of Environmental Sciences, beijing normal university, beijing 100875, china) 北京市环境保护所北京 1 0 0 0 3 北京师范大学北京 1 0 0 875

With the swift development of economy, the water quality of beijing is becoming worse day by day and hampers the sustainable development obviously. In this paper, the current conditions of the municipal wastewater structure in the industrial sectors are analysed and discussed in terms of the indicators, such as direct wastewater\|discharge coefficient, complete wastewater\|discharge coefficient, direct discharge coefficient of COD and complete discharge coefficient of COD, by taking a year of 1990s as the base year. Some countermeasures are studied and the corresponding recommendations are put forward in order to improve the water environment in beijing. This provides a scientific ground for coordinating the relationship between the aquatic environment and economic growth in this city.

关键词： direct wastewater discharge coefficient (DWDC) complete wastewater discharge coefficient (CWDC) direct discharge coefficient of COD(DDC of COD) complete discharge coefficient of COD (CDC of COD)

DISSIPATIVITY AND EXPONENTIAL STABILITY OF θ-METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG

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Journal of Computational mathematics 2003年第6期21卷 715-726页

作者： hong-jiong Tian (Department of mathematics, Shanghai {2. university, Shanghai 2.02.4, china) department of mathematics Shanghai Normal University 上海 200234

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.

关键词： Singular perturbation θ-methods Dissipativity Exponential stability.

Estimates for eigenvalues of Laplacian operator with any order

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Science china mathematics 2007年第8期50卷 1078-1086页

作者： Fa-en WU~(1+) Lin-fen CAO~2.1 Department of mathematics,beijing Jiaotong {2.,beijing 100044,china 2.Department of mathematics,Henan normal {2.,Xinxiang 453007,china Department of mathematics Beijing Jiaotong University Beijing China Department of mathematics Henan Normal University Xinxiang China

Let D be a bounded domain in an n-dimensional Euclidean space ? n . Assume that $$0 < \lambda _1 \leqslant \lambda _2.\leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ are the eigenvalues of the Dirichlet Lapl... 详细信息

Let D be a bounded domain in an n-dimensional Euclidean space ? n . Assume that $$0 2.\leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ are the eigenvalues of the Dirichlet Laplacian operator with any order l: $$\left\{ \begin{gathered} ( - \vartriangle )^l u = \lambda u, in D \hfill \\ u = \frac{{\partial u}}{{\partial \vec n}} = \cdots = \frac{{\partial ^{l - 1} u}}{{\partial \vec n^{l - 1} }} = 0, on \partial D \hfill \\ \end{gathered} \right.$$ . Then we obtain an upper bound of the (k+1)-th eigenvalue λ k+1 in terms of the first k eigenvalues. $$\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )} \leqslant \frac{2.{n}[4l(n + 2. - 2.]^{\tfrac{2.{2.} \left\{ {\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{2.{2.} \lambda _i^{\tfrac{{l - 1}}{l}} \sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{2.{2.} \lambda _i^{\tfrac{2.{l}} } } } \right\}^{\tfrac{2.{2.} $$ . This ineguality is independent of the domain D. Furthermore, for any l ? 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2.considered by Qing-Ming Cheng and hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.

关键词： Dirichlet problem eigenvalue estimate Laplacian operator 35P15 58G2.

Characterization for commutators of n-dimensional fractional Hardy operators

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Science china mathematics 2007年第10期50卷 1418-1426页

作者： Zun-wei FU~(1,2. Zong-guang LIU~3 Shan-zhen LU~(1+) hong-bin WANG~3 ~1 School of Mathematical Sciences,beijing {2. university,beijing 100875,china ~2.Department of mathematics,Linyi {2. university,Linyi 2.6005,china ~3 Department of mathematics,china university of Mining and Technology (beijing),beijing 100083,china School of Mathematical Sciences Beijing Normal University Beijing China Department of mathematics Linyi Normal University Linyi China Department of mathematics China University of Mining and Technology (Beijing) Beijing China

In this paper,it was proved that the commutator H_(β,b)generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L^(p1)(R^n)to L^(p2.(R^n)if and only if b is a CMO(R^... 详细信息

关键词： n-dimensional fractional Hardy operator, commutator, C(M)O function, homogeneous Herz space

Existence and Multiplicity of Periodic Solutions of the Second-Order Differential Equations with Jumping Nonlinearities

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Acta Mathematica Sinica,English Series 2002年第3期18卷 615-624页

作者： Zai hong WANG Department of mathematics, Capital {2. university. beijing 100037. P. R. china Department of mathematics Capital Normal University Beijing P. R. China

We provide sufficient conditions for the existence and multiplicity of periodic solutions for Duffing's equations with jumping nonlinearities under resonance conditions.

关键词： Periodic solution Jumping nonlinearities Spectrum curve Time map

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration

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Science china mathematics 2008年第3期51卷 343-350页

作者： hong WenMing School of Mathematical Sciences, Laboratory of mathematics and Complex Systems, Ministry of Education,{2. normal university, {2. 100875, china School of Mathematical Sciences Laboratory of Mathematics and Complex Systems Ministry of Education Beijing Normal University Beijing China

Moderate deviations for the quenched mean of the super-Brownian motion with random immigration are proved for 3≤d≤6, which fills in the gap between central limit theorem(CLT)and large deviation principle(LDP).

关键词： super-Brownian motion large deviation moderate deviation random immigration Primary 60J80 Secondary 60F05

On the stability of diffusion processes with state-dependent switching

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Science china mathematics 2006年第9期49卷 1258-1274页

作者： XI Fubao & ZHAO Liqin Department of mathematics, beijing Institute of Technology, beijing 100081, china School of Mathematical Sciences, beijing normal university, beijing 100875, china Department of mathematics Beijing Institute of Technology Beijing China School of Mathematical Sciences Beijing Normal University Beijing China

In this paper we consider the stability for diffusion processes with state-dependent switching. We first prove their Feller continuity by the coupling methods. Furthermore, we also prove their strong Feller continuity and their exponential ergodicity under some reasonable conditions. Finally, we append a very brief discussion about the regularity of these processes.

关键词： Feller continuity, coupling, strong Feller continuity, exponential ergodicity,state-dependent switching.