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Estimates for eigenvalues of Laplacian operator with any order

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 university,Beijing 100044,{2. 2.0},Henan normal university,Xinxiang 453007,{2. 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.

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ON THE RANGE OF RANDOM WALKS IN RANDOM ENVIRONMENT　

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Chinese Annals of mathematics,Series B 1995年第1期16卷 131-138页

作者： ZHOU XIANYIN(department of mathematics, Beijing {2. university, Beijing 100875, china.) 北京师范大学北京 100875

The range of random walk on Z￣d in symmetric random environment is investigated. As results, it is proved that the strong law of large numbers for the range of random walk on Zdin some random environments holds if d... 详细信息

关键词： Law of large numbers Ergodic theorem Range of random walk Random environment Effective resistance.

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Pairing problem of generators in Kac-Moody algebras

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Chinese Science Bulletin 1998年第22期43卷 1872-1879页

作者： LU Caihui department of mathematics, Capital normal {2., Beijing 100037, china department of mathematics Capital Normal University Beijing China

The following result is proved: in any Kac Moody algebra g(A) , (ⅰ) given any non central element h in the Cartan subalgebra  , or (ⅱ) given any real root vector x β, β∈Δ re . There exists y∈g(A) such that th... 详细信息

关键词： generator root system real root vector.

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Moment bounds for IID sequences under sublinear expectations

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Science china mathematics 2011年第10期54卷 2155-2160页

作者： HU Feng1,2.1department of mathematics,Qufu {2. university,Qufu 2.3165,china 2.chool of mathematics,Shandong university,Jinan 2.0100,china department of mathematics Qufu Normal University Qufu China School of mathematics Shandong University Jinan China

With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality,we get the following result:For any continuous functionsatisfying the growth condition |(x) | C(1 + |x|p) for some C > 0,p 1 depending on ,the central limit theorem under sublinear expectations obtained by Peng still holds.

关键词： moment bound sublinear expectation IID random variables G-normal distribution central limit theorem

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On quasi-convex mappings of order α in the unit ball of a complex Banach space

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Science china mathematics 2006年第11期49卷 1451-1457页

作者： LIU Taishun & XU Qinghua department of mathematics, Huzhou Teachers College, Huzhou, 313000, china {2. of mathematics, Jiangxi normal university, Nanchang, 33002., china department of mathematics Huzhou Teachers College Huzhou China department of mathematics Jiangxi Normal University Nanchang China

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order a in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings. Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space X. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order a defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.

关键词： coefficient estimation the homogeneous expansion quasi-convex mapping quasi-convex mapping of order a.

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Existence and Application of Optimal Markovian Coupling with Respect to Non-Negative Lower Semi-Continuous Functions

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Acta Mathematica Sinica,English Series 2000年第2期16卷 261-270页

作者： Shaoyi Zhangdepartment of mathematics.Beijing {2. university,Beijing 100875,P.R.china department of mathematics,Hubei {2. university,Huangshi 435002.P.R.china department of mathematics Beijing Normal University Beijing 100875 P. R. China department of mathematics Hubei Normal University Huangshi 435002 P. R. China

In this paper,for two given transition probabilities,the existence of the optimal Markovian coupling with respect to a non-negative lower semi-continuous function is *** an application of this result,the well-known Strassen’s theorem is ***,it is proved that the existence of an order-preserving Markovian coupling of two given jump processes is equivalent to their stochastical comparability.

关键词： Transition probability Markovian coupling Jump process

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Littlewood-Paley operators with variable kernels

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Science china mathematics 2006年第5期49卷 639-650页

作者： CHEN Jiecheng, DING Yong & FAN Dashan department of mathematics, Zhejiang university (Xixi Campus), Hangzhou 31002.,{2. School of Mathematical Sciences, Beijing normal university, Beijing 100875,{2. department of mathematics, university of Wisconsin-Milwaukee, Milwaukee WI 532.1, USA department of mathematics, Central{2. normal university, Wuhan 430074,{2. department of mathematics Zhejiang University (Xixi Campus) Hangzhou China School of Mathematical Sciences Beijing Normal University Beijing China department of mathematics University of Wisconsin-Milwaukee Milwaukee USA department of mathematics Central China Normal University Wuhan China

In this paper we study a certain directional Hilbert transform and the boundedness on some mixed norm spaces. As one of applications, we prove the Lp-boundedness of the Littlewood-Paley operators with variable kernels... 详细信息

关键词： directional Hilbert transform, Sobolev spaces, Littlewood-Paley operator,variable kernel.

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Harmonic maps with potential from complete manifolds

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Chinese Science Bulletin 1998年第21期43卷 1780-1786页

作者： CHEN Qun mathematics department, Central{2. normal university, Wuhan 430079,{2. Institute of mathematics, Fudan university, Shanghai 2.0433,{2. mathematics department Central China Normal University Wuhan China Institute of mathematics Fudan University Shanghai China

Harmonic maps with potential from complete manifolds are considered. This is a new kind of maps more general than the usual harmonic maps relating to many interesting problems such as equilibrium system of ferromagnetic spin chain and Neumann motion. Aiming at the general properties, the author derives basic gradient estimates and then Liouville type results for these maps, which are interesting in constrast to those of the usual harmonic maps for the presence of potentials.

关键词： harmonic maps potential ferromagnetic chain gradient estimate.

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Regular n-simplices in R^n with vertices in Z^(n)

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Science china mathematics 2004年第1期47卷 104-113页

作者： XIA Jianguo & QIN Hourongdepartment of mathematics, Nanjing {2. university, Nanjing 2.0097, china (email: jgxia@***) department of mathematics, Nanjing university, Nanjing 2.0093, china Dept. of Math. Nanjing Normal Univ. Nanjing 210097 China Dept. of Math. Nanjing Univ. Nanjing 210093 China

In this paper the Ⅰ and Ⅱ regular n-simplices are introduced. We prove that the sufficient and necessary conditions for existence of an Ⅰ regular n-simplex in Rn are that if n is even then n = 4m(m + 1), and if n is odd then n = 4m + 1 with that n + 1 can be expressed as a sum of two integral squares or n = 4m - 1, and that the sufficient and necessary condition for existence of a Ⅱ regular n-simplex in Rn is n = 2.2.- 1 or n = 4m(m+1)(m 6 N). The connection between regulars-simplex in Rn and combinational design is given.

关键词： regular n-simplex in Rn, quadratic form.

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Optimal value functions of generalized semi-infinite min-max programming on a noncompact set

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Science china mathematics 2005年第2期48卷 261-276页

作者： WANG Changyu, YANG Xiaoqi & YANG Xinmin Institute oi Operations Research, Qufu normal university, Qufu 2.3165,{2. department of Applied mathematics, Dalian university of Technology, Dalian 11602.,{2. department of Applied mathematics, The Hong Kong Polytechnic university, Hong Kong,{2. department of mathematics, Chongqing normal university, Chongqing 400047,{2. Institute of Operations Research Qufu Normal University Qufu China department of Applied mathematics Dalian University of Technology Dalian China department of Applied mathematics The Hong Kong Polytechnic University Hong Kong China department of mathematics Chongqing Normal University Chongqing China

In this paper, we study optimal value functions of generalized semi-infinite min-max programming problems on a noncompact set. Directional derivatives and subdifferential characterizations of optimal value functions are given. Using these properties,we establish first order optimality conditions for unconstrained generalized semi-infinite programming problems.

关键词： optimal value function, directional derivative, subdifferential, semi-infinite programming, first order optimality condition.

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