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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 {2.,Beijing 100044,china 2.department of mathematics,Henan Normal {2.,Xinxiang 453007,china department of mathematics Beijing Jiaotong University Beijing 100044 China department of mathematics Henan Normal University Xinxiang 453007 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 $$0 < \lambda _1 \leqslant \lambda _2 \leqslant \cdots \leqslant \lambda _k \leqslant \cdots $$ 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. $$\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )} \leqslant \frac{1}{n}[4l(n + 2l - 2)]^{\tfrac{1}{2}} \left\{ {\sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{{l - 1}}{l}} \sum\limits_{i = 1}^k {(\lambda _{(k + 1)} - \lambda _i )^{\tfrac{1}{2}} \lambda _i^{\tfrac{1}{l}} } } } \right\}^{\tfrac{1}{2}} $$ . 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 MODULI NUMBER OF PLANE CURVE SINGULARITIES WITHONE CHARACTERISTIC PAIR

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Chinese Annals of mathematics,Series B 1999年第4期20卷 407-412页

作者： CHEN HAO(department of mathematics, Zhongshan {2., Guangzhou 5102.5, china) Zhongshan Univ Dept Math Guangzhou 510275 Peoples R China

The author gives another linear-algebraic proof of the famous result of Zariski, Delorme,Briancon- Granger- Maisonobe about the moduli number of plane curve singularities with the same topological type as Xa + Yb = 0 (i.e.,with one characteristic pair). Since the original proof depends very much on the division theorem of Briancon, it cannot be generalized to higher dimensions. It is hopeful that the proof here will be applied to the higher dimensional cases.

关键词： Moduli number Plane curve Singularity Characteristic pair

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THE FUNCTION d_k(n) AT CONSECUTIVE INTEGERS

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Chinese Annals of mathematics,Series B 1999年第4期20卷 473-478页

作者： MENG ZAIZHAO(department of mathematics, Peking {2., Beijing 100871, china) Peking Univ Dept Math Beijing 100871 Peoples R China

Let dk(n) denote the k-fold iterated divisor function (k ≥ 2.. It is proved that for sufficiently large x) dk(n) = dk(n + 1) holds for >> x(log log x)-3 integers n ≤x.

关键词： Divisor function Multiplicative function Congruence

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A novel kinetic procedure for the determination of iron in environmental samples

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Journal of Environmental Sciences 2000年第2期12卷 84-87页

作者： HE Rong\|huan,WANG Jian\|hua (department of Chemistry, yantai Normal {2., yantai 2.402.,china)

A novel kinetic procedure for the determination of trace iron(Ⅲ) was proposed based on the aerial oxidation of Gallocyanine by oxygen dissolved in water at pH 7 0. The calibration graph is linear for 0 05—0.6 μg/ml, and the detection limit is 0 01μg/ml (Fe). The present procedure is selective for iron(Ⅲ) and very easy to control compared with kinetic procedures for iron. It had been used to the determination of trace iron in environmental samples, and the results were satisfactory.

关键词： kinetic spectrophotometry Gallocyanine iron determination CLC number: X515 Document code: A

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AN ITERATION METHOD FOR INCOMPRESSIBLEVISCOUS/INVISCID COUPLED PROBLEM VIA A SPECTRALAPPROXIMATION

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Journal of Computational mathematics 1999年第4期17卷 379-396页

作者： Chuan-jn Xu(department of mathematics, Xiamen {2., Xiamen 361005, china) 厦门大学数学系福建厦门 361005

An efficient iteration-by-subdomain method (known as the Schwarz alternating algorithm) for incompressible viscous/inviscid coupled model is presented. Appropriate spectral collocation approximations are proposed. The convergence analysis show that the iterative algorithms converge with a rate independent of the polynomial degree used.

关键词： coupled equations Navier-Stokes equations Euler equations cllocation approximation Schwarz alternating algorithm

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Strong uniqueness of best approximations in spaces of bounded linear operators

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Science china mathematics 2004年第3期47卷 339-351页

作者： LI Chong HE Jinsu department of mathematics, Zhejiang {2. Hangzhou 31002., china department of mathematics, Zhejiang Normal {2. Jinhua 32.004, china 1. department of mathematics Zhejiang University 310027 Hangzhou China 2. department of mathematics Zhejiang Normal University 321004 Jinhua China

The present paper is concerned with problems of the strong uniqueness of the best approximation and the characterization of a uniqueness element in operator *** results on the strong uniqueness of the best approximation operatorfrom RS-sets are proved and the uniqueness element of a sun in the compactoperator space from c0 to c0 is characterized by the strict Kolmogorov's *** recent results due to Lewicki and others are extended and improved.

关键词： best approximation operator, RS-set, sun, strong uniqueness, uniqueness element.

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SPECTRAL AND PSEUDOSPECTRAL APPROXIMATIONS IN TIME FOR PARABOLIC EQUATIONS

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Journal of Computational mathematics 1998年第2期16卷 107-120页

作者： Fa-yong Zhang (department of mathematics, Heilongjiang {2., Harbin 150080, china) Heilongjiang Univ Dept Math Harbin 150080 Peoples R China

In this paper, spectral and pseudospectral methods are applied to both time and space variables for parabolic equations. Spectral and pseudospectral schemes are given, and error estimates are obtained for approximate ... 详细信息

关键词： spectral approximation pseudospectral approximation parabolic equation error estimate

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The Design of Lens Imaging System byMeans of Fractional Fourier Transform

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Chinese Journal of Lasers 2002年第2期11卷 27-32页

作者： CHEN Jiannong XU Qiang (department of Physics, yantai Teachers’ {2., yantai 2.402., china)D.R.Selviah(department of Electronic and Electrical Engineering, {2. College London, Torrington Place, London WC1E 7JE, UK) department of Physics Yantai Teachers' University Yantai 264025 department of Electronic and Electrical Engineering University College London Torrington Place 英国 London WC1E 7JE

The relation between the 2.d fractional Fourier transform and the imaging process of an optical system is discussed. By changing the coordinate scales of the input plane in respect to the magnification of the optical imaging system, the fractional Fourier transform can be a powerful tool in designing specific imaging system. The Gaussian imaging formula of single lens is obtained by using the tool. Finally the procedures are generalized for designing a double-lens imaging system through an example.

关键词： lens imaging system fractional Fourier transform Gaussian imaging formula

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The proof of FFD(L)≤11/9OPT(L) +7/9

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Chinese Science Bulletin 1997年第15期42卷 1262-1265页

作者： LI Rongheng, YUE Minyi1. department of mathematics, Hunan Normal {2., Changsha 410081, china 2. Institute of Applied mathematics, Chinese Academy of Sciences, Beijing 100080, china department of mathematics Hunan Normal University Changsha 410081 China Institute of Applied mathematics Chinese Academy of Sciences Beijing 100080 China

THE one-dimensional bin-packing problem is defined as follows: for a given list L={p1, p2.…, Pn}, where 0i≤1 denotes the item and its size as well, we are to pack all the items in-to bins, each of which has a capaci... 详细信息

THE one-dimensional bin-packing problem is defined as follows: for a given list L={p1, p2./sub>,…, Pn}, where 0i≤1 denotes the item and its size as well, we are to pack all the items in-to bins, each of which has a capacity 1, and the goal is to minimize the number of bins *** first-fit-decreasing （FFD） algorithm is a famous approximate algorithm for the bin-pack-ing problem. The FFD algorithm first sorts all the list into non-increasing order and then pro-cesses the pieces in that order by placing each item into the first bin into which it fits.

关键词： bin-packing minimal counter-example weigtht-function FFD-algorithm.

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MIXED FINITE ELEMENT METHODS FOR A STRONGLYNONLINEAR PARABOLIC PROBLEM

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Journal of Computational mathematics 1999年第2期17卷 209-220页

作者： Yan-ping Chen(department of mathematics, Nanjing {2., Naming 2.0008, china Institute ofComputational and Applied mathematics, Xiangtan {2., Hunan 411105, china) 南京大学数学系江苏南京

A mixed finite element method is developed to approximate the solution of a strongly nonlinear second-order parabolic problem. The existence and uniqueness of the approximation are demonstrated and L-2.error estimates... 详细信息

关键词： finite element method nonlinear parabolic problem

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