The Mechanical Quadrature Methods and Their Extrapolation for Solving BIE of Steklov Eigenvalue Problems

By means of the potential theory Steklov eigenvalue problems are transformed into general eigenvalue problems of boundary integral equations (BIE) with the logarithmic singularity.Using the quadrature rules, the paper presents quadrature methods for BIE of Steklov eigenvalue problem, which possess high accuracies O(h^3) and low computing complexities. Moreover, an asymptotic expansion of the errors with odd powers is shown. Using h^3-Richardson extrapolation, we can not only improve the accuracy order of approximations, but also derive a posterior estimate as adaptive algorithms. The efficiency of the algorithm is illustrated by some examples.     

线性空间C[a,b]上的线性相关性研究

研究了线性空间C[a,b]上的线性相关性,给出了衡量C[a,b]上n个函数线性相关性程度的量以及线性相关的充分必要条件.     

Kinetic Energy Partition Method for Competing Modes

We present a new basis set expansion method for quantum dynamics systems with two competing modes where the interaction potentials are equally dominant. The new idea introduced here is a kinetic energy partition scheme instead of the usual division of the potential energy. The partition results in two kinetic energy terms with their effective masses. By distributing each partial kinetic energy to the respective potential, the full Hamiltonian can be expressed as the sum of the two competing modes. The solution procedure is illustrated by using a system consisting of a particle under the action of two harmonic potentials with different equilibrium distances and force constants. Next we apply this method to obtain the potential energy curves for the prototype hydrogen molecule ion. This new expansion converges very fast to the exact solutions for both eigenvalues and eigenfunctions.     

Augmentation of the Generalised n×n Eigenvalue Equation to a Generalised (n+1)×(n+1) Eigenvalue Equation

Generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) where B and S ^b are n×n Hermitian matrices while S ^b is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H| _k = _k S| _k (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and S ^b , respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues _k and the eigenvectors | _k of the augmented eigenvalue equation can be expressed in terms of the eigenvalues _i and the eigenvectors | _i of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n ²). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n ³) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.     

非线性椭圆型方程的共振问题

在对一类二阶椭圆型带有非线性项的方程边值问题解的存在性的研究中,一般采用非线性泛函分析的方法,利用Mountain Pass原理,来进行研究,但前提是所考察方程对应的泛函要有一定的紧性,也即通常的(PS)条件。然而,很多情形中泛函是不满足(PS)条件的,例如方程中非线性项具有临界指数或方程在共振情况下所带的非线性项为次线性。本文所考虑的即是后者情况下方程解的存在性。     

关于Riemann流形中的2-调和子流形

讨论了黎曼流形中的 2-调和子流形,获得了这类子流形的第二基本形式模长平方和Ricci曲率的pinching定理:设M是n+p维黎曼流形N的具有平行平均曲率向量的n维 2-调和子流形,如果N的截面曲率的上、下确界分别记为KN和KN,则当M的第二基本形式模长平方s[ (n-1)KN-KN+nH2 ]时,M是极小子流形。     

多夹层板壳的非线性理论及应用(Ⅱ)──正交异性材料扁壳的基本方程

本文把文[1]建立的多夹层壳体的中小转动一阶大挠度理论,具体地运用到多夹层扁壳中去给出了正交异性材料的多夹层扁壳的大挠度问题平衡方程和边界条件及其特例,宏观各向同性材料多夹层扁壳的大挠度方程.     

增强谐波辐射的一种新途径

提出用ＭＯＰＡ构型来增强谐波辐射。通过这个构型，人们可以用低能量电子束获得多种短波长和高功率相干光源。     

Sharp Estimates for the Size of Balls in the Complement of a Hypersurface

In this paper, we make estimates for the radius of balls contained in some component of the complementary of a complete hypersurface into a space form, generalizing and improving analogous radius estimates for embedded compact hypersurfaces obtained by Blaschke, Koutroufiotis and the authors. The results are obtained using an algebraic lemma and a tangency principle related with the length of the second fundamental form. The algebraic lemma also is used to improve a result for graphs due to Hasanis–Vlachos. The first author dedicates this work to his parents José (in memoriam) and Herondina