The correction operator for the canonical interpolation operator of the Adini element and the lower bounds of eigenvalues

In this paper,we develop a correction operator for the canonical interpolation operator of the Adini element.We use this new correction operator to analyze the discrete eigenvalues of the Adini element method for the fourth order elliptic eigenvalue problem in the three dimensions.We prove that the discrete eigenvalues are smaller than the exact ones.     

Eigenvalue approximation from below using non-conforming finite elements

This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators,with emphasis on obtaining lower bounds. In addition,this article also contains some new materials for eigenvalue approximations of the Laplace operator,which include:1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally,we list several unsolved problems.     

MODIFIED MORLEY ELEMENT METHOD FOR A FOURTH ORDER ELLIPTIC SINGULAR PERTURBATION PROBLEM

This paper proposes a modified Morley element method for a fourth order ellipticsingular perturbation problem. The method also uses Morley element or rectangle Morleyelement, but linear or bilinear approximation of finite element functions is used in the lowerpart of the bilinear form. It is shown that the modified method converges uniformly in theperturbation parameter.     

Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods

We introduce some ways to compute the lower and upper bounds of the Laplace eigenvalue problem.By using the special nonconforming finite elements,i.e.,enriched Crouzeix-Raviart element and extended Q1ro t,we get the lower bound of the eigenvalue.Additionally,we use conforming finite elements to do the postprocessing to get the upper bound of the eigenvalue,which only needs to solve the corresponding source problems and a small eigenvalue problem if higher order postprocessing method is implemented.Thus,we can obtain the lower and upper bounds of the eigenvalues simultaneously by solving eigenvalue problem only once.Some numerical results are also presented to demonstrate our theoretical analysis.     

Lower bound of the sum of positive eigenvalues of a graph

1.IntroductionA(general)graphGisagraphwithloopsormultiedges,whileasimplegraphhasneitherloopsnormultiedges.ForagivengraphG,theedgesetandtheadjacencymatrixofGaredenotedbyE(G)andA(G)respectively.Moreover,weuseS(G),P(G)andq(G)todenotethesumofpositiveeigenvalues,thepositiveandnegativeinertiaindicesofA(G).In[l],C.DelormepointedoutthatS(G)5#E(G)withequalityohlyformatchings.Hefurtherputforwardthequestion:WhatisthelowerboundofS(G)foragiven#E(G)?Isit~?Inthispaperwegiveanaffirmativeanswertoth…     

向量互补问题和极小元问题

本文不仅引入了非空集的有效下界的概念,而且引入了新的严格C-伪单调映射,在C-伪单调性的假设下,文内证明了向量互补问题（VCP）的可行集及其相关集合的有效下界和下有效点的存在性。     

一类反应扩散方程的爆破时间下界估计

该文讨论了一类反应项为非线性非局部热源且热汇具有时间系数的反应扩散方程,分别在Dirichlet、Neumann或Robin边界条件下,在有界区域中的爆破行为.若解可能在有限时间发生爆破,通过构造合适的辅助函数,对时间系数给出适当的条件,利用Sobolev、H?lder不等式及Payne和Schaefer积分不等式等技巧,得出了解的爆破时间下界的估计.     

A NOTE ON THE BACKWARD ERRORS FOR INVERSE EIGENVALUE PROBLEMS

In this note,we consider the backward errors for more general inverse eigenvalus prob-lems by extending Sun‘‘‘‘s approach.The optimal backward errors defined for diagonal-ization matrix inverse eigenvalue problem with respect to an approximate solution,and the upper and lower bounds are derived for the optimal backward errors.The results may be useful for testing the stability of practical algorithms.     

On area-efficient drawings of rectangular duals for VLSI floor-plan

In this paper, we consider a problem to seek a rectangular dual and its area-efficient drawing such that can be drawn in the smallest area among all rectangular duals under the constraints imposed not only on the area and the minimum dimension of each face but also on the length of abutment between two adjacent faces. Since the problem is hard to solve, we tackle this problem in an exhaustive manner by using an algorithm to enumerate all the rectangular duals. In order to make this exhaustive method efficient, we propose the following two algorithms working under the constraints stated above; an algorithm to find an area-efficient drawing of a given rectangular dual, and an algorithm to estimate a lower bound to the area required to draw a given rectangular dual. We also show some esperimental results to demonstrate the effectiveness of the lower bound. The area-efficient drawing of can be used as a VLSI floor-plan by regarding each inner face of as an area for a block to be placed.     

一类非线性抛物系统解的爆破现象

研究了具有依赖于时间的系数的非线性抛物方程解的爆破现象.对已知数据项进行一定的假设并设置一些辅助函数,应用微分不等式技术,得到了方程的解发生爆破的条件.当爆破发生时,分别推导了方程在二维区域和三维区域上解的爆破时间的下界.     

A Sharpening of the Taikov Lower Bound in the Inequality between the C- and L-Norms for Trigonometric Polynomials

Mathematical Notes -     

An Improved Lower Bound for the Multimedian Location Problem

We consider the problem of locating, on a network, n new facilities that interact with m existing facilities. In addition, pairs of new facilities interact. This problem, the multimedian location problem on a network, is known to be NP-hard. We give a new integer programming formulation of this problem, and show that its linear programming relaxation provides a lower bound that is superior to the bound provided by a previously published formulation. We also report results of computational testing with both formulations.     

谱元法求解Helmholtz方程透射特征值问题

研究了Helmholtz方程透射特征值问题,提出一种Chebyshev谱元法求解,该方法兼具了有限元法处理边界及区域的灵活性和谱方法的快速收敛特性.运用加权余量原理,得到了Chebyshev谱元法用于透射特征值问题的基本理论以及数学公式,将原问题转化为二次特征值问题.最后通过数值实验算例验证了Chebyshev谱元法的有效性.