Using principal eigenvectors of adjacency matrices with added diagonal weights to compose centrality measures and identify bowtie structures for a digraph

Principal eigenvectors of adjacency matrices are often adopted as measures of centrality for a graph or digraph. However, previous principal-eigenvector-like measures for a digraph usually consider only the strongly connected component whose adjacency submatrix has the largest eigenvalue. In this paper, for each and every strongly connected component in a digraph, we add weights to diagonal elements of its member nodes in the adjacency matrix such that the modified matrix will have the new unique largest eigenvalue and corresponding principal eigenvectors. Consequently, we use the new principal eigenvectors of the modified matrices, based on different strongly connected components, not only to compose centrality measures but also to identify bowtie structures for a digraph.     

矩阵的广义特征值及特征向量的神经网络方法

矩阵特征值及特征向量计算在实际问题中有广泛的应用.应用神经网络方法来计算广义特征值及对应的特征向量,给出了相应的算法,并对给出的算法在数学上进行了严格证明.并用实例验证了其正确性.     

一种新颖的用于光腔模式及光束传输模拟的特征向量法

根据有限元法单元划分的思想，提出了一种新颖的模拟光腔模式及光束传输的特征向量法. 该方法的关键之处在于基于衍射积分理论构造了一种新的光束传输矩阵，通过求解特征矩阵方程可一次性得到谐振腔的一系列特征向量，每一列特征向量即代表了腔镜上光场的一个确定模式的振幅及相位分布. 并可采用该方法模拟光场传输到腔内或腔外任意地方的场分布. 该方法将传统方法中大量的迭代过程转化成为本征积分方程特征向量的求解过程，并与初值取值无关，且可一次性求得多个模式分布，从而可方便地分析谐振腔的模式鉴别能力. 特征向量法对圆形镜共焦 关键词： 谐振腔 特征向量法 模式分布     

秩等于1的矩阵的有关性质

对秩等于1的矩阵的结构、乘法与乘幂运算、特征值与特征向量和对角化问题进行了讨论.     

甲烷—空气燃烧中NOx生成的简化模型研究

用特征值－特征向量法对甲烷－空气燃烧时ＮＯＸ生成的化学反应机理进行了简化，得到了仅由１５个基元以应组成的简化反应模型。对比计算结果表明，该模型能在较宽燃空比内与全面模型很好吻合，因而可望用于燃烧过程数值模拟时对ＮＯＸ生成的预测     

关于反中心对称矩阵的某些性质探讨

利用反中心对称矩阵的定义以及翻转矩阵等技巧,给出了反中心对称矩阵的伴随矩阵、特征值及特征向量的一些新结论.     

航天光学遥感器在轨调制传递函数神经网络评价方法

通过对航天光学遥感器在轨调制传递函数模型和遥感图像的分析,找出遥感图像中与调制传递函数有关的特征信息,采用神经网络为工具,完成利用遥感器传输下来的任意一幅地面景物图像进行调制传递函数的评价。首先模拟出包含不同调制传递函数等级的遥感图像,组成训练样本集,再从图像中分别提取出直接与调制传递函数有关的特征参量和与景物结构有关的特征参量,作为神经网络的输入,网络通过对训练样本集中模拟出的大量调制传递函数已知的遥感图像训练后,当再次输入一幅调制传递函数未知的遥感图像时,便能够正确估计出其调制传递函数值。这种方法不需要在地面铺设靶标或预先获得调制传递函数已知的同一地面景物的航空图像作为参考,只需获得任意一幅地面景物图像即可完成对遥感器调制传递函数的评价。实验结果表明,当不考虑噪声对调制传递函数的影响时,对调制传递函数的评价误差约为6%,而在考虑噪声时,评价误差约为9%。     

Stability of principal components

In this article we deal with the problem of stability of the conclusions from principal components analysis over repeated samples. We define a measure of stability for each component and investigate some of the measures properties. We then obtain the maximum likelihood estimators (MLEs) of the measures, and derive their joint limiting distributions. The MLEs of the measures turn out to be asymptotically unbiased and jointly have the multivariate normal distribution. Modified estimators are also found to reduce the amount of bias in the MLEs. To facilitate interpretation of the measures we define stability confidence level as coverage probability, and associate with each measure a stability confidence level to describe the measure in terms of probability. Finally, we investigate the stability of the components via a simulation study and compare the performance of the MLEs and the modified estimators in terms of bias and precision. This work was sponsored by a grant from the Office of Vice-President for Research at Kuwait University under project number SS049.     

A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem

The inexact Rayleigh quotient iteration (RQI) is used for computing the smallest eigenpair of a large Hermitian matrix. Under certain condition, the method was proved to converge quadratically in literature. However, it is shown in this paper that under the original given condition the inexact RQI may not quadratically converge to the desired eigenpair and even may misconverge to some other undesired eigenpair. A new condition, called the uniform positiveness condition, is given that can fix misconvergence problem and ensure the quadratic convergence of the inexact RQI. An alternative to the inexact RQI is the Jacobi-Davidson (JD) method without subspace acceleration. A new proof of its linear convergence is presented and a sharper bound is established in the paper. All the results are verified and analyzed by numerical experiments. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471074, 10771116) and the Doctoral Program of the Ministry of Education of China (Grant No. 20060003003)     

基于系统空间结构的约当规范形广义特征向量的求解方法

基于系统空间结构的分析 ,给出了求解系统约当规范形广义特征向量的新方法 .与 [1 ]中给出的方法相比 ,本文的方法更简单直接 .最后举例说明了方法的应用 .