三维电阻抗成像的体积元方法的数值模拟和分析

电阻抗成像是一类椭圆方程反问题,本文在三维区域上对其进行数值模拟和分析.对于椭圆方程Neumann边值正问题,本文提出了四面体单元上的一类对称体积元格式,并证明了格式的半正定性及解的存在性;引入单元形状矩阵的概念,简化了系数矩阵的计算;提出了对电阻率进行拼接逼近的方法来降低反问题求解规模,使之与正问题的求解规模相匹配;导出了误差泛函的Jacobi矩阵的计算公式,利用体积元格式的对称性和特殊的电流基向量,将每次迭代中需要求解的正问题的个数降到最低.一系列数值实验的结果验证了数学模型的可靠性和算法的可行性.本文所提出的这些方法,已成功应用于三维电阻抗成像的实际数值模拟.     

广义Minkowski不等式的进一步改进

针对"亚正定阵理论(Ⅱ)"一文的广义Minkowski不等式不成立问题,在已有的修正结果基础上,给出一种完整的修正结果;并更正了"亚正定阵的几个开问题及一些不等式"一文中有关的错误结论;用反例说明"广义正定矩阵的进一步研究"一文中的有关结论是错误的,给出完整的修正结果.     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Conditioning of finite difference schemes for a singularly perturbed convection-diffusion parabolic equation

In the case of the boundary value problem for a singularly perturbed convection-diffusion parabolic equation, conditioning of an ε-uniformly convergent finite difference scheme on a piecewise uniform grid is examined. Conditioning of a finite difference scheme on a uniform grid is also examined provided that this scheme is convergent. For the condition number of the scheme on a piecewise uniform grid, an ε-uniform bound O(δ ₁ ^?2 lnδ ₁ ^?1 + δ ₀ ^?1 ) is obtained, where δ₁ and δ₀ are the error components due to the approximation of the derivatives with respect to x and t, respectively. Thus, this scheme is ε-uniformly well-conditioned. For the condition number of the scheme on a uniform grid, we have the estimate O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ); this scheme is not ε-uniformly well-conditioned. In the case of the difference scheme on a uniform grid, there is an additional error due to perturbations of the grid solution; this error grows unboundedly as ε → 0, which reduces the accuracy of the grid solution (the number of correct significant digits in the grid solution is reduced). The condition numbers of the matrices of the schemes under examination are the same; both have an order of O(ε^?1δ ₁ ^?2 + δ ₀ ^?1 ). Neither the matrix of the ε-uniformly convergent scheme nor the matrix of the scheme on a uniform grid is ε-uniformly well-conditioned.     

Localization of the eigenvalues of a pencil of positive definite matrices

Let A and B be real square positive definite matrices close to each other. A domain S on the complex plane that contains all the eigenvalues λ of the problem Az = λBz is constructed analytically. The boundary ?S of S is a curve known as the limacon of Pascal. Using the standard conformal mapping of the exterior of this curve (or of the exterior of an enveloping circular lune) onto the exterior of the unit disc, new analytical bounds are obtained for the convergence rate of the minimal residual method (GMRES) as applied to solving the linear system Ax = b with the preconditioner B.     

A system of equations for describing cocyclic Hadamard matrices

Given a basis for 2‐cocycles over a group G of order , we describe a nonlinear system of 4t‐1 equations and k indeterminates over , whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that ( ) is a solution of the system if and only if the 2‐cocycle gives rise to a cocyclic Hadamard matrix . Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups and . A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups , we have found that it suffices to check t instead of the 4t rows of , to decide the Hadamard character of the matrix (for a special class of cocycles f). © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 276–290, 2008     

Asymptotic enumeration of dense 0-1 matrices with specified line sums

Let s=(s₁,s₂,…,sm) and t=(t₁,t₂,…,tn) be vectors of non-negative integers with . Let B(s,t) be the number of m×n matrices over {0,1} with jth row sum equal to sj for 1?j?m and kth column sum equal to tk for 1?k?n. Equivalently, B(s,t) is the number of bipartite graphs with m vertices in one part with degrees given by s, and n vertices in the other part with degrees given by t. Most research on the asymptotics of B(s,t) has focused on the sparse case, where the best result is that of Greenhill, McKay and Wang (2006). In the case of dense matrices, the only precise result is for the case of equal row sums and equal column sums (Canfield and McKay, 2005). This paper extends the analytic methods used by the latter paper to the case where the row and column sums can vary within certain limits. Interestingly, the result can be expressed by the same formula which holds in the sparse case.     

A note on the largest eigenvalue of a large dimensional sample covariance matrix

Let {v_ij; i, J = 1, 2, …} be a family of i.i.d. random variables with E(v₁₁⁴) = ∞. For positive integers p, n with p = p(n) and p/n → y > 0 as n → ∞, let M_n = (1/n) V_n V_n^T , where V_n = (v_ij)_{1 ≤ i ≤ p, 1 ≤ j ≤ n}, and let λ_max(n) denote the largest eigenvalue of M_n. It is shown that a.s. This result verifies the boundedness of E(v₁₁⁴) to be the weakest condition known to assure the almost sure convergence of λ_max(n) for a class of sample covariance matrices.     

多个相异性矩阵的数据分析

本文给出了分析多个相异性矩阵的三种方法.首先找到了一种图表示,使我们对所有相异性矩阵有一个总体的了解;其次定义了一个新的相异性矩阵,它可以看作是对所有原始相异性矩阵的一个折衷处理;最后提出了一种MIMU方法.在文中我们还对由上述方法得到的坐标图进行了比较.     

On the robustness of least squares procedures in regression models

The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.