线性流形上广义中心反对称矩阵的最佳逼近

本讨论在线性流形上广义反对称矩阵的最佳逼近，给出了若干有意义的结果。     

一类行列式的插值解法

应用Lagrange插值的思想给出一类典型行列式的统一解法.     

一类反对称正交反对称矩阵逆特征值问题

设P为一给定的对称正交矩阵,记AARnP={A∈Rn×n‖AT=-A,(PA)T=-PA}.讨论了下列问题:问题给定X∈Cn×m,Λ=diag(λ1,λ2,…,λm).求A∈AARPn使AX=XΛ.问题设A~∈Rn×n,求A*∈SE使‖A~-A*‖=infA∈SE‖A~-A‖,其中SE为问题的解集合,‖.‖表示Frobenius范数.研究了AARPn中元素的通式,给出了问题解的一般表达式,证明了问题存在唯一逼近解A*,且得到了此解的具体表达式.     

线性流形上反对称正交对称矩阵反问题的最小二乘解

该文讨论了线性流形上矩阵方程AX=B反对称正交对称反问题的最小二乘解及其最佳逼近问题．给出了最小二乘问题解集合的表达式，得到了给定矩阵的最佳逼近问题的解，最后给出计算任意矩阵的最佳逼近解的数值方法及算例．     

矩阵方程AXB=C的反对称解和极小范数反对称解

建立了求矩阵方程AXB=C反对称解的迭代方法.使用该方法不仅能够判断反对称解的存在性,而且在有反对称解时,能够在有限步迭代计算之后得到反对称解.选取特殊的初始矩阵,可求得极小范数反对称解.     

线性流形上矩阵方程B^TXB=D的反对称解

该文讨论了两类线性流形上矩阵方程B^TXB=D的反对称解和反对称最佳逼近解存在的条件,给出了通解的一般表达式,同时解决了解对给定矩阵的唯一最佳逼近问题.     

矩阵基础上的行列式理论

首先证明矩阵的伴随矩阵存在性定理,其次利用递推定义方法建立行列式概念,由此可构建矩阵基础上的行列式理论。     

关于外积与反对称矩阵的关系探讨与应用

本文基于外积与反对称矩阵之间的关系,探讨三维空间中的绕轴旋转、投影和反射变换的矩阵描述,将复杂的几何变换,转换为容易编程的矩阵乘法．并进一步将其推广到一般曲面的变换中,具有一定实用价值．     

线性流形上对称正交反对称矩阵反问题的最小二乘解

设P是n阶对称正交矩阵,如果n阶矩阵A满足AT=A和(PA)T=-PA,则称A为对称正交反对称矩阵,所有n阶对称正交反对称矩阵的全体记为SARnp.令S={A∈SARnp f(A)=‖AX-B‖=m in,X,B〗∈Rn×m本文讨论了下面两个问题问题Ⅰ给定C∈Rn×p,D∈Rp×p,求A∈S使得CTAC=D问题Ⅱ已知A~∈Rn×n,求A∧∈SE使得‖A~-A∧‖=m inA∈SE‖A~-A‖其中SE是问题Ⅰ的解集合.文中给出了问题Ⅰ有解的充要条件及其通解表达式.进而,指出了集合SE非空时,问题Ⅱ存在唯一解,并给出了解的表达式,从而得到了求解A∧的数值算法.     

线性流形上反对称正交对称矩阵反问题的最小二乘解

该文讨论了线性流形上矩阵方程AX=B反对称正交对称反问题的最小二乘解及其最佳逼近问题. 给出了最小二乘问题解集合的表达式, 得到了给定矩阵的最佳逼近问题的解, 最后给出计算任意矩阵的最佳逼近解的数值方法及算例.     

On the Determinant-like Function and the Vector Determinant

A generalisation of the determinant to rectangular matrices, known as the determinant-like function, has its magnitude defined previously. In this paper, we show that the determinant-like function is a rotation of the vector determinant.We further propose that this rotation is an identity transformation and thus the determinant-like function is in fact the same as the vector determinant. From this, we derive some properties of the determinant-like function.     

Primal-Dual Path-Following Algorithms for Determinant Maximization Problems With Linear Matrix Inequalities

Primal-dual path-following algorithms are considered for determinant maximization problem (maxdet-problem). These algorithms apply Newton's method to a primal-dual central path equation similar to that in semidefinite programming (SDP) to obtain a Newton system which is then symmetrized to avoid nonsymmetric search direction. Computational aspects of the algorithms are discussed, including Mehrotra-type predictor-corrector variants. Focusing on three different symmetrizations, which leads to what are known as the AHO, H..K..M and NT directions in SDP, numerical results for various classes of maxdet-problem are given. The computational results show that the proposed algorithms are efficient, robust and accurate.     

求解函数解析式的解方程组法

针对函数方程f(x)+bf(g(x))=h(x),其中f(x)为待求函数,b为任意实数,h(x)为已知函数,g(x)满足gn(x)=x.经过多次迭代换元,构建由待求函数构成的线性方程组,运用Cramer法则,可得出该函数方程有唯一解的充要条件为bn≠(-1)n,且此时可解出f(x)=1/1-(1-b)nn-1Σi=0(-b)ih(gi(x)).     

Antisymmetric flows on planar graphs

We prove that every oriented planar graph admits a homomorphism to the Paley tournament P₂₇₁ and hence that every oriented planar graph has an antisymmetric flow number and a strong oriented chromatic number of at most 271. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 200–210, 2006     

Influence Function and Efficiency of the Minimum Covariance Determinant Scatter Matrix Estimator

The minimum covariance determinant (MCD) scatter estimator is a highly robust estimator for the dispersion matrix of a multivariate, elliptically symmetric distribution. It is relatively fast to compute and intuitively appealing. In this note we derive its influence function and compute the asymptotic variances of its elements. A comparison with the one step reweighted MCD and with S-estimators is made. Also finite-sample results are reported.     

Hardy Inequality for Antisymmetric Functions

Functional Analysis and Its Applications - We consider Hardy inequalities on antisymmetric functions. Such inequalities have substantially better constants. We show that they depend on the lowest...     

浅析用函数思想解线性代数问题

函数思想方法是解决数学问题的一种重要的思想方法.本文通过六个问题的函数解法阐述函数思想在解线性代数问题中的运用.     

The Fredholm Determinant Representation for the Partition Function of the Six-Vertex Model

The six-vertex model with domain-wall boundary conditions is considered. The Fredholm determinant representation for the partition function of the model is obtained. The kernel of the corresponding integral operator depends on Laguerre polynomials. Bibliography: 13 titles.