Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

k＜1时Cartan-Hartogs域上的极值问题

本文讨论两种类型的极值问题,其中一种类型的极值问题可以认为是复平面上经典的Schwarz引理在高维的一个推广;另一种类型的极值是某空间上的度量,可以用来考虑域的双全纯等价分类问题．在本文中,k＜1时Cartan-Hartogs域与单位超球间的极值与极值映照被得到。     

Minimal circumscribed Hermitian ellipsoid of Hartogs domains and an application to an extremal problem

In this paper we construct circumscribed Hermitian ellipsoids of Hartogs domains of least volume and as an application, we obtain the Carath′eodory extremal mappings between the Hartogs domains and the unit ball, and also give an explicit formula for calculating the extremal values.     

Classification of Razor Blades to the filtration equation—the sublinear case

We study nonnegative solutions of the filtration equation u_t=Δ?(u) in , where ? is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at |x|=0 for t>0, and start with zero initial data. We first prove a nonexistence result when ? is too sublinear and we show an axial trace result in the other case: there exist a time τ≡τ(u) and a Radon measure λ on [0,τ) such that

     

四阶非奇异截断复矩矩阵的延拓问题

讨论Curto-Fialkow所给出的四阶截断复矩问题,即给一个复数序列γ≡γ~（（4））：γ_（00）,γ_（0）1,γ_（10）,γ_（02）,γ_（11）,γ_（20）,γ_（03）,γ_（12）,γ_（21）,γ_（30）,γ_（04）,γ_（13）,γ_（22）,γ_（31）,γ_（40）,其中γ_（00）〉0,γ_（ij）=y_（ji）,找到一个正的Borel测度使得γ_（ij）=∫-izz~jdμ（0≤i＋j≤4）成立;得到了四阶非奇异截断复矩矩阵M（2）的平坦延拓存在的充分必要条件及在特殊情况下的解,并举例进行了验证.     

Extremal problems on the Hua domain of the first type

In this paper we study some extremal problems between the Hua domain of the first type and the unit ball.     

Nonnegative realization of spectra having negative real parts

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list σ except for one (the Perron eigenvalue) have real parts smaller than or equal to zero.     

Hermitian非自反矩阵的两类反问题

记J为一广义反射矩阵,HAJn×n为关于J的n阶Hermitian非自反矩阵的集合.本文考虑如下两个问题:问题Ⅰ给定X,B∈n×m,求A∈HAJn×n,使得‖AX-B‖=min.问题Ⅱ给定X∈n×m,B∈n×n,求A∈HAJn×n,使得XHAX=B.首先利用奇异值分解讨论问题Ⅰ的解的通式,然后利用广义奇异值分解得到了问题Ⅱ有解的充分必要条件和解的通式,最后给出问题Ⅰ和Ⅱ的逼近解的具体表达式.     

Changing poles in the rational Lanczos method for the Hermitian eigenvalue problem

Applications such as the modal analysis of structures and acoustic cavities require a number of eigenvalues and eigenvectors of large‐scale Hermitian eigenvalue problems. The most popular method is probably the spectral transformation Lanczos method. An important disadvantage of this method is that a change of pole requires a complete restart. In this paper, we investigate the use of the rational Krylov method for this application. This method does not require a complete restart after a change of pole. It is shown that the change of pole can be considered as a change of Lanczos basis. The major conclusion of this paper is that the method is numerically stable when the poles are chosen in between clusters of the approximate eigenvalues. Copyright © 2001 John Wiley & Sons, Ltd.     

Hausdorff moment problem and fractional moments: A simplified procedure

The purpose of this paper is the recovering of a probability density function with support [0, 1] from the knowledge of its sequence of moments, i.e. the classical Hausdorff moment problem. To avoid the well-known ill-conditioning, firstly the moment curve is calculated from the assigned sequence of moments; next the unknown density is approximated by Maximum Entropy (MaxEnt) technique selecting some proper points on the moment curve. Exploiting convergence in entropy, a simplified quick procedure is suggested to recover the approximate density. An application to Laplace Transform inversion is illustrated.     

Linear restriction problem of Hermitian reflexive matrices and its approximation

In this paper, we consider a linear restriction problem of Hermitian reflexive matrices and its approximation. By using the properties and structure of Hermitian reflexive matrices and the special properties of reflexive vectors and anti-reflexive vectors, we convert the linear restriction problem to an equivalence problem trickily, which is a special feature of this paper and is a different method from other articles. Then we solve this problem completely and also obtain its optimal approximate solution. Moreover, an algorithm provided for it and the numerical examples show that the algorithm is feasible.     

Existence theorems for hyperbolic differential inclusions in Banach algebras

In this paper, an existence theorem for a certain hyperbolic differential inclusions in Banach algebras is proved under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions for Carathéodory as well as discontinuous hyperbolic differential inclusions is also proved under certain monotonicity conditions.     

关于Hermitian矩阵的特征的注记

应用有限个简单的检验矩阵,给出了复矩阵A为Hermitian矩阵的充要条件.     

一类Hermitian鞍点矩阵的特征值估计

本文研究了一类大型稀疏Hermitian鞍点线性系统Az=(B E E* 0)(x y)=(f g)=b系数矩阵的特征值,其中B∈C~(p×p)是Hermitian正定阵矩阵,E∈C~(p×q)是列降秩.本文分别给出了该系数矩阵正特征值与负特征值界的一个估计式,同时通过数值算例验证本文所给出的特征值界的估计是合理且有效的.     

Necessary and sufficient conditions for the existence of complementary solutions and characterizations of the matrix classesQ andQ 0

We simplify a result by Mangasarian on the existence of solutions to the linear complementarity problem. The simplified condition gives a new geometric interpretation of the result. When used to characterize the matrix classesQ andQ ₀, our condition suggests a finitely checkable sufficient condition forP andP ₀.This work was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173, and by general research development funds provided by the Georgia Institute of Technology.     

Asymptotics of orthogonal polynomials beyond the scope of Szegő’s theorem

First, we give a simple proof of a remarkable result due to Videnskii and Shirokov: let B be a Blaschke product with n zeros; then there exists an outer function φ, φ(0) = 1, such that ‖(Bφ)′‖ ? Cn, where C is an absolute constant. Then we apply this result to a certain problem of finding the asymptotics of orthogonal polynomials.