RESEARCH ANNOUNCEMENTS On a Generalization of the c-numerical Range

Let Cn×n be the set of　n×n complex matrices and (?)n　the set of　orthonormal　n-tuples ofvectors in Cn.For a vector c in Cn　and a matrix A in Cn×n,the c-numerical range of　A is theset     

COMPUTING THE NEAREST BISYMMETRIC POSITIVE SEMIDEFINITE MATRIX UNDER THE SPECTRAL RESTRICTION

Let A and C denote real n × n matrices. Given real n-vectors x1, ... ,xm, m ≤ n, and a set of numbers L = {λ1,λ2,... ,λm}. We describe (I) the set (?) of all real n × n bisymmetric positive seidefinite matrices A such that Axi is the "best" approximate to λixi, i = 1,2,...,m in Frobenius norm and (II) the Y in set (?) which minimize Frobenius norm of ||C - Y||.An existence theorem of the solutions for Problem I and Problem II is given and the general expression of solutions for Problem I is derived. Some sufficient conditions under which Problem I and Problem II have an explicit solution is provided. A numerical algorithm of the solution for Problem II has been presented.     

Ｙ-数值半径的范数性质(英文)

Given an n×n complex matrix A and an n-dimensional complex vector y=(ν1 , ··· , νn ), the y-numerical radius of A is the nonnegative quantity ry(A)=max{n∑j=1ν*jAx︱:Axj︱: x*jxj=1,xj ∈Cn}.Here Cn is an n-dimensional linear space overthe complex field C. For y = (1, 0, ··· , 0) it reduces to the classical radius r(A) =max {|x*Ax|: x*x=1}.We show that ry is a generalized matrix norm if and only ifn∑j=1νj≠ 0.Next, we study some properties of the y-numerical radius of matrices andvectors with non-negative entries.     

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

SO(n)-Invariant Special Lagrangian Submanifolds of C~(n 1) with Fixed Loci

Let SO(n) act in the standard way on Cn and extend this action in the usual way to Cn 1 =C Cn. It is shown that a nonsingular special Lagrangian submanifold L (?) Cn 1 that is invariant under this SO(n)-action intersects the fixed C (?) Cn 1 in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A(?)C lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear pde and ultimately calls on the work of Gerard and Tahara to prove the existence of the extension.     

AN IMPROVEMENT OF A RESULT OF IVOCHKINA AND LADYZHENSKAYA ON A TYPE OF PARABOLIC MONGE-AMP\`ERE EQUATION

For the initial-hotmdary value problem about a type of parabolic Monge-Ampere equation of the form (IBVP): {-Dtu + (deD^2xu)^1/n = f(x,t), (x,t) ∈ Q = Ω×(0,T)}, u(x,t) =Ф(x,t)(x,t) ∈δpQ}, where Ω is a bounded convex domain in R^n, the result in [4] by Ivochkina and Ladyzheokaya is improved in the sense that, under assumptions that the data of the problempossess lower regularity and satisfy lower order compatibility conditions than than in [4], the existence of classical solution to (IBVP) is still established (see Theorem 1.1 below). This cannot be reallzed by only using the method in [4]. The main additional effort the authors have done is a kind of nonlinear perturbation.     

Matrix Algebra Motivated by Essentially Stochastic Matrices

A matrix of order n whose row sums are all equal to 1 is called an essentially stochastic matrix (see Johnsen [4]). We extend this notion as the following. Let F be a field of characteristic 0 or a prime greater than n. M_n(F) denotes the set of all n×n matrices over F. Let t be an elernent of F. A matrix A=(a_(ij)) in M_n(F) is called essentially t-stochastic' provided its row sums are each equal to t. We denote by R_n(t) the set of all essentially t-stochastic matrices over F. We shall mainly study R_n(0) and. Our main references are Johnson [2,4] and Kim [5].     

ON THE CONVERGENCE OF COMPLEX SOR

1　IntroductionConsider the following linear algebraic systemAX =b ( 1 .1 )with A∈Cn× n,b∈Cn,A=D-L-U,where D is diagonal,L and U are strictly lower andupper triangular matrices. And A is consistently ordered as defined by Young ( see [5] ) . Inother words,A isa particular weakly cyclic ofindex p=3 matrix( p-cyclic matrix.Asfor thediscussion when p=2 ,see[1 ] ) .The relationship between the eigenvaluesμ of the Jacobiiterative matrix B and the eigenvaluesλ of its associated successi…     

Least-Squares Solutions of the Matrix Equation A^T XA = B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n 1-j,n 1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A~TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.     

INCOMPLETE SEMI-ITERATIVE METHODS FOR SOLVING SINGULAR LINEAR OPERATOR EQUATIONS IN BANACH SPACE WITH APPLICATIONS IN MARKOV CHAIN MODELING

1　 IntroductionLet Cbe the open complex plane,let X be a complex Banach space.The set of allbounded linear operators from X into X is denoted by B[X] which is also a Banach space.If X=Cn,the n-dimensional Euclidean space,then B[X] is the set of all n×n matrices,denoted by Cn,n. We denote the spectrum of an operator T∈ B[X] byσ( T) and its resol-vent operator R( λ,T) =( λI-T) - 1 ,where I is the identity operator andλ∈C.The spectral radius of T is denoted by r( T) .N( T) and…