REMARKS ON BRUALDI＇S THEOREM FOR SPECTRUM INCLUSION REGION OF AN IRREDUCIBLE MATRIX

This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition for a boundary point of Brualdi‘s inclusion region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.     

Discrete chaos in Banach spaces

This paper is concerned with chaos in discrete dynamical systems governed by continuously Frech@t differentiable maps in Banach spaces. A criterion of chaos induced by a regular nondegenerate homoclinic orbit is established. Chaos of discrete dynamical systems in the n-dimensional real space is also discussed, with two criteria derived for chaos induced by nondegenerate snap-back repellers, one of which is a modified version of Marotto's theorem. In particular, a necessary and sufficient condition is obtained for an expanding fixed point of a differentiate map in a general Banach space and in an n-dimensional real space, respectively. It completely solves a long-standing puzzle about the relationship between the expansion of a continuously differentiable map near a fixed point in an n-dimensional real space and the eigenvalues of the Jacobi matrix of the map at the fixed point.     

一个新的m×m矩阵Kaup-Newell谱问题及其相应的可积分解

A new m × m matrix Kaup-Newell spectral problem is constructed from a normal 2 × 2 matrix Kaup-Newell spectral problem, a new integrable decomposition of the Kaup-Newell equation is presented. Through this process, we find the structure of the r-matrix is interesting.     

—个新的m×m矩阵Kaup-Newell谱问题及其相应的可积分解(英文)

A new m×m matrix Kaup-Newell spectral problem is constructed from a normal 2×2 matrix Kaup-Newell spectral problem,a new integrable decomposition of the Kaup-NeweU equation is presented.Through this process,we find the structure of the r-matrix is interesting.     

GLOBAL EXPONENTIAL STABILITY IN HOPFIELD AND BIDIRECTIONAL ASSOCIATIVE MEMORY NEURAL NETWORKS WITH TIME DELAYS

Without assuming the boundedness, strict monotonicity and differentiability of the activation functions, the authors utilize the Lyapunov functional method to analyze the global convergence of some delayed models. For the Hopfield neural network with time delays, a new sufficient condition ensuring the existence, uniqueness and global exponential stability of the equilibrium point is derived. This criterion concerning the signs of entries in the connection matrix imposes constraints on the feedback matrix independently of the delay parameters. From a new viewpoint, the bidirectional associative memory neural network with time delays is investigated and a new global exponential stability result is given.     

A REMARK ON IMPLICITIZING RATIONAL CURVES WITH BASE POINTS

A simple relationship between the Bezout matrix corresponding to a rational curve with base points and the Bezout matrix corresponding to the same rational curve except that whose base points are eliminated is clarified. Based on this relationship,the author proves that the implicit equation of a rational curve with base points is the largest rton-zero leading principal minor of the gezout resultant corresponding to the rational curve assuming that the rational curve doesn‘t have triva/base point 0,and thus provides a simple approach to Jmplicitze rational curves with base points. Furthermore，as a by-product ，art algorithm is presented to compute the base points of a rational curve.     

A RELAXED HSS PRECONDITIONER FOR SADDLE POINT PROBLEMS FROM MESHFREE DISCRETIZATION＊

In this paper, a relaxed Hermitian and skew-Hermitian splitting （RHSS） preconditioner is proposed for saddle point problems from the element-free Galerkin （EFG） discretization method. The EFG method is one of the most widely used meshfree methods for solving partial differential equations. The RHSS preconditioner is constructed much closer to the coefficient matrix than the well-known HSS preconditioner, resulting in a RHSS fixed-point iteration. Convergence of the RHSS iteration is analyzed and an optimal parameter, which minimizes the spectral radius of the iteration matrix is described. Using the RHSS pre- conditioner to accelerate the convergence of some Krylov subspace methods （like GMRES） is also studied. Theoretical analyses show that the eigenvalues of the RHSS precondi- tioned matrix are real and located in a positive interval. Eigenvector distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are obtained. A practical parameter is suggested in implementing the RHSS preconditioner. Finally, some numerical experiments are illustrated to show the effectiveness of the new preconditioner.     

AN EFFECTIVE SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM FOR NONLINEAR OPTIMIZATION PROBLEMS

In this paper,a new globally convergent algorithm for nonlinear optimization prablems with equality and inequality constraints is presented. The new algorithm is of SQP type which determines a search direction by solving a quadratic programming subproblem per itera-tion. Some revisions on the quadratic programming subproblem have been made in such a way that the associated constraint region is nonempty for each point x generated by the algorithm, i. e. , the subproblems always have optimal solutions. The new algorithm has two important properties. The computation of revision parameter for guaranteeing the consistency of quadratic sub-problem and the computation of the second order correction step for superlinear convergence use the same inverse of a matrix per iteration, so the computation amount of the new algorithm will not be increased much more than other SQP type algorithms; Another is that the new algorithm can give automatically a feasible point as a starting point for the quadratic subproblems pe     

The Haar Wavelet Analysis of Matrices and Its Applications

It is well known that Fourier analysis or wavelet analysis is a very powerful and useful tool for a function since they convert time-domain problems into frequency-domain problems. Are there similar tools for a matrix? By pairing a matrix to a piecewise function,a Haar-like wavelet is used to set up a similar tool for matrix analyzing, resulting in new methods for matrix approximation and orthogonal decomposition. By using our method, one can approximate a matrix by matrices with different orders. Our method also results in a new matrix orthogonal decomposition, reproducing Haar transformation for matrices with orders of powers of two. The computational complexity of the new orthogonal decomposition is linear. That is, for an m × n matrix, the computational complexity is O(mn). In addition,when the method is applied to k-means clustering, one can obtain that k-means clustering can be equivalently converted to the problem of finding a best approximation solution of a function. In fact, the results in this paper could be applied to any matrix related problems.In addition, one can also employ other wavelet transformations and Fourier transformation to obtain similar results.     

A note for preconditioning nonsymmetric matrices

In this paper, we consider preconditioners for generalized saddle point systems with a nonsymmetric coefficient matrix. A constraint preconditioner for this systems is constructed, and some spectral properties of the preconditioned matrix are given. Our results extend the corresponding ones in [3] and [4].     

Multivariate refinement equation with nonnegative masks

This paper is concerned with multivariate refinement equations of the type where (?) is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported nonnegative sequence on Zs, and M is an s×s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation.     

ON HERMITIAN POSITIVE DEFINITE SOLUTIONS OF MATRIX EQUATION X-A^＊X^-2 A=I

The Hermitian positive definite solutions of the matrix equation X-A^＊X^-2 A=I are studied. A theorem for existence of solutions is given for every complex matrix A. A solution in case A is normal is given. The basic fixed point iterations for the equation are discussed in detail. Some convergence conditions of the basic fixed point iterations to approximate the solutions to the equation are given.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

A Note on Generic Fiedler Matrices

In this paper, we first show that a generic m×n Fiedler matrix may have 2m-n-1 kinds of factorizations which are very complicated when m is much larger than n. In this work, two special cases are examined, one is an m×n Fiedler matrix being factored as a product of (m - n) Fiedler matrices, the other is an m×(m - 2) Fiedler matrix's factorization. Then we discuss the relation among the numbers of parameters of three generic m×n, n×p and m×p Fiedler matrices, and obtain some useful results.     

树映射的ω-极限集

Let T be a tree and f be a continuous map form T into itself.We show mainly in this paper that a point x of T is an ω-limit point of f if and only if every open neighborhood of x in T contains at least nx 1 points of some trajectory,where nx equals the number of connected components of T/{x}.Then,for any open subset Gω(f) in T,there exists a positive integer m=m(G) such that at most m points of any trajectory lie outside G.This result is a generalization of the related result for maps of the interval.     

On KKT points of Celis-Dennis-Tapia subproblem

The Celis-Dennis-Tapia(CDT) problem is a subproblem of the trust region algorithms for the constrained optimization. CDT subproblem is studied in this paper. It is shown that there exists the KKT point such that the Hessian matrix of the Lagrangian is positive semidefinite, if the multipliers at the global solution are not unique. Next the second order optimality conditions are also given, when the Hessian matrix of Lagrange at the solution has one negative eigenvalue. And furthermore, it is proved that all feasible KKT points satisfying that the corresponding Hessian matrices of Lagrange have one negative eigenvalue are the local optimal solutions of the CDT subproblem.     

OPERATOR-VALUED FREE FISHER INFORMATION OF RANDOM MATRICES

The free Fisher information of an operator random matrix is studied. When the covariance of a random matrix is a conditional expectation, the free Fisher information of such a matrix is the double of this conditional expectation’s Watatani index.     

On d-row (Column) Antimagic Matrices and Subset Partitions

An m × k matrix is said to be a d-row(column) antimagic matrix if its row-sums(column-sums)form an arithmetic progression with a difference d.The goal of this paper is to obtain the existence theorems and construction methods of some d-row(column) antimagic matrices.Using these results we give the necessary and sufficient condition for the existence of an(m,d)-partition of [1,mk].     

ON THE BEST APPROXIMATION MATRIX PROBLEM AND MATRIX FOURIER SERIES

In this paper the concept of positive definite bilinear matrix moment functional. acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b], is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Kiemann -Lebesgue matrix property and the bessel-parseval matrix inequality are proved. The concept of total set vjith respect to a positive definite matrix functional is introduced , and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional, is established.     

ISOGENOUS OF THE ELLIPTIC CURVES OVER THE RATIONALS

AbstractAn elliptic curve is a pair (E,O), where ?is a smooth projective curve of genus 1 and O is a point of E, called the point at infinity. Every elliptic curve can be given by a Weierstrass equationE:y2 a1xy a3y = x3 a2x2 a4x a6.Let Q be the set of rationals. E is said to be dinned over Q if the coefficients ai, i = 1,2,3,4,6 are rationals and O is defined over Q.Let E/Q be an elliptic curve and let E(Q)tors be the torsion group of points of E denned over Q. The theorem of Mazur asserts that E(Q)tors is one of the following 15 groupsE(Q)tors　Z/mZ, m = 1,2,..., 10,12,Z/2Z × Z/2mZ, m = 1,2,3,4.We say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there is an isogeny, i.e. a morphism : E E' such that (O) = O, where O is the point at infinity.We give an explicit model of all elliptic curves for which E(Q)tors is in the form Z/mZ where m= 9,10,12 or Z/2Z × Z/2mZ where m = 4, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit m