A Hermitian Curvature Flow

Chinese Annals of Mathematics, Series B - A Hermitian curvature flow on a compact Calabi-Yau manifold is proposed and a regularity result is obtained. The solution of the flow, if exists, is a...     

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

     

FAST PARALLELIZABLE METHODS FOR COMPUTING INVARIANT SUBSPACES OF HERMITIAN MATRICES

We propose a quadratically convergent algorithm for computing the invariant subspaces of an Hermitian matrix. Each iteration of the algorithm consists of one matrix-matrix multiplication and one QR decomposition. We present an accurate convergence analysis of the algorithm without using the big O notation. We also propose a general framework based on implicit rational transformations which allows us to make connections with several existing algorithms and to derive classes of extensions to our basic algorithm with faster convergence rates. Several numerical examples are given which compare some aspects of the existing algorithms and the new algorithms.     

Curves covered by the Hermitian curve

A family of maximal curves is investigated that are all quotients of the Hermitian curve. These curves provide examples of curves with the same genus, the same automorphism group and the same Weierstrass semigroup at a generic point, but that are not isomorphic.     

UNIFORM ESTIMATES OF SOLUTIONS OF δ^--EQUATION ON STEIN MANIFOLDS

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman-Leray-Norguet formula for （p, q） differential forms on an open set with C^1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of δ^--equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of δ^--equation for （p, q） differential forms on a Stein manifold.     

A few uncaught universal Hermitian forms

We will complete the list of universal binary Hermitian forms over imaginary quadratic fields by investigating three Hermitian forms missed by previous researchers.

     

Reducing complex matrices to condensed forms by unitary congruence transformations

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix.     

Spheres in Hermitian Symmetric Spaces and Flag Manifolds

This paper contains a proof of the following property of compact irreducible Hermitian symmetric spaces. If H=G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H)=|E ₁,...,E _m is the fixed point set of T on H, then for each pair E _i , E _j there is a 2-dimentional sphere N _ij H such that E _i and E _j are antipodal points of N _ij.     

Backward errors for eigenvalues and eigenvectors of Hermitian,skew-Hermitian,H-even and H-odd matrix polynomials

We discuss the perturbation analysis for eigenvalues and eigenvectors of structured homogeneous matrix polynomials with Hermitian, skew-Hermitian, H-even and H-odd structure. We construct minimal structured perturbations (structured backward errors) such that an approximate eigenvalue and eigenvector pair (finite or infinite eigenvalues) is an exact eigenvalue eigenvector pair of an appropriately perturbed structured matrix polynomial. We present various comparisons with unstructured backward errors and previous backward errors constructed for the non-homogeneous case and show that our results generalize previous results.