Po-Fang Hsieh

A global analysis of matrices of functions of several variables

Let x = (x1, xs , 0-e) x,) be an r-dimensional vector, where q are real or complex numbers, and let 9 be the domain defined by I xj I < l (j = 1,2, *.., r). (l-1) Consider a family 3 of functions of x which are defined on 9, and consider an n by TZ matrix A(x) whose elements belong to this family 9. Let p(h; x) be the characteristic polynomial of A(x): p(h; x) = det (A(x) Al,), (1.2) where 1, is the TZ by TZ identity matrix, h is a complex variable, and det means the determinant. We assume that the polynomial p(X; X) is factored as PC& 4 = P&t x) PA& 4 for every x in 9, (1.3) where

Construction of a fundamental matrix solution at a singular point of the first kind by means of the SN decomposition of matrices

Abstract It is known that any matrix can be decomposed into a diagonalizable part and a nilpotent part. We call this the SN decomposition. We can derive the SN decomposition quite easily with a computer. Generalizing the SN decomposition to particular matrices of infinite order, we explain basic steps of construction of a linear transformation which reduces a given system of linear meromorphic ordinary differential equations to a normal form at a singular point of the first kind. Some examples are given utilizing Mathematica. We also show that the same idea produces a block-diagonalization of a given system at a singular point of the second kind.

Fundamental Theorems of Ordinary Differential Equations

In this chapter, we explain the fundamental problems of the existence and uniqueness of the initial-value problem

Singularities of the Second Kind

\frac{{d\vec y}}{{dt}} = \vec f\left( {t,\vec y} \right), \vec y\left( {{t_0}} \right) = {\vec c_0}

The Second-Order Differential Equation \frac{{{{d}^{2}}x}}{{d{{t}^{2}}}} + h\left( x \right)\frac{{dx}}{{dt}} + g\left( x \right) = 0

(P) in the case when the entries of

Boundary-Value Problems of Linear Differential Equations of the Second-Order

\frac{{d\vec{y}}}{{dt}} = f\left( {t,\vec{y}} \right),{\text{ }}\vec{y}\left( {{{t}_{0}}} \right) = {{\vec{c}}_{0}}