Univalence of Nonlinear Mappings : A Qualitative Approach

This note is a sequel to the previous one published in this journal (Vol. 30, No.1). In that article, we used one of mean value theorems to prove the univalence of a nonlinear mapping based on the qualitative regularity ofthe Jacobian matrix. The qualitative regularity is a property of a matrix whose regularity is shown to be valid by using only the sign patterns of mappings involved. In this note, we extend the result into a vector space over an integral domain. The vectors themselves are of n-tuples of elements in the integral domain. This integral domain is totally ordered, and some natural properties are assumed concerning this order. First two lennnata are given, and the first one is in fact a sort of mean value theorem for mappings from a direct product of discrete spaces into a discrete space, and utilizes mathematical induction. The second lemma depends on the fact that theory of matrices and determinants can be constructed also on a ring except for inverse matrix. Finally, our main proposition derives from the very integrity of a given domain. Another merit of the result is that the domain of a mapping need not be convex, and can be even a set oflattice points.

研究ノート (Note)