A Theorem on Homotopy Paths,
Abstract
Consider the set of points x a member of R to the n + 1st power satisfying H(x) = 0, where H: R to the n + 1st power yields R to the nth power is a C squared function and 0 is a regular value. This set, 1/H (0), is a C to the first power one-dimensional manifold, and each component can be described by a curve x(theta). In this note a theorem is proved which is directly related to and motivated by a result on piecewise linear functions. This theorem relates the signs of the derivates x dot(i) (theta) to the signs of the determinants of submatrices of the Jacobian matrix H'. Applications to solving nonlinear equations are given.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 01, 1977
- Accession Number
- ADA044666
Entities
People
- C. B. Garcia
- F. J. Gould
Organizations
- University of Chicago