Some geometric properties of successive difference substitutions

Abstract

This paper provides a new, geometric perspective to study successive difference substitutions, and proves that the sequence of the successive difference substitution sets is not convergent. An interesting result that a given k-dimensional rational hyperplane can be transformed to a k-dimensional coordinate hyperplane of new variables by finite steps of successive difference substitutions is presented. Moreover, a sufficient condition for the sequence of the successive difference substitution sets of a form being not terminating is obtained. That is, a class of polynomials which cannot be proved to be positive semi-definite by the successive difference substitution method are obtained.

Stability Analysis of Linear Uncertain Systems via Checking Positivity of Forms on Simplices

“…≥ q jm ≥ 0}. (11) f 1 (q) or f 2 (q) may be not homogenous on q, if so, we need to homogenize them, i.e., we introduce a new variable q 0 , and let h 1 (q 0 , q 1 , . .…”