  This paper mainly studies nonnegativity decision of forms based on variable
substitutions. Unlike existing research, the paper regards simplex subdivisions
as new perspectives to study variable substitutions, gives some subdivisions of
the simplex T_n, introduces the concept of convergence of the subdivision
sequence, and presents a sufficient and necessary condition for the convergent
self-similar subdivision sequence. Then the relationships between subdivisions
and their corresponding substitutions are established. Moreover, it is proven
that if the form F is indefinite on T_n and the sequence of the successive
L-substitution sets is convergent, then the sequence of sets {SLS^(m)(F)} is
negatively terminating, and an algorithm for deciding indefinite forms with a
counter-example is obtained. Thus, various effective substitutions for deciding
positive semi-definite forms and indefinite forms are gained, which are beyond
the weighted difference substitutions characterized by "difference".
