Equisingularity of sections, $(t^r)$ condition, and the integral closure of modules

This paper uses the theory of integral closure of modules to study the sections of both real and complex analytic spaces. The stratification conditions used are the (t^) conditions introduced by Thom and Trotman. Our results include a new simple proof showing how the (t^r) conditions improve under Grassman modification, and a characterization of the (t^r) conditions using the multiplicity of a submodule of the Jacobian module of the singularity. This gives numerical criteria for Verdier Equisingularity of families of sections of the space.