Abstract In this chapter we discuss differentiable structures on topological manifolds. particular, transversality, tubular neighbourhoods, index theory, the degree and theorems by Poincaré Hopf Bobylev Krasnosel’skiĭ.

This paper proposes an optimized topology control scheme

Last lecture, we spent a considerable amount of effort defining manifolds. We like manifolds because they are locally Euclidean. So, even though it is hard for us to reason about them globally, we know what to do in small neighborhoods. It turns out that this ability is all we really need. This is rather fortunate, because we suddenly have spaces with more interesting structure than the Euclide...

‎‎Let $R$ be a commutative ring with identity and let $M$ be an $R$-module‎. ‎We define the primary spectrum of $M$‎, ‎denoted by $mathcal{PS}(M)$‎, ‎to be the set of all primary submodules $Q$ of $M$ such that $(operatorname{rad}Q:M)=sqrt{(Q:M)}$‎. ‎In this paper‎, ‎we topologize $mathcal{PS}(M)$ with a topology having the Zariski topology on the prime spectrum $operatorname{Spec}(M)$ as a sub...