this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...

Abstract In this paper we discuss the convergence of distances associated to converging structures Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under mild controllability assumption limit vector-fields structure, equi-Lipschitz that converge uniformly compact subsets, locally Carnot-Carathéodory distance. case in which distance is boundedly compact...

‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in ...

For an algebra A of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that A separates points in the sense that for each distinct pair x, y ∈ X, there exists an f ∈ A such that f(x) 6= f(y). If A does not separate points, it is known that there exists an algebra  on a compact Hausdorff space (X̂, τ̂) that does separate points such that th...