High Quality Content by WIKIPEDIA articles! In mathematics, and in particular singularity theory an Ak, where k ? 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold. Let f : Rn ? R be a smooth function. We denote by ? the infinite dimensional space of all such functions. Let diff denote the infinite dimensional Lie group of diffeomorphisms Rn ? Rn, and diff the infinite dimensional Lie group of diffeomorphisms R ? R. The product group diff x diff acts on ? in the following way: let ? : Rn ? Rn and ? : R ? R be diffeormorphisms and f : Rn ? R any smooth function.

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