Ak singularity
Introduction In mathematics, and in particular singularity theory, an ''' singularity''', where  is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.Let  be a smooth function. We denote by  the infinite-dimensional space of all such functions. Let  denote the infinite-dimensional Lie group of diffeomorphisms  and  the infinite-dimensional Lie group of diffeomorphisms  The product group  acts on  in the following way: let  and  be diffeomorphisms and  any smooth function. We define the group action as follows::The orbit of , denoted , of this group action is given by:The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in  and a diffeomorphic change of coordinate in  such that one member of the orbit is carried to any other. A function  is said to have a type -singularity if it lies in the orbit of:where  and  is an integer.By a '''normal form''' we mean a particularly simple representative of any given orbit. The above expressions for  give normal forms for the type -singularities. The type -singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of .This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish  from .
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