Ak Singularity

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 $f: \R^n \to \R$ be a smooth function. We denote by $\Omega (\R^n,\R)$ the infinite-dimensional space of all such functions. Let $\operatorname(\R^n)$ denote the infinite-dimensional Lie group of diffeomorphisms $\R^n \to \R^n,$ and $\operatorname(\R)$ the infinite-dimensional Lie group of diffeomorphisms $\R \to \R.$ The product group $\operatorname(\R^n) \times \operatorname(\R)$ acts on $\Omega (\R^n,\R)$ in the following way: let $\varphi : \R^n \to \R^n$ and $\psi : \R \to \R$ be diffeomorphisms and $f: \R^n \to \R$ any smooth function. We define the group action as follows:
:$(\varphi,\psi)\cdot f := \psi \circ f \circ \varphi^$
The orbit of , denoted , of this group action is given by
:$\mbox(f) = \ \ .$

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
:$f(x_1,\ldots,x_n) = 1 + \varepsilon_1x_1^2 + \cdots + \varepsilon_x^_ \pm x_n^$
where $\varepsilon_i = \pm 1$ 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 .

References

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Singularity theory

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