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{diff}(\R^n) denote the infinite-dimensional Lie group of diffeomorphisms \R^n \to \R^n, and \operatorname{diff}(\R) the infinite-dimensional Lie group of diffeomorphisms \R \to \R. The product group \operatorname{diff}(\R^n) \times \operatorname{diff}(\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^{-1} The orbit of , denoted , of this group action is given by : \mbox{orb}(f) = \{ \psi \circ f \circ \varphi^{-1} : \varphi \in \mbox{diff}(\R^n), \psi \in \mbox{diff}(\R ) \} \ . 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.…

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{diff}(\R^n) denote the infinite-dimensional Lie group of diffeomorphisms \R^n \to \R^n, and \operatorname{diff}(\R) the infinite-dimensional Lie group of diffeomorphisms \R \to \R. The product group \operatorname{diff}(\R^n) \times \operatorname{diff}(\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^{-1} The orbit of , denoted , of this group action is given by \mbox{orb}(f) = \{ \psi \circ f \circ \varphi^{-1} : \varphi \in \mbox{diff}(\R^n), \psi \in \mbox{diff}(\R ) \} \ . 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_{n-1}x^{2}_{n-1} \pm x_n^{k+1} 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 .

Text derived from https://en.wikipedia.org/w/index.php?oldid=1366566348 under CC BY-SA 4.0; modifications have been made.