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In mathematics, and in particular singularity theory an ''A''''k'', where ''k'' ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold. Let ''f'' : R''n'' → R be a smooth function. We denote by Ω(R''n'',R) the infinite-dimensional space of all such functions. Let diff(R''n'') denote the infinite-dimensional Lie group of diffeomorphisms R''n'' → R''n'', and diff(R) the infinite-dimensional Lie group of diffeomorphisms R → R. The product group diff(R''n'') × diff(R) acts on Ω(R''n'',R) in the following way: let φ : R''n'' → R''n'' and ψ : R → R be diffeormorphisms and ''f'' : R''n'' → R any smooth function. We define the group action as follows: : The orbit of ''f'', denoted orb(''f''), 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 R''n'' and a diffeomorphic change of coordinate in R such that one member of the orbit is carried to any other. A function ''f'' is said to have a type ''A''''k''-singularity if it lies in the orbit of : where and ''k'' ≥ 0 is an integer. By a normal form we mean a particularly simple representative of any given orbit. The above expressions for ''f'' give normal forms for the type ''A''''k''-singularities. The type ''A''''k''-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 ''f''. This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish ε''i'' = +1 from ε''i'' = −1. == References == * 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ak singularity」の詳細全文を読む スポンサード リンク
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