In mathematics, and in particular singularity theory, an Ak singularity, where k ≥ 0 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 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 and a diffeomorphic change of coordinate in such that one member of the orbit is carried to any other. A function f is said to have a type Ak-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 Ak-singularities. The type Ak-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 εi = +1 from εi = −1.
In mathematics, and in particular singularity theory, an Aksingularity, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The...
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formed. The word ak = 'so..., so if...' provides many examples: akτin (= ak-τ-in) - 'so...', 'so if...', 'yea, if...' akmsin (= ak-m-s-in) - 'so if he...
stem. These suffixes are in many cases fused with the article (-a for singular and -ak for plural), which in general is required to close a noun phrase in...
has a singularity at a point z when za = 1, and also when za2 = 1. By the induction suggested by the above equations, f must have a singularity at each...
QR decomposition Ak = Qk Rk where Qk is an orthogonal matrix (i.e., QT = Q−1) and Rk is an upper triangular matrix. We then form Ak+1 = Rk Qk. Note that...
classifier, which always carries a definite/indefinite reading. The singularity or plurality of the noun is determined by the addition of the classifier...
the same in both singular and plural. Nouns such as nkyene (salt), ani (eye), sika (money), etc., are written the same in both singular and plural. The...
differential geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem. His first main...
exact sequence is a bounded exact sequence in which only the groups Ak, Ak+1, Ak+2 may be nonzero. For example, the following chain complex is a short...
Home Army (Polish: Armia Krajowa, pronounced [ˈarmja kraˈjɔva]; abbreviated AK) was the dominant resistance movement in German-occupied Poland during World...
(the subject) is third person singular (he/she/it). Zuek egunkariak erosten dizkidazue. Zu-ek you-ERG(PL) egunkari-ak newspaper-PL erosten buy-GER di-zki-da-zue...
dynamical systems; it is also a particular special case of more general singularity theory in geometry. Bifurcation theory studies and classifies phenomena...
confluence of punk, bluegrass, and subterranean hip-hop that gave life to a singular brand of Americana rap." Thompson further praised the album's impact on...
subdivided into microdistricts (Russian: микрорайоны, singular микрорайон, Turkmen: etrapçalar, singular etrapça). These are administrative units that possess...
nouns are required to be inflected if they are plural. Nouns which are singular are unmarked. Menominee has four grammatical persons: first, second, third...
null pronoun pro is the subject 'I,' while the clause-initial ak is the first person singular subject agreement morpheme. On the other hand, those who have...
PAN *aku) Nominative 2 (e.g., PAN *=ku, *[S]aku) Accusative (e.g., PAN *i-ak-ən) Genitive 1 (e.g., PAN *=[a]ku) Genitive 2 (e.g., PAN *(=)m-aku) Genitive...
Missile tracking and control Bass Tilt – MR-123, fire control radar of the AK-630 close-in weapon system Bell Clout – Electronic warfare jamming radar Bell...