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14-773: [REDACTED] Look up unfolding in Wiktionary, the free dictionary. Unfolding may refer to: Mathematics [ edit ] Unfolding (functions) , of a manifold Unfolding (geometry) , of a polyhedron Deconvolution Other uses [ edit ] Unfolding (DSP implementation) Unfolding (music) , in Schenkerian analysis Unfolding (sculpture) , by Bernhard Heiliger located near Milwaukee, Wisconsin, United States Equilibrium unfolding , in biochemistry See also [ edit ] Unfold (disambiguation) Unfoldment (disambiguation) Topics referred to by

28-676: A smooth k {\displaystyle k} -dimensional manifold, and consider the family of mappings (parameterised by N {\displaystyle N} ) given by F : M × N → R . {\displaystyle F:M\times N\to \mathbb {R} .} We say that F {\displaystyle F} is a k {\displaystyle k} -parameter unfolding of f {\displaystyle f} if F ( x , 0 ) = f ( x ) {\displaystyle F(x,0)=f(x)} for all x . {\displaystyle x.} In other words

42-647: A versal unfolding. Every versal unfolding has the property that Im ⁡ ( Φ ) ⋔ orb ⁡ ( f ) {\displaystyle \operatorname {Im} (\Phi )\pitchfork \operatorname {orb} (f)} , but the converse is false. Let x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} be local coordinates on M {\displaystyle M} , and let O ( x 1 , … , x n ) {\displaystyle {\mathcal {O}}(x_{1},\ldots ,x_{n})} denote

56-400: Is a smooth mapping from M {\displaystyle M} to R {\displaystyle \mathbb {R} } and so belongs to the function space C ∞ ( M , R ) . {\displaystyle C^{\infty }(M,\mathbb {R} ).} As we vary the parameters of the unfolding, we get different elements of the function space. Thus,

70-773: Is different from Wikidata All article disambiguation pages All disambiguation pages Unfolding (functions) Let M {\displaystyle M} be a smooth manifold and consider a smooth mapping f : M → R . {\displaystyle f:M\to \mathbb {R} .} Let us assume that for given x 0 ∈ M {\displaystyle x_{0}\in M} and y 0 ∈ R {\displaystyle y_{0}\in \mathbb {R} } we have f ( x 0 ) = y 0 {\displaystyle f(x_{0})=y_{0}} . Let N {\displaystyle N} be

84-410: Is that where " ⋔ {\displaystyle \pitchfork } " denotes " transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliates C ∞ ( M , R ) {\displaystyle C^{\infty }(M,\mathbb {R} )} – how the resulting functions will vary. There is an idea of

98-539: Is the case with unfoldings, x {\displaystyle x} and y {\displaystyle y} are called variables, and a , {\displaystyle a,} b , {\displaystyle b,} and c {\displaystyle c} are called parameters, since they parameterise the unfolding. In practice we require that the unfoldings have certain properties. In R {\displaystyle \mathbb {R} } , f {\displaystyle f}

112-547: The group of diffeomorphisms of M {\displaystyle M} etc., acts on C ∞ ( M , R ) . {\displaystyle C^{\infty }(M,\mathbb {R} ).} The action is given by ( ϕ , ψ ) ⋅ f = ψ ∘ f ∘ ϕ − 1 . {\displaystyle (\phi ,\psi )\cdot f=\psi \circ f\circ \phi ^{-1}.} If g {\displaystyle g} lies in

126-402: The orbit of f {\displaystyle f} under this action then there is a diffeomorphic change of coordinates in M {\displaystyle M} and R {\displaystyle \mathbb {R} } , which takes g {\displaystyle g} to f {\displaystyle f} (and vice versa). One property that we can impose

140-478: The ring of smooth functions. We define the Jacobian ideal of f {\displaystyle f} , denoted by J f {\displaystyle J_{f}} , as follows: Then a basis for a versal unfolding of f {\displaystyle f} is given by the quotient This quotient is known as the local algebra of f {\displaystyle f} . The dimension of

154-993: The functions f : M → R {\displaystyle f:M\to \mathbb {R} } and F : M × { 0 } → R {\displaystyle F:M\times \{0\}\to \mathbb {R} } are the same: the function f {\displaystyle f} is contained in, or is unfolded by, the family F . {\displaystyle F.} Let f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } be given by f ( x , y ) = x 2 + y 5 . {\displaystyle f(x,y)=x^{2}+y^{5}.} An example of an unfolding of f {\displaystyle f} would be F : R 2 × R 3 → R {\displaystyle F:\mathbb {R} ^{2}\times \mathbb {R} ^{3}\to \mathbb {R} } given by As

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168-646: The local algebra is called the Milnor number of f {\displaystyle f} . The minimum number of unfolding parameters for a versal unfolding is equal to the Milnor number; that is not to say that every unfolding with that many parameters will be versal. Consider the function f ( x , y ) = x 2 + y 5 {\displaystyle f(x,y)=x^{2}+y^{5}} . A calculation shows that This means that { y , y 2 , y 3 } {\displaystyle \{y,y^{2},y^{3}\}} give

182-414: The same term [REDACTED] This disambiguation page lists articles associated with the title Unfolding . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Unfolding&oldid=824202466 " Category : Disambiguation pages Hidden categories: Short description

196-527: The unfolding induces a function Φ : N → C ∞ ( M , R ) . {\displaystyle \Phi :N\to C^{\infty }(M,\mathbb {R} ).} The space diff ⁡ ( M ) × diff ⁡ ( R ) {\displaystyle \operatorname {diff} (M)\times \operatorname {diff} (\mathbb {R} )} , where diff ⁡ ( M ) {\displaystyle \operatorname {diff} (M)} denotes

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