# Dictionary Definition

unfolding n : a developmental process; "the
flowering of ante-bellum culture" [syn: flowering]

# User Contributed Dictionary

## English

### Verb

unfolding- present participle of unfold

# Extensive Definition

In mathematics, an unfolding of a function
is a certain family of functions.

Let M be a smooth
manifold and consider a smooth mapping f : M \to \mathbb. Let
us assume that for given x_0 \in M and y_0 \in \mathbb we have
f(x_0) = y_0 . Let N be a smooth k-dimensional manifold, and
consider the family of mapping (parameterised by N) given by F : M
\times N \to \mathbb . We say that F is a k-parameter unfolding of
f if F(x,0) = f(x) for all x. In other words the functions f : M
\to \mathbb and F : M \times \ \to \mathbb are the same: the
function f is contained in , or is unfolded by, the family F.

Let f : \mathbb^2 \to \mathbb be given by f(x,y)
= x^2 + y^5. An example of an unfolding of f would be F : \mathbb^2
\times \mathbb^3 \to \mathbb given by

- F((x,y),(a,b,c)) = x^2 + y^5 + ay + by^2 + cy^3.

In application we require that the unfoldings
have certain nice properties. \mathbbotice that f is a smooth
mapping from M to \mathbb and so belongs to the function
space C^(M,\mathbb). As we vary the parameters of the unfolding
we get different elements of the function space. Thus, the
unfolding induces a function \Phi : N \to C^(M,\mathbb). The space
\mbox(M) \times \mbox(\mathbb), where \mbox(M) denotes the group
of diffeomorphisms of M etc,
acts
on C^(M,\mathbb). The action is given by (\phi,\psi) \cdot f = \psi
\circ f \circ \phi^. If g lies in the orbit of f
under this action then there is a diffeomorphic change of
coordinates in M and \mathbb which takes g to f (and vise versa).
One nice property that we may like to impose is that

- \mbox(\Phi) \pitchfork \mbox(f)

There is an idea of a versal unfolding. Every
versal unfolding has the property that \mbox(\Phi) \pitchfork
\mbox(f) , but the converse is false. Let x_1,\ldots,x_n be local
coordinates on M, and let \mathcal(x_1,\ldots,x_n) denote the
ring
of smooth functions. We define the Jacobian ideal
of f, denoted by J_f as follows:

- J_f := \left\langle \frac, \ldots, \frac \right\rangle.

- \frac

- \frac = \ \ .

- F((x,y),(a,b,c)) = x^2 + y^5 + ay + by^2 + cy^3

Sometimes unfoldings are called deformations,
versal unfoldings are called versal deformations, etc.

An important object associated to an unfolding is
its bifurcation set. This set lives in the parameter space of the
unfolding, and gives all parameter values for which the resulting
function has degenerate singularities.

## References

- V. I. Arnold, S. M. Gussein-Zade & A. N. Varchenko, Singularities of differentiable maps, Volume 1, Birkhäuser, (1985).
- J. W. Bruce & P. J. Giblin, Curves & singularities, second edition, Cambridge University press, (1992).

# Synonyms, Antonyms and Related Words

advancing, amplification, anthesis, apocalypse, apparition, appearance, appearing, arising, avatar, baring, bloom, blooming, blossom, blossoming, blow, blowing, bradytelic, coming, coming into being,
coming-forth, demonstration, developing, development, dilation, disclosing, disclosure, discovering, discovery, display, efflorescence, elaboration, emergence, enactment, enlargement, epiphany, evolution, evolutional, evolutionist, evolutionistic, evolvement, evolving, exhibit, exhibition, expansion, expatiation, explication, expose, exposition, exposure, florescence, flowerage, flowering, forthcoming, full bloom,
genetic, growth, horotelic, incarnation, issuance, laying bare, manifestation, materialization,
materializing,
maturational,
maturative, maturing, occurrence, ontogenetic, opening, ostentation, patefaction, performance, phylogenetic, physiogenetic, presentation, presentment, production, progress, progressing, progression, projection, realization, removing the
veil, representation, retrospective, revealing, revealment, revelation, rise, rising, show, showing, showing forth, showing
up, showup, stripping, tachytelic, theophany, uncloaking, uncovering, unfoldment, unmasking, unrolling, unveiling, unwrapping, upgrowth, varnishing day,
vernissage,
working-out