<s>
In	O
mathematics	O
,	O
an	O
unfolding	B-Algorithm
of	O
a	O
smooth	O
real-valued	O
function	O
ƒ	O
on	O
a	O
smooth	O
manifold	O
,	O
is	O
a	O
certain	O
family	O
of	O
functions	O
that	O
includesƒ	O
.	O
</s>
<s>
As	O
is	O
the	O
case	O
with	O
unfoldings	B-Algorithm
,	O
and	O
are	O
called	O
variables	O
,	O
and	O
and	O
are	O
called	O
parameters	O
,	O
since	O
they	O
parameterise	O
the	O
unfolding	B-Algorithm
.	O
</s>
<s>
In	O
practice	O
we	O
require	O
that	O
the	O
unfoldings	B-Algorithm
have	O
certain	O
properties	O
.	O
</s>
<s>
In	O
,	O
is	O
a	O
smooth	O
mapping	O
from	O
to	O
and	O
so	O
belongs	O
to	O
the	O
function	B-Algorithm
space	I-Algorithm
As	O
we	O
vary	O
the	O
parameters	O
of	O
the	O
unfolding	B-Algorithm
,	O
we	O
get	O
different	O
elements	O
of	O
the	O
function	B-Algorithm
space	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
unfolding	B-Algorithm
induces	O
a	O
function	O
The	O
space	O
,	O
where	O
denotes	O
the	O
group	O
of	O
diffeomorphisms	O
of	O
etc.	O
,	O
acts	O
on	O
The	O
action	O
is	O
given	O
by	O
If	O
lies	O
in	O
the	O
orbit	O
of	O
under	O
this	O
action	O
then	O
there	O
is	O
a	O
diffeomorphic	O
change	O
of	O
coordinates	O
in	O
and	O
,	O
which	O
takes	O
to	O
(	O
and	O
vice	O
versa	O
)	O
.	O
</s>
<s>
This	O
property	O
ensures	O
that	O
as	O
we	O
vary	O
the	O
unfolding	B-Algorithm
parameters	O
we	O
can	O
predict	O
–	O
by	O
knowing	O
how	O
the	O
orbit	O
foliates	O
–	O
how	O
the	O
resulting	O
functions	O
will	O
vary	O
.	O
</s>
<s>
There	O
is	O
an	O
idea	O
of	O
a	O
versal	O
unfolding	B-Algorithm
.	O
</s>
<s>
The	O
minimum	O
number	O
of	O
unfolding	B-Algorithm
parameters	O
for	O
a	O
versal	O
unfolding	B-Algorithm
is	O
equal	O
to	O
the	O
Milnor	O
number	O
;	O
that	O
is	O
not	O
to	O
say	O
that	O
every	O
unfolding	B-Algorithm
with	O
that	O
many	O
parameters	O
will	O
be	O
versal	O
.	O
</s>
<s>
is	O
a	O
versal	O
unfolding	B-Algorithm
.	O
</s>
<s>
A	O
versal	O
unfolding	B-Algorithm
with	O
the	O
minimum	O
possible	O
number	O
of	O
unfolding	B-Algorithm
parameters	O
is	O
called	O
a	O
miniversal	O
unfolding	B-Algorithm
.	O
</s>
<s>
An	O
important	O
object	O
associated	O
to	O
an	O
unfolding	B-Algorithm
is	O
its	O
bifurcation	O
set	O
.	O
</s>
<s>
This	O
set	O
lives	O
in	O
the	O
parameter	O
space	O
of	O
the	O
unfolding	B-Algorithm
,	O
and	O
gives	O
all	O
parameter	O
values	O
for	O
which	O
the	O
resulting	O
function	O
has	O
degenerate	O
singularities	O
.	O
</s>
<s>
Sometimes	O
unfoldings	B-Algorithm
are	O
called	O
deformations	O
,	O
versal	O
unfoldings	B-Algorithm
are	O
called	O
versal	O
deformations	O
,	O
etc	O
.	O
</s>
