Digital Signal Processing Reference
In-Depth Information
C
n
Now a cochain complex structure
(
{
(
A
)
}
,
d
)
in
C
(
A
)
is defined as it follows.
C
0
C
1
d
:
(
A
)
→
(
A
)
is defined by
dZ
(
X
)
=−
XZ
+
ZX
.
C
n
C
n
+
1
When
n
is positive
d
:
(
A
)
→
(
A
)
is defined by
X
1
,...,
X
j
,...,
j
(
d
θ)(
X
1
,...,
X
n
+
1
)
=
n
(
−
1
)
[
X
j
(θ(
X
n
+
1
))
1
≤
j
≤
i
=
j
θ(..
X
j
,...,
−
X
j
X
i
,...,
X
n
+
1
)
X
1
,...,
X
j
,...,
+
(θ(
X
n
,
X
j
))
X
n
+
1
]
The linear map
d
satisfies
d
2
0. For details regarding the last claim the reader
is referred to [
17
]. Thus the couple
=
(
{
V
n
}
n
∈Z
,
d
)
is a cochain complex. It is the
so-called
A
-valued KV complex of the algebra
A
.
Its cohomology is called
A
-valued
KV cohomology of the locally flat manifold
(
M
,
D
)
.
=
(
X
(
),
)
Actually both associator and KV anomaly of A
M
D
are real homoge-
neous maps of degree
2
from C
2
to C
3
(
)
(
).
A
A
4.2.1 H
2
(A) and the Rigidity Problem
Actually rigidity of a mathematical structure and solidness in mechanics look alike.
Probability theory may be understood as useful tools to match with the solidness of
informations, (viz results of random experiences). Roughly speaking, geometry deals
with invariants under dynamics. Topology deals with proximity concepts. Intuitively
rigidity suggests invariance under small perturbations. Colloquially speaking, iso-
morphic models are the same model. So, rigidity of a parametrized model means no
change under small variation of the parameter. A conjecture of Muray Gerstenhaber
says this, [
10
]:
Every restrict theory of deformation generates its proper cohomology theory.
What a non specialist might keep is the following. Whenever a deformation
process DEF is controlled by a cohomology theory whose cohomology is denoted
H
(
, non null elements of
H
2
are non trivial deformations, viz irre-
versible deformations. So 2-cocycles look like infinitesimal deformations of the
process DEF and elements of
H
2
DEF
)
(
DEF
)
(
DEF
)
may be regarded as obstructions to the
rigidity request.
Statistical models are parameters for deciding after random considerations.
So, to be concerned with the rigidity of statistical invariants is a relevant request.
The framework of statistical models is locally flat (or affinity flat) geometry. They
