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where A
i
and A
iC1
are the two adjacent fuzzy sets to which the weight
w
ij
belongs,
with centers c
A
i
and c
A
i
C
1
, respectively (Fig.
10.2
). The diagonal elements of the
W
i
are taken to be 0 (no self-feedback in neurons is considered), and
matrices
denotes that the membership value of the element
w
ii
;iD 1; ;3is indifferent.
The matrices which have as elements the membership values of the weights
w
ij
are denoted by M
i
and the associated jjL
1
jj are calculated. Each L
1
norm is divided
by the number of the non-diagonal elements of the matrices M
i
. This results to
0
1
0
1
0
A
i
12
a
A
i
0a
A
i
12
a
A
i
13
13
@
A
@
A
C
12
0
A
i
C
1
12
C
13
C
23
3
a
A
i
12
0a
A
i
12
C
13
23
C
1
3
a
A
i
W D
23
23
a
A
i
13
a
A
i
13
a
A
i
C
1
a
A
i
23
0
0
0
1
0
1
23
a
A
i
C
1
13
12
a
A
i
C
1
0
A
i
12
0
A
i
13
@
A
C
@
A
12
13
C
23
C
1
3
12
13
23
C
2
3
0
A
i
C
1
23
a
A
i
12
0
A
i
23
a
A
i
12
C
a
A
i
C
1
13
a
A
i
C
1
13
a
A
i
C
1
23
a
A
i
23
0
0
0
1
0
1
0
A
i
C
1
12
0
A
i
C
1
12
a
A
i
13
a
A
i
13
@
A
C
@
A
C
12
C
13
C
23
C
1
3
12
C
13
23
C
2
3
a
A
i
C
1
12
0
A
i
C
1
23
a
A
i
C
1
12
0
A
i
23
13
a
A
i
C
1
a
A
i
13
a
A
i
23
a
A
i
0
0
0
@
1
A
C
0
@
1
A
23
0
A
i
C
1
12
a
A
i
C
1
0
A
i
C
1
12
a
A
i
C
1
13
13
C
12
13
C
23
C
2
3
a
A
i
C
1
12
12
13
23
C
3
3
a
A
i
C
1
12
0
A
i
C
1
23
0
A
i
23
a
A
i
C
1
13
a
A
i
C
1
13
a
A
i
C
1
a
A
i
23
0
0
23
The following lemma holds [
165
]:
Lemma 3.
The
jjL
1
jj
of the matrices
M
i
(i.e.
P
iD1
P
jD1
j
ij
j
) divided by the
number of the non-diagonal elements , i.e.
N.N 1/
and by
2
N1
, where
N
is
the number of neurons, equals unity.
X
N
X
jD1
j
ij
jD1
N
1
N.N 1/2
N1
(10.8)
iD1
Proof.
There are 2
N1
couples of matrices M
i
. Due to the strong fuzzy partition
there are always two matrices M
i
and M
j
with complementary elements, i.e. .
w
ij
/
and 1 .
w
ij
/. Therefore the sum of the corresponding L
1
norms jjM
i
jj C jjM
j
jj
normalized by the number of the non-zero elements (i.e., N.N 1/) equals
unity. Since there are 2
N1
couples of L
1
norm sums
jjM
i
jj C jjM
j
jj
it holds
2
N
1
P
2
N
1
iD1
jjM
i
jj D 1. This normalization procedure can be used to derive the
membership values of the weight matrices W
i
.
Remark 1.
The decomposition of the matrix W into a group of matrices W
i
reveals
the existence of non-observable attractors. According to Lemmas
1
and
2
these
attractors coincide with the eigenvectors
v
i
of the matrices
W
i
. Thus the patterns
