Biomedical Engineering Reference
In-Depth Information
where.
N
′ .is.the.size.of.
S
ck
,.that.is,.the.number.of.possible.combinations.of.the.
cutting.and.pasting.positions.such.that.the.transposon.and.moving.regions.
in.
ξ .only.contain.the.don't-care.bit.(*)..If.
ξ ∈
Z
,.
C
row
′
.is.positive.or.zero;.oth-
erwise,.it.is.negative.or.zero.
ProofofLemma4.11
Based.on.the.definition.of.
( )
i
b
mn
.given.in.Equation.(4.60),.consider
∑
∑
( )
i
( )
i
b
−
b
mn
mn
ξ
∈
Z
ξ
∈
Z
′
m
m
L L
−
L L
−
g
g
1
∑
∑
∑
(
)
.
(A.19)
=
∆ ξ
,
M
;
ξ
,
M
′
∆ ξ
(
,
R
;
ξ
,
R
)
i
c
,
k
n
c
,
k
i
c
,
k
n
c
,
k
(
L L
−
+
1)
3
n
n
n
n
n
n
n
n
g
c
=
0
c
=
0
k
∉κ
(
c
m
n
n
n
)
∑
∑
(
)
×
∆ ξ
,
I
;
ξ
,
G
−
∆ ξ
(
,
I
;
ξ
,
G
)
i
c
,
k
m
c
,
c
,
k
i
c
,
k
m
c
,
c
,
k
n
n
m n
n
n
n
m n
n
ξ
∈
Z
ξ
∈
Z
′
.
m
m
where.
(
κ = + .
Again,.for.any.particular.
c
)
(
c
,
c
L
]
n
n
n
g
ξ
∈
Z
.(
Z
odd
.or.
Z
even
),.the.following.three.cases.
m
are.obtained:
.
a.. For. any. cutting. position.
c
m
,. which. leaves. some. actual. bits. outside.
,
G
c
m n n
.in.
ξ
,.it.is.possible.to.flip.the.rightmost.of.those.actual.bits.
to.ind.a.corresponding.
c
,
k
ξ
∈
Z
′
,.such.that
m
′
∑
∑
(
)
∆ ξ
,
I
;
ξ
,
G
−
∆ ξ
(
,
I
;
ξ
,
G
)
=
0
i
c
,
k
m
c
,
c
,
k
i
c
,
k
m
c
,
c
,
k
n
n
m n
n
n
n
m n
n
ξ
∈
Z
ξ
∈
Z
′
m
m
.
b.. For.any.cutting.position.
c
m
,.which.cuts.all.the.actual.bits.of.
ξ .inside.
G
c
n
,.and.for.any.pasting.position.
k
n
,.if.some.actual.bits.in.
G
c
n
.
,
c
,
k
,
c
,
k
m n
m n
are.compared.with.don't-care.bits.in.
I
c
n n
of. ξ ,.it.is.possible.to.flip.the.
rightmost.of.those.actual.bits.to.find.a.corresponding.
,
ξ
∈
Z
′
,.such.
m
′
that
∑
∑
(
)
∆ ξ
,
I
;
ξ
,
G
−
∆ ξ
(
,
I
;
ξ
,
G
)
=
0
i
c
,
k
m
c
,
c
,
k
i
c
,
k
m
c
,
c
,
k
n
n
m n
n
n
n
m n
n
ξ
∈
Z
ξ
∈
Z
′
m
m
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