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⎛
⎛
−
max
κ
α
j
,μ
α
j
ρ
w
j
⎞
1
ρ
m
1
⎝
GIFPWAP
w
(α
1
,α
2
,...,α
m
)
=
⎝
1
⎠
(
8
)
−
,
j
=
1
ρ
⎞
⎠
⎛
j
ρ
w
j
⎞
1
m
1
−
1
min
λ
α
j
,
⎝
1
⎠
1
−
−
−
v
α
j
=
1
where
κ
α
j
+
λ
α
j
≤
1,
j
=
1
,
2
,...,
m
.
⎛
⎛
−
min
κ
α
j
,μ
α
j
ρ
w
j
⎞
1
ρ
m
1
⎝
⎝
1
⎠
GIFPWAQ
w
(α
1
,α
2
,...,α
m
)
=
(
9
)
−
,
j
=
1
ρ
⎞
⎠
⎛
α
j
ρ
w
j
⎞
1
m
1
−
1
max
λ
α
j
,
⎝
1
⎠
1
−
−
−
v
j
=
1
where
κ
α
j
+
λ
α
j
≤
1,
j
=
1
,
2
,...,
m
.
Proof
Now we prove (2) (the others can be proven similarly). We first prove the
following equation by using mathematical induction on
m
:
w
1
F
n
ρ
w
2
F
n
ρ
w
m
F
n
ρ
κ
α
1
,λ
α
1
(α
1
)
⊕
κ
α
2
,λ
α
2
(α
2
)
⊕···⊕
κ
α
m
,λ
α
m
(α
m
)
⎛
⎝
⎛
w
j
⎞
⎠
1
1
ρ
m
−
μ
F
n
⎝
1
=
−
,
κ
α
j
,λ
α
j
(
α
j
)
j
=
1
ρ
⎞
⎠
⎛
ρ
w
j
⎞
⎠
1
1
1
m
⎝
1
1
−
−
−
−
μ
F
n
(1.337)
κα
j
,λα
j
(
α
j
)
j
=
1
where
−
1
−
κ
α
j
−
λ
α
j
n
κ
α
j
+
λ
α
j
1
μ
F
n
)
=
μ
α
j
+
κ
α
j
π
α
j
(1.338)
κ
α
j
,λ
α
j
(
α
j
−
1
−
κ
α
j
−
λ
α
j
n
κ
α
j
+
λ
α
j
1
v
F
n
)
=
v
α
j
+
λ
α
j
π
α
j
(1.339)
κ
α
j
,λ
α
j
(
α
j
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