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α
σ (
j
)
is the
j
th largest of
m
where
J
∗
,
n
κ
α
σ (
j
)
,λ
α
σ (
j
)
ω
i
J
∗
,
n
κ
α
i
,λ
α
i
(
α
i
)
(
=
,
,...,
)
i
1
2
m
.
(8)
GIFPHAP
w
,ω
(α
1
,α
2
,...,α
m
)
w
1
P
n
α
σ (
1
)
ρ
w
2
P
n
α
σ (
2
)
ρ
=
⊕
κ
α
σ (
1
)
,λ
α
σ (
1
)
κ
α
σ (
2
)
,λ
α
σ (
2
)
w
m
P
n
α
σ (
m
)
ρ
1
ρ
⊕···⊕
κ
α
σ (
m
)
,λ
α
σ (
m
)
α
σ (
j
)
is the
j
th largest of
m
,
P
n
κ
α
σ (
j
)
,λ
α
σ (
j
)
κ
α
σ (
j
)
+
λ
α
σ (
j
)
≤
=
,
,...,
where
1,
j
1
2
ω
i
P
n
m
κ
α
i
,λ
α
i
(α
i
) (
i
=
1
,
2
,...,
m
)
.
(9)
GIFPHAQ
w
,ω
(α
1
,α
2
,...,α
m
)
w
1
Q
n
α
σ (
1
)
ρ
w
2
Q
n
α
σ (
2
)
ρ
=
⊕
κ
α
σ (
1
)
,λ
α
σ (
1
)
κ
α
σ (
2
)
,λ
α
σ (
2
)
w
m
Q
n
α
σ (
m
)
ρ
1
ρ
⊕···⊕
κ
α
σ (
m
)
,λ
α
σ (
m
)
α
σ
(
j
)
is the
j
th largest
m
,
Q
n
κ
α
σ (
j
)
,λ
α
σ (
j
)
where
κ
α
σ (
j
)
+
λ
α
σ (
j
)
≤
1,
j
=
1
,
2
,...,
ω
i
Q
n
κ
α
i
,λ
α
i
(
α
i
)
(
=
,
,...,
)
of
m
i
1
2
m
.
Let
α
σ(
j
)
=
(μ
α
σ(
j
)
,
v
α
σ(
j
)
),
j
=
1
,
2
,...,
m
, then, similar to Theorem 1.45, we
have
Theorem 1.56
(Xia and Xu 2010)
GIFPHAD
w
,ω
(α
1
,α
2
,...,α
m
)
(
1
)
⎛
⎛
−
μ
α
σ(
j
)
+
κ
α
σ(
j
)
π
α
σ(
j
)
ρ
w
j
⎞
1
ρ
m
1
⎝
⎝
1
⎠
=
−
,
j
=
1
ρ
⎞
⎠
⎛
−
κ
α
σ(
j
)
π
α
σ(
j
)
ρ
w
j
⎞
1
m
1
−
1
α
σ(
j
)
−
1
⎝
1
⎠
1
−
−
−
v
j
=
1
GIFPHAF
w
,ω
(α
1
,α
2
,...,α
m
)
(
2
)
⎛
⎛
w
j
⎞
⎠
1
1
ρ
m
⎝
−
μ
F
n
⎝
1
=
−
,
κ
α
σ (
j
)
,λ
α
σ (
j
)
(
α
σ (
j
)
)
j
=
1
ρ
⎞
⎠
⎛
ρ
w
j
⎞
⎠
1
1
1
m
⎝
1
1
−
−
−
−
v
F
n
κ
α
σ (
j
)
,λ
α
σ (
j
)
(
α
σ (
j
)
)
j
=
1
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