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γ i = 1 ( 1 + 1 β i )( 1 + 1 β k + 1 ) γ i = 1 ( 1 μ β i )( 1 μ β k + 1 )
γ i = 1 ( 1 + 1 β i )( 1 + 1 β k + 1 )( 1 + 1 β k + 1 ) + 1 i = 1 ( 1 μ β i )( 1 μ β k + 1 )
=
,
2 i = 1 v
γ
β i v
β k
+
1
1 + 1 )
1 v
β k + 1
γ i = 1 ( 1 + 1 )( 1 v
+ γ(γ 1 ) i = 1 v
β i ))
β i v
β k + 1
(1.256)
k + 1
i = 1 ( 1 + 1 β i ) k + 1
γ k + 1
i = 1 ( 1 μ β i )
i = 1 v
β i
=
,
k + 1
i
1 ( 1 + 1 β i ) + 1 ) k + 1
k + 1
i
1 ( 1 + 1 )( 1 v β i )) + 1 ) k + 1
1 ( 1 μ β i )
1 v β i
=
i
=
=
i
=
=
+
which indicates that Eq. ( 1.251 ) holds for n
k
1. Thus Eq. ( 1.249 ) holds for all
n . Since
w i α i
β i
= β i ,
v β i ) =
1
w i
1
w i
γμ α
γμ α
+
1
)
μ α )) λ +
α
μ α )) λ +
α
(
1
+
1
)(
1
1
(
1
+
1
)(
1
1
=
1
w i
1
w i ,
γμ α
γμ α
+
1
)
+
1
)
μ α )) λ +
α
μ α )) λ +
α
(
1
+
1
)(
1
1
(
1
+
1
)(
1
1
w i
v α ) λ (
v α ) λ
(
+
)
1
1
1
γ
(
1
+
1
)
v α ) λ +
1
)(
1
v α ) λ
1
1
w i
w i
(
1
+
1
)
v α ) λ (
1
v α ) λ
(
1
+
1
)
v α ) λ (
1
v α ) λ
+
)
+
)
1
1
(
1
+
1
)
v α ) λ +
1
)(
1
v α ) λ
(
1
+
1
)
v α ) λ +
1
)(
1
v α ) λ
( 1 + 1 )( 1 μ α )) λ +
w i
( 1 + 1 )( 1 μ α )) λ μ α
w i
2
1 α
=
(
α w i
) (
μ α )) λ μ α w i ,
μ α )) λ +
1
+
1
)(
1
2
1
+
1
1
+
1
)(
1
γ ( 1 + 1 ) v α ) λ ( 1 v α ) λ w i
(
v α ) λ w i
) (
v α ) λ w i
v α ) λ +
v α ) λ (
1
+
1
)
2
1
)(
1
+
1
1
+
1
)
1
(1.257)
then we have
n
i = 1 (
n
i = 1 (
1
+
1
β i )
1
μ β i )
1
(
α i w i
(
μ α i )) λ μ α i w i
n
μ α i )) λ +
2
1
+
1
)(
1
1
1
+
1
)(
1
=
+
1
)
(
α i w i
) (
μ α i )) λ μ α i w i
μ α i )) λ +
2
1
+
1
)(
1
1
+
1
1
+
1
)(
1
i = 1
(
α i w i
(
μ α i )) λ μ α i w i
n
μ α i )) λ +
2
1
+
1
)(
1
1
1
+
1
)(
1
1
(
α i w i
) (
μ α i )) λ μ α i w i
μ α i )) λ +
2
1
+
1
)(
1
1
+
1
1
+
1
)(
1
i = 1
γ i = 1 (
α i w i
n
(
μ α i )) λ μ α i w i
μ α i )) λ +
2
1
+
1
)(
1
1
γ
1
+
1
)(
1
i
=
1
=
i = 1 (
α i w i
) (
μ α i )) λ μ α i w i
μ α i )) λ +
2
1
+
1
)(
1
1
+
1
1
+
1
)(
1
(1.258)
k
k
1 (
1
+
1
β i ) +
1
)
1 (
1
μ β i )
i
=
i
=
1
(
α i w i
(
μ α i )) λ μ α i w i
n
μ α i )) λ +
2
1
+
1
)(
1
1
1
+
1
)(
1
=
+
1
)
(
α i w i
) (
μ α i )) λ μ α i w i
1
+
1
)(
1
μ α i )) λ +
2
1
+
1
1
+
1
)(
1
i
=
1
 
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