Appl
.Mat
h.J.Chi
ne
s
eUni
v.Se
r
.B
2005,20(4):491498
A DERI
VATI
VEFREEALGORI
THM FOR
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MI
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Vol
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Le
mma2.For∀j∈ {1,2,...,p},t
he
r
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xi
s
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v
v
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n
r
at
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ht
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ee
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r
at
i
onalnumbe
rb
uc
ht
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i
i
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mma4.Δk=ρ Δ1=ρ1 ρ2
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r
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ow wi
t
h-M.
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t
r
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ve t
o ge
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he i
mpor
t
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l
us
i
on i
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e anal
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r
n
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al
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t
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e
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n
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he
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xi
s
tωk,j= (ωk,j,1,ωk,j,2,...,ωk,j,n) ∈Z ,βk,j= (βk,j,1,
T
n
βk,j,2,...,βk,j,n)∈Z s
uc
ht
hat
n
n
Г1(k) Г2(k)
xk,j = b
e
ρ1
ρ2 Δ1Σ βk,j,ie
0,1,2,...,N},
xΣ ω
k,j
,i
i+ b
i j∈ {
i
=1
(2)
i
=1
andf
ore
ve
r
yωk,j,it
he
r
ee
xi
s
tj
i
uc
ht
hatωk,j,i=µj0,i0,whe
r
eГ1(k),Г2(k)∈Z.
0,
0s
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oofi
sbyi
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t
i
on.Fork= 1,Le
mma3i
mpl
i
e
st
hec
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l
us
i
on.Now as
s
ume
(2)hol
∈{0,1,...,N}.The
.
dsf
orxk,j,j
r
ear
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ourpos
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i
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ome
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nt
hi
sc
as
e,wewr
i
t
e
n
n
i
=1
i
=1
Г(k) Г2(k)
xk+ 1,j = b
e
ρ1
ρ2 Δ1Σ βk+ 1,j,ie
,
xΣ ω
k+ 1,j
,i
i+ b
i
(3)
whe
r
eωk+ 1,j,i=ωk,j,iandβk+ 1,j,i=βk,j,i∈Z.
2.xk+ 1,ji
spr
oduc
e
dbyc
r
os
s
i
ngxk,j1 andxk,j2 ati
nde
xi
I
nt
hi
sc
as
e,
0.
i
0
i
0
n
xk+ 1,j =b
x(
Σ ωk,j1,iei+
i
=1
Σω
e)+ b
ρ
k,j
ii
2,
i
=i
0+ 1
n
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1
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[0.000000,-1.000000]
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[u.3qXX44,u.3qXX44,u.3qXX44,u.3qXX44,u.3qXX44,
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5
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i
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s-186.731.
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x4 2
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3
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e
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r
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i
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mi
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e
r
s
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s
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t
i
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2
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i
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i
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gi
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hi
sf
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i
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mi
z
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