On Cut-Approach

Muroran-IT Academic Resources Archive
Title
Author(s)
Citation
Issue Date
URL
On Cut-Approach
Kinokuniya, Yoshio
室蘭工業大学研究報告. Vol.5 No.2, pp.929-934, 1966
1966-08-25
http://hdl.handle.net/10258/3287
Rights
Type
Journal Article
See also Muroran-IT Academic Resources Archive Copyright Policy
Muroran Institute of Technology
O n Cut~Approach
*
nd
v
d
n
・
1
u
tk
o
n
'I
K
QU
O
O
1
n
Y
A
b
s
t
r
a
c
t
Ap
a
i
ro
fm
u
t
u
a
l
l
y cut-ωnjugate p
r
o
p
e
r
t
i
e
s，w
h
iじha
r
et
ob
ee
x
c
l
u
s
i
v
e
l
yd
i
s
t
i
n
g
u
i
s
h
e
do
na
n
y
r
e五r
s
t
l
yd
e
f
i
n
e
c
l
. Byt
h
e
s
ep
r
o
p
e
r
t
i
e
saf
u
n
c
l
a
m
e
n
l
a
i
ι
h
e
o
r
e
mi
si
n
c
l
u
c
e
da
n
c
lt
h
e
n
s
e
to
fp
o
i
n
t
s，a
s
o
m
ei
m
p
o
r
t
a
n
ti
n
v
e
s
t
i
g
a
t
i
o
n
sa
r
cc
l
e
v
e
l
o
p
e
c
lo
v
e
ru
l
l
r
a
f
u
n
c
t
i
o
n
so
f乱 s
e
t
.
1
. Set1
ヲ
unction
I
nt
h
i
spaper byas
e
tj
i
m
c
t
i
o
nwemean anonn
e
g
a
t
i
v
ea
d
d
i
t
i
v
ef
u
n
c
t
i
o
no
f
as
e
t
;i
.
e
.i
ffi
sas
e
tf
u
n
c
t
i
o
nフ f
o
r any two s
e
t
s ]陛1 and ]U~ o
fp
o
i
n
t
si
na
talways e
f
f
e
c
t
st
h
a
t
f
i
n
i
t
e
d
i
m
e
n
s
i
o
n
a
le
u
c
l
i
d
i
a
ns
p
a
c
e E，i
ヲ
四
。
J
P
l
;
U.
:
.
1
'
12)= f
(
l
t
L
J+
f(
既)-f(~~11 n1
1
1
2) •
C
:
:
I
fa s
e
tf
u
n
c
t
i
o
nfr
n
a
y，f
o
rany s
u
b
s
e
t p'c及
巴 beexpressedi
nt
h
eform
0f(P)
f(F)= !
(
1
.1
)
PEP
fi
sa
η
a
p
p
l
i
c
αt
i
o
ni
n!
J
!
;o
r，i
fnotan a
p
p
l
i
c
a
t
i
o
n，an u
l
t
r
aて{
u
n
c
t
i
o
n
.
J
In t
h
ep
r
e
v
i
o
u
s paper1
t
h
ep
r
e
s
e
n
ta
u
t
h
o
rs
t
a
t
e
d aboutt
h
ee
m
p
i
r
i
c
i
s
tview
e
tf
u
n
c
t
i
o
n
s may be o
b
s
e
r
v
e
c
li
n new w
a
y
s
.I
nt
h
e
o
fa
n
a
l
y
s
i
s throughwhich s
e
m
p
i
r
i
c
i
s
tt
h
e
o
r
yo
fa
n
a
l
y
s
i
s i
ft
h
er
e
l
a
t
i
o
n(
1
.1
) were t
obe t
r
u
e， i
tml
叫， f
o
r
anye
n
u
r
n
e
r
a
b
l
ep
a
r
t
i
t
i
o
n (.:.1'Ikh~l ム・ of J
1
'
/
， bet
h
a
t
ラ
ラ
f(
1
t
l
)= J
:
f(
凪)ヲ
which c
l
i
r
e
c
t
l
yr
n
e
a
n
st
h
a
tfi
sc
o
m
p
l
e
t
e
l
ya
d
d
i
t
i
v
ei
nJ
t
L Thεrefore t
h
a
ta s
e
t
f
u
n
c
t
i
o
nfi
s an u
l
t
r
a
f
u
n
c
t
i
o
ni
se
q
u
i
v
a
l
e
n
tt
oi
tt
h
a
tfi
sn
o
t everywhere c
o
r
n
p
l
e
t
e
l
ya
d
d
i
t
i
v
e(
i
nJ
Y
I
)
. Inthefollowing，ar
e
c
o
n
s
t
r
u
c
t
i
v
es
t
u
c
l
yo
fu
l
t
r
af
u
n
c
t
i
o
n
s
w
i
t
h some r
e
s
u
l
t
s、
w
i
l
lbe s
t
a
t
e
d
.
ラ
同
幽
z
. Prindple of Cut-Approach
For ag
i
v
e
np
r
o
p
e
r
t
yTt
obe t
e
s
t
e
df
o
ra s
e
仁 i
f
(V~lJIr，
NCE)((..M~N & Nct)[
>M c.
t
)
Ti
ss
a
i
c
lt
obep
r
o
g
r
e
s
s
i
v
e
，andi
f
(v
J
t
I
， NCE) ((J1'I~JY & .
.
'
1
'
Ic.
t
)[
>.Nct)，
，
r
t
obe r
e
g
r
e
s
s
i
v
e
. I
fi
ti
s，f
o
rany s
e
t1
. i
n E，n
e
c
e
s
s
a
r
yt
h
a
t
持組関谷芳雄
(
4
2
3
)
Yo
s
h
i
oKinokuη'ya
930
MctVMctj
andi
ft
h
er
e
l
a
t
i
o
n
s及CctandM c q cannot s
i
m
u
l
t
a
n
e
o
u
s
l
y be o
b
s
e
r
v
e
d t
h
e
nT
and qa
r
es
a
i
dt
obem
u
t
u
a
l
l
yc
u
t
c
o
n
j
u
g
a
t
e
. Then t
h
ef
o
l
l
o
w
i
n
gf
a
c
ti
sd
i
r
e
c
t
l
y
五n
i
t
i
o
n
:
o
b
t
a
i
n
e
dby t
h
ede
Proposition2
.1
. 1
fTω1
dqarec
u
t
c
o
n
j
u
g
a
t
epr
.
ψe
r
t
i
e
sandT i
sr
e
g
r
e
s
s
i
v
e(
r
e
s
p
.p
r
o
g
r
e
s
s
i
v
e
)t
h
e
n qi
sα1りr
o
g
r
e
s
s
i
v
e(何学・ r
e
g
r
・
'
e
s
s
i
v
e
)ρr
o
p
e
r
t
y
.
I
f Ti
sr
e
g
r
e
s
s
i
v
e and q i
sp
r
o
g
r
ε
s
s
i
v
e，making up a p
a
i
ro
fc
u
tc
o
n
j
u
g
a
t
e
p
r
o
p
e
r
t
i
e
s andi
f
ラ
ラ
開
ラ
c及r
andM c q
B)ct，B，
ラ
t
h
e
nt
h
e
r
emay e
x
i
s
ttwo s
e
q
u
e
n
c
e
so
fs
e
t
s (Bk) and (Ak) such t
h
a
t
R)CB2C .
園 CA
二
及f
2CA)C
ラ
Bkct and Akcq ん
(=1，
2，.
一
).
I
nt
h
i
sc
a
s
ei
fi
ti
sn
o
tt
h
a
t
ラ
B =UB/ c =
nAk=A
we have
o
i
d
.
A - B手 v
(
2
.
1
)
h
a
t
When Bcq，wemaychoose such t
A2= A3= ・・・ = B
s
ot
h
a
t A =B. When Act，wemaychoose sucht
h
a
t
B2= B
3= ・一 = A
s
ot
h
a
t A=B. T
h
e
r
e
f
o
r
e wemay，f
o
rt
h
ec
a
s
eo
f(
2
.1
)s
u
p
p
o
s
et
h
a
t
ラ
ラ
Bct and Acq.
(
2
.
2
)
I
nc
a
s
eo
f(
2
.
2
)，as
e
tcanbe i
n
s
e
巾 db
etween A and B and i
t，o
fc
o
u
r
s
e，has
. Thisb
e
i
n
gs
o，at
r
a
n
s
i
n
d
u
c
t
i
v
e modemay bede
五ned
onep
r
o
p
e
r
t
yo
f Tandq
t
or
e
a
c
h ac
o
n
s
t
r
u
c
t
i
o
n such t
h
a
t
(
V
C
ε I) (B，
ct) (Ii
sas
i
m
p
l
e
o
r
d
e
r
e
ds
e
to
fi
n
d
i
c
e
s
)，
(
V
C，KεI) (c<K!>B，
CB<)，
(
V
A
εA
)(
ム cq) (
Ai
sas
i
m
p
l
e
o
r
d
e
r
e
ds
e
to
fi
n
d
i
c
e
s
)，
(VA，μ
εA
) (A<μi
)A，
:
:
;
2Aμ
)
ラ
andi
f B = UB，and A ニ
nA" thenwehaveoneofthecases
r(
i
i
) Bct，Acq and(VG)(BcG[
>Gcq).
(
i
) B ニム o
Thec
a
s
e(
i
i
)canr
e
a
d
i
l
ybet
u
r
n
e
di
n
t
ot
h
ec
a
s
e(
i
)
. I
na
d
d
i
t
i
o
n
，i
nt
h
ee
m
p
i
r
i
c
i
s
t
t
h
e
r
emuste
x
i
s
tenumerable s
u
b
s
e
t
so
fi
n
d
i
c
e
s(
C
k
)and(
ん
)sucht
h
a
t
t
h
e
o
r
y，
B ニ UBι k and A= nA.ik ・
(
4
2
4
)
9
3
1
OnC
u
t
A
p
p
r
o
a
c
h
Thuswehave:
Proposition 2
.
2
.(
P
r
i
n
c
争l
eofC
u
t
A
p
p
r
o
a
c
h
)
. 1fTαnd q aJで c
u
t
c
o
j
u
r
t
i
e
sand1
:
1i
sr
e
g
アe
s
s
i
v
e
，andi
fM 亡 q，thenthereexistt
日 oe
numerable
g
a
t
e戸ゆe
s
e
q
u
e
n
c
e
st
:
:
?
戸
、 s
u
b
s
e
t
s(B1.) and(Ak
)s
u
c
ht
h
a
t
B1C B
...cA2c A1c M
，
2C
Bkctand A"cq ん
(=1，
2，.
一
)，
ωl
dond
e
n
o
t
i
n
gαsB =U BkandA =nA，
o weh
aveA=B.
The s
e
tB o
b
t
a
i
n
e
di
nt
h
eabove，i
sc
a
l
l
e
da Tc
u
to
ft
h
es
e
tJ
t
I
， and t
hen
，
)i
sc
a
l
l
e
da c
u
t
a
l
ゆr
o
a
c
hfrombelow and(A，
，
) ac
u
t
a
p
ρroach
t
h
es
e
q
u
e
n
c
e
s (B，
fromabovewith r
e
s
p
e
c
tt
o T(
o
rq
)
.
回
3
. Cut-Amosphere
When
。
<c<f(Jt
r
)
l
e
tu
s de
五neT andqsuch t
h
a
t
(Fcþ) 三 (f(F)~c)
and (Fcq)三 (
f
(F)>c)，
t
h
e
nTand q a
r
ea
p
p
a
r
e
n
t
i
yc
u
t
c
o
n
j
u
g
a
t
e and T(
r
e
s
p
.q
)i
sar
e
g
r
e
s
s
i
v
e(
r
e
s
p
.
p
r
o
g
r
e
s
s
i
v
e
)p
r
o
p
e
r
t
y
. So a
p
p
l
y
i
n
gt
h
ep
r
i
n
c
i
p
l
eo
fc
u
t
a
p
p
r
o
a
c
h wemayh
a
v
e
，
)a
nd(Ak) s
u
c
ht
h
a
t
c
u
t
a
p
p
r
o
a
c
h
e
s(B，
ラ
ラ
BjCBzC.
‘ CA
C
;
;
:
;A1C]
J
;
I
z
f(B1.)ミ
ミ f(B止 1
)~ c<f(Ak
+
l
)<f(Ak
)ん
(=1，
2，
…)
(
3
.
1
)
x
p
e
c
tt
h
a
t Bニ A. I
nt
h
i
sc
a
s
e i
fl
i
m
andi
fB U B
kand A nA"， wecan e
f(81.) sandl
i
mf(Ak)=α，we have
二
ラ
二
二
(
3
.
2
)
ß~c ミミ α
and i
n
v
e
r
s
e
l
yi
fci
san a
r
b
i
t
r
a
r
yv
a
l
u
e found i
nt
h
er
e
l
a
t
i
o
n(
3
.
2
)，t
h
er
e
l
a
t
i
o
n
，
)and(A
，
，
)
. T
herefore whens
手仏
(
3
.
1
)i
st
oh
o
l
donf
o
rt
h
esames
e
q
u
e
n
c
e
s(B，
t
h
es
t
a
t
eo
b
s
e
r
v
e
dthrought
h
el
i
m
i
t
i
n
gp
r
o
c
e
d
u
r
e
ラ
ラ
(Ak-Bj) ム
( j→∞)
(
3
.
3
)
mustg
i
v
e an a
t
m
o
s
p
h
e
r
i
cs
t
a
t
ei
np
o
i
n
tt
h
a
tt
h
el
i
m
i
t
i
n
gv
a
l
u
eo
ff(Ak- Bj
)i
s
e
s
p
i
t
eo
ft
h
ef
a
c
t
t
obecounteda
se
q
u
a
lt
ot
h
ep
o
s
i
t
i
v
enumberα sd
l
i
m(A，，-Bj)= v
o
i
d
.
Sot
h
es
t
a
t
ei
n
d
i
c
a
t
e
dby (
3
.3
)i
sc
a
l
l
e
da c
u
t
-αtmo
ゅ
んe
r
ewhens*
α
.
I
fr
Q
J<f(P)くむ(三 i
n
f
i
n
i
t
e
s
i
m
a
l
)f
o
re
v
e
r
yp
o
i
n
tP i
nM
， fi
ss
a
i
dt
o be
ρowdelッ i
n]
J
f
. I
fwedemands=αin (
3
.
2
)，i
tmustbet
h
a
tt
h
es
e
q
u
e
n
c
e
s(Bk
)
and (Ak) a
r
ed
e
x
t
e
r
o
u
s
l
y chosen t
os
a
t
i
s
f
yt
h
ec
o
n
d
i
t
i
o
n
. For t
h
i
sp
u
r
p
o
s
e，i
t
mightbe h
e
l
p
f
u
lt
os
u
p
p
o
s
et
h
a
t，i
nc
a
s
eo
fapowderyf
u
n
c
t
i
o
n，
f themassvalue
(
4
2
5
)
yo
s
h
i
oK
i
n
o
k
u
n
i
y
a
9
3
2
0 shou
lda
tanyr
a
t
ebe thoughta
sanaccumulationo
fi
n
f
i
n
i
t
e
s
i
m
a
lq
u
a
t
i
f
C
2
J
I
)>
t
i
e
sf
(P) (PE.
1
J
1
)
. Then，i
ff(N)>
f(
I
l
'
)，by t
r
a
n
s
f
e
r
r
i
n
gp
o
i
n
t
s from _1
yt
o]
1
'
，
t
h
ed
i
f
f
e
r
e
n
c
eo
ff
v
a
l
u
e might be d
e
c
r
e
a
s
e
du
n
t
i
li
tv
a
n
i
s
h
e
s
. Suchap
r
i
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z
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孔s
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e
c
h
.，Hokkaido
(
R
e
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e
i
刊
(
4
2
7
)
934
Yoshio Kinokuniya
References
1
) Kinokuniya，Y.: Mem. Muroran l
n
s
t
. Tech. 5(
1
)，341-347(
1
9
6
5
)
.
As ap
r
e
l
i
m
i
n
a
r
y guide:
Kinokuniya，Y.: Mem. MuroranI
n
s
t
. Tech. 4(
2
)，491-496(
1
9
6
3
)
(
4
2
8
)

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