I
nte
rna
t
io
na
l J
o
urna
l o
f
E
lect
rica
l a
nd
Co
m
p
ute
r
E
ng
in
ee
ring
(
I
J
E
CE
)
Vo
l.
9
,
No
.
4
,
A
u
g
u
s
t
201
9
,
p
p
.
3
2
5
6
~3
2
6
1
I
SS
N:
2
0
8
8
-
8708
,
DOI
: 1
0
.
1
1
5
9
1
/
i
j
ec
e
.
v9
i
4
.
pp
3
2
5
6
-
3261
3256
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ia
e
s
co
r
e
.
co
m/
jo
u
r
n
a
ls
/in
d
ex
.
p
h
p
/
I
JE
C
E
Densely
ho
m
o
g
en
eo
us fu
zz
y
spa
ces
Sa
m
er
A
l G
ho
ur
De
p
a
rt
m
e
n
t
o
f
M
a
th
e
m
a
ti
c
s a
n
d
S
tatisti
c
s,
Jo
rd
a
n
Un
iv
e
rsity
o
f
S
c
i
e
n
c
e
a
n
d
T
e
c
h
n
o
l
o
g
y
,
J
o
rd
a
n
Art
icle
I
nfo
AB
ST
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
J
u
l 2
6
,
2
0
1
8
R
ev
i
s
ed
A
p
r
2
,
2
0
1
9
A
cc
ep
ted
A
p
r
1
1
,
2
0
1
9
W
e
e
x
ten
d
th
e
c
o
n
c
e
p
t
o
f
b
e
in
g
d
e
n
se
l
y
h
o
m
o
g
e
n
e
o
u
s
to
in
c
l
u
d
e
f
u
z
z
y
to
p
o
lo
g
ica
l
sp
a
c
e
s.
W
e
p
ro
v
e
th
a
t
o
u
r
e
x
ten
sio
n
is
a
g
o
o
d
e
x
ten
sio
n
i
n
t
h
e
se
n
se
o
f
L
o
we
n
.
W
e
p
ro
v
e
t
h
a
t
a
-
c
u
t
t
o
p
o
lo
g
ica
l
sp
a
c
e
(
X
,
ℑ
a
)
o
f
a
DH
f
u
z
z
y
to
p
o
lo
g
ica
l
sp
a
c
e
(
,
ℑ
)
is DH i
n
g
e
n
e
ra
l
o
n
ly
f
o
r
=
0
.
K
ey
w
o
r
d
s
:
C
u
t to
p
o
lo
g
ies
Den
s
el
y
h
o
m
o
g
e
n
eo
u
s
Fu
zz
y
C
DH
Go
o
d
ex
ten
s
io
n
Co
p
y
rig
h
t
©
2
0
1
9
In
stit
u
te o
f
A
d
v
a
n
c
e
d
E
n
g
i
n
e
e
rin
g
a
n
d
S
c
ien
c
e
.
Al
l
rig
h
ts re
se
rv
e
d
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Sa
m
er
A
l G
h
o
u
r
,
Dep
ar
t
m
en
t o
f
Ma
th
e
m
at
ics a
n
d
Statis
tics
,
J
o
r
d
an
Un
iv
er
s
it
y
o
f
Scie
n
ce
a
n
d
T
ec
h
n
o
lo
g
y
,
I
r
b
id
2
2
1
1
0
,
J
o
r
d
an
E
m
ail: a
l
g
o
r
e@
j
u
s
t.e
d
u
.
j
o
1.
I
NT
RO
D
UCT
I
O
N
As
d
ef
in
ed
in
[
1
]
,
th
e
n
o
tio
n
o
f
a
f
u
zz
y
s
et
in
a
s
et
X
is
a
f
u
n
ctio
n
f
r
o
m
X
in
to
th
e
clo
s
e
d
in
ter
v
a
l
[
0
,
1
]
.
A
cc
o
r
d
in
g
l
y
,
C
h
a
n
g
[
2
]
in
tr
o
d
u
ce
d
th
e
n
o
tio
n
o
f
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
o
n
a
n
o
n
-
e
m
p
t
y
s
et
X
as
a
co
llectio
n
o
f
f
u
zz
y
s
et
s
o
n
X
,
clo
s
ed
u
n
d
er
ar
b
itra
r
y
s
u
p
r
e
m
a
an
d
f
i
n
ite
i
n
f
i
m
a
an
d
co
n
tain
i
n
g
t
h
e
co
n
s
ta
n
t
f
u
zz
y
s
ets
0
an
d
1
.
Ma
th
e
m
aticia
n
s
e
x
ten
d
ed
m
an
y
to
p
o
lo
g
ical
co
n
ce
p
ts
to
in
clu
d
e
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es
s
u
c
h
a
s
:
s
e
p
ar
atio
n
ax
io
m
s
,
co
n
n
ec
ted
n
e
s
s
,
co
m
p
ac
t
n
es
s
a
n
d
m
etr
izab
ilit
y
.
Se
v
er
al
f
u
zz
y
h
o
m
o
g
en
eit
y
co
n
ce
p
t
s
w
er
e
d
is
cu
s
s
ed
in
[
3
-
1
1
]
.
A
s
ep
ar
ab
le
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
τ
)
is
co
u
n
tab
le
d
e
n
s
e
h
o
m
o
g
en
eo
u
s
(
C
DH)
[
1
2
]
if
g
iv
e
n
a
n
y
t
w
o
co
u
n
tab
le
d
en
s
e
s
u
b
s
e
ts
A
an
d
B
o
f
(
X
,
τ
)
th
er
e
is
a
h
o
m
eo
m
o
r
p
h
is
m
f
:
(
X
,
τ
)
→
(
X
,
τ
)
s
u
ch
t
h
at
f
(
A
)
=
B
.
T
h
e
s
tu
d
y
o
f
C
DH
to
p
o
lo
g
ical
s
p
ac
es
an
d
th
eir
r
elate
d
co
n
ce
p
ts
is
s
till
a
h
o
t
ar
ea
o
f
r
esear
ch
,
a
s
ap
p
ea
r
s
in
[
1
3
-
2
0
]
an
d
o
th
er
p
ap
er
s
.
R
ec
en
tl
y
,
a
u
th
o
r
s
in
[
9
]
ex
ten
d
ed
C
DH
to
p
o
lo
g
ical
p
r
o
p
er
ty
to
i
n
cl
u
d
e
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es.
T
h
ey
p
r
o
v
ed
th
at
th
e
ir
ex
te
n
s
io
n
is
a
g
o
o
d
ex
ten
s
io
n
in
th
e
s
e
n
s
e
o
f
L
o
w
en
,
a
n
d
p
r
o
v
ed
th
at
a
-
cu
t
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
a
)
o
f
a
C
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
)
is
C
DH
in
g
en
er
a
l
o
n
l
y
f
o
r
a
=
0
.
Fo
r
th
e
p
u
r
p
o
s
e
o
f
d
ea
lin
g
w
it
h
n
o
n
-
s
ep
ar
ab
le
to
p
o
lo
g
ical
s
p
ac
es,
au
t
h
o
r
s
in
[
2
1
]
m
o
d
i
f
ied
th
e
d
ef
in
i
tio
n
o
f
C
DH
to
p
o
lo
g
ic
al
s
p
ac
es
as
f
o
llo
w
s
:
A
s
u
b
s
et
A
o
f
a
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
τ
)
is
ca
lled
a
σ
-
d
is
cr
ete
s
et
if
it
i
s
t
h
e
u
n
io
n
o
f
co
u
n
tab
l
y
m
a
n
y
s
e
ts
,
ea
ch
w
it
h
th
e
r
elativ
e
to
p
o
lo
g
y
,
b
ei
n
g
a
d
i
s
cr
ete
to
p
o
lo
g
ical
s
p
ac
e.
A
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
τ
)
is
d
en
s
el
y
h
o
m
o
g
e
n
eo
u
s
(
DH)
p
r
o
v
id
ed
(
X
,
τ
)
h
as
a
σ
-
d
is
cr
ete
s
u
b
s
et
w
h
ich
i
s
d
en
s
e
in
(
X
,
τ
)
an
d
if
A
an
d
B
ar
e
tw
o
s
u
ch
σ
-
d
is
cr
ete
s
u
b
s
et
s
o
f
(
X
,
τ
)
th
er
e
is
a
h
o
m
eo
m
o
r
p
h
is
m
f
:
(
X
,
τ
)
→
(
X
,
τ
)
s
u
c
h
t
h
at
f
(
A
)
=
B
.
I
t is
k
n
o
w
n
t
h
at
C
DH
a
n
d
DH
t
o
p
o
lo
g
ical
co
n
ce
p
ts
ar
e
in
d
ep
en
d
en
t.
T
h
e
s
t
u
d
y
o
f
DH
to
p
o
lo
g
ical
s
p
ac
es
is
co
n
ti
n
u
ed
in
[
2
2
-
2
8
]
an
d
o
t
h
er
p
ap
e
r
s
.
As
a
m
a
in
g
o
al
o
f
th
e
p
r
esen
t
w
o
r
k
w
e
w
ill
s
h
o
w
h
o
w
th
e
d
e
f
i
n
itio
n
o
f
DH
to
p
o
lo
g
ical
s
p
ac
es
ca
n
b
e
m
o
d
if
ied
in
o
r
d
er
to
d
ef
in
e
a
g
o
o
d
ex
ten
s
io
n
o
f
it
in
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es.
W
e
w
i
ll
g
i
v
e
r
elatio
n
s
h
ip
s
b
et
w
ee
n
C
D
H
an
d
DH
f
u
zz
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Den
s
ely
h
o
mo
g
en
e
o
u
s
fu
z
z
y
s
p
a
ce
s
(
S
a
mer A
l G
h
o
u
r
)
3257
T
h
r
o
u
g
h
o
u
t
th
i
s
p
ap
e
r
,
if
is
a
s
et,
th
e
n
|
|
=
w
ill
d
e
n
o
te
it
s
ca
r
d
in
alit
y
.
W
e
w
r
ite
ℚ
(
r
esp
.
ℕ)
to
d
en
o
te
th
e
s
et
o
f
all
r
atio
n
al
n
u
m
b
er
s
(
r
esp
.
n
atu
r
al
n
u
m
b
er
s
)
.
T
h
e
clo
s
u
r
e
o
f
a
f
u
zz
y
s
et
in
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
w
il
l
b
e
d
en
o
ted
b
y
(
)
.
A
s
s
o
ciat
ed
with
a
g
i
v
e
n
to
p
o
lo
g
ical
s
p
ac
e
(
,
)
an
d
ar
b
itra
r
y
s
u
b
s
et
o
f
,
w
e
d
en
o
te
th
e
r
elativ
e
to
p
o
lo
g
y
o
n
b
y
τ
,
th
e
clo
s
u
r
e
o
f
b
y
(
)
an
d
th
e
b
o
u
n
d
ar
y
o
f
b
y
(
)
.
to
p
o
lo
g
ical
s
p
ac
es a
s
w
el
l a
s
w
e
w
ill d
ea
l
w
it
h
c
u
t to
p
o
lo
g
ical
s
p
ac
e
s.
2.
P
RE
L
I
M
I
NARIE
S
I
n
th
i
s
p
ap
er
w
e
s
h
all
f
o
llo
w
t
h
e
n
o
tatio
n
s
a
n
d
d
ef
i
n
itio
n
s
o
f
[
2
9
]
an
d
[
3
0
]
.
I
f
(
X
,
τ
)
is
a
to
p
o
lo
g
ical
s
p
ac
e,
th
en
th
e
cla
s
s
o
f
all
lo
w
er
s
e
m
i
-
co
n
ti
n
u
o
u
s
f
u
n
ctio
n
s
f
r
o
m
(
,
)
to
(
[
0
,
1
]
,
)
,
w
h
er
e
is
th
e
u
s
u
al
E
u
clid
ea
n
to
p
o
lo
g
y
o
n
[
0
,
1
]
,
is
a
f
u
zz
y
to
p
o
lo
g
y
o
n
.
T
h
is
f
u
zz
y
to
p
o
lo
g
y
is
d
en
o
te
d
b
y
(
)
.
T
h
e
f
o
llo
w
i
n
g
d
e
f
i
n
itio
n
s
an
d
p
r
o
p
o
s
itio
n
s
w
il
l b
e
u
s
ed
in
t
h
e
s
eq
u
el:
Def
i
n
itio
n
2
.
1
.
[
9
]
L
et
b
e
a
n
o
n
-
e
m
p
t
y
s
et,
b
e
a
n
o
n
-
e
m
p
t
y
s
u
b
s
et
o
f
an
d
b
e
a
co
llecti
o
n
o
f
f
u
zz
y
p
o
in
t
s
in
.
T
h
en
ℚ
(
)
w
ill d
e
n
o
te
th
e
s
e
t
ℚ
(
)
=
{
:
is
a
fuz
zy
poi
n
t
wi
th
∈
a
n
d
r
∈
ℚ
∩
(
0
,
1
)
}
.
The
s
uppor
t
of
,
de
n
ote
d
by
(
)
,
is
de
fin
e
d
by
(
)
=
{
:
∈
for
s
ome
}
.
Def
i
n
itio
n
2
.
2
.
[
2
1
]
A
s
u
b
s
et
o
f
a
to
p
o
lo
g
ical
s
p
ac
e
(
,
)
is
c
alled
a
-
d
is
cr
ete
s
et
i
f
it
is
t
h
e
u
n
io
n
o
f
co
u
n
tab
l
y
m
an
y
s
ets,
ea
ch
w
it
h
th
e
r
elati
v
e
to
p
o
lo
g
y
,
b
ein
g
a
d
is
cr
ete
to
p
o
lo
g
ical
s
p
ac
e
.
Def
i
n
itio
n
2
.
3
.
[
2
1
]
A
to
p
o
lo
g
ical
s
p
ac
e
(
,
)
is
ca
lled
d
en
s
el
y
h
o
m
o
g
e
n
eo
u
s
(
DH)
if
f
h
as a
σ
-
d
is
cr
ete
d
en
s
e
s
u
b
s
et.
I
f
an
d
ar
e
t
w
o
-
d
is
cr
ete
d
en
s
e
s
u
b
s
et
s
o
f
,
th
en
t
h
er
e
is
a
h
o
m
eo
m
o
r
p
h
i
s
m
ℎ
:
(
,
)
→
(
,
)
s
u
c
h
th
at
ℎ
(
)
=
.
Def
i
n
itio
n
2
.
4
.
[
3
1
]
A
s
s
o
ciate
d
w
it
h
a
g
i
v
e
n
f
u
zz
y
to
p
o
lo
g
ic
al
s
p
ac
e
(
X
,
ℑ
)
an
d
ar
b
itra
r
y
s
u
b
s
et
o
f
,
w
e
d
ef
i
n
e
t
h
e
in
d
u
ce
d
f
u
zz
y
to
p
o
lo
g
y
o
n
o
r
th
e
r
elativ
e
to
p
o
lo
g
y
o
n
by
ℑ
=
{
∣
:
λ
∈
ℑ
}
.
Def
i
n
itio
n
2
.
5
.
[
9
]
A
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
)
is
s
aid
to
b
e
s
e
m
i
-
d
is
cr
ete
if
f
f
o
r
an
y
x
∈
X
,
th
er
e
ex
is
t
s
a
f
u
zz
y
p
o
in
t
o
r
a
f
u
zz
y
cr
i
s
p
p
o
in
t
f
o
r
s
o
m
e
a
w
it
h
∈
ℑ
.
Def
i
n
itio
n
2
.
6
.
[
3
2
]
L
et
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
a
n
d
let
b
e
a
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
.
T
h
en
is
s
aid
to
b
e
Den
s
e(
I
)
if
f
o
r
ev
er
y
n
o
n
-
ze
r
o
f
u
zz
y
o
p
en
s
et
λ
th
er
e
ex
is
ts
∈
s
u
ch
t
h
at
∈
.
Den
s
e
(
I
I
)
if
(
⋃
∈
)
=
1
.
Def
i
n
itio
n
2
.
7
.
[
9
]
A
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
is
ca
lled
s
ep
ar
ab
le
if
f
th
er
e
e
x
is
t
s
a
co
u
n
t
ab
le
d
en
s
e(
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
.
Def
i
n
it
io
n
2
.
8
.
[
3
3
]
A
p
r
o
p
er
ty
Ƥ
o
f
a
f
u
zz
y
to
p
o
l
o
g
ical
s
p
ac
e
is
s
aid
to
b
e
a
g
o
o
d
ex
ten
s
io
n
o
f
th
e
p
r
o
p
er
ty
Ƥ
in
cla
s
s
ical
to
p
o
lo
g
y
i
f
f
w
h
e
n
ev
er
th
e
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
is
to
p
o
lo
g
icall
y
g
e
n
er
ated
,
s
a
y
b
y
(
,
)
,
th
en
(
,
(
)
)
h
as p
r
o
p
er
t
y
Ƥ
if
f
(
,
)
h
as
p
r
o
p
er
ty
Ƥ
.
Def
i
n
itio
n
2
.
9
.
[
3
4
]
L
et
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
an
d
∈
[
0
,
1
)
.
T
h
e
to
p
o
lo
g
y
{
⁻
¹
(
,
1
]
:
∈
ℑ
}
o
n
X
is
ca
lled
-
cu
t
to
p
o
lo
g
ical
s
p
ac
e
o
f
(
,
ℑ
)
an
d
w
ill
b
e
d
en
o
ted
b
y
ℑ
.
T
h
e
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
)
w
i
ll
b
e
ca
lled
-
cu
t
to
p
o
lo
g
ical
s
p
ac
e
o
f
(
,
ℑ
)
.
Def
in
itio
n
2
.
1
0
.
[
9
]
A
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
is
s
aid
to
b
e
co
u
n
tab
le
d
en
s
e
h
o
m
o
g
en
eo
u
s
; d
en
o
ted
C
DH;
i
f
f
(
,
ℑ
)
is
s
ep
ar
ab
le.
I
f
an
d
ar
e
t
w
o
co
u
n
tab
le
d
en
s
e(
I
)
co
llectio
n
s
o
f
f
u
z
z
y
p
o
in
ts
o
f
,
th
en
t
h
er
e
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
s
u
ch
t
h
at
ℎ
(
(
)
)
=
(
)
.
P
r
o
p
o
s
itio
n
2
.
1
1
.
[
9
]
L
et
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
a
n
d
let
b
e
a
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
X
.
T
h
en
w
e
h
a
v
e
t
h
e
f
o
llo
w
i
n
g
I
f
is
d
en
s
e
(
I
)
,
th
e
n
ℚ(
S(
)
)
is
d
en
s
e(
I
I
)
.
I
f
is
d
en
s
e
(
I
I
)
,
th
en
ℚ(
S(
)
)
is
d
en
s
e(
I
)
.
P
r
o
p
o
s
itio
n
2
.
1
2
.
[
9
]
L
et
(
,
)
b
e
a
to
p
o
lo
g
ical
s
p
ac
e,
⊆
,
an
d
b
e
a
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
.
T
h
en
w
e
h
av
e
t
h
e
f
o
llo
w
i
n
g
I
f
is
d
en
s
e
i
n
(
,
)
,
th
en
ℚ(
)
is
d
en
s
e(
I
)
in
(
,
(
)
)
.
I
f
is
d
en
s
e(
I
)
in
(
,
(
)
)
,
th
en
(
)
is
d
en
s
e
in
(
,
)
.
P
r
o
p
o
s
itio
n
2
.
1
3
.
[
3
5
]
L
et
(
,
₁
)
an
d
(
,
₂
)
b
e
tw
o
to
p
o
lo
g
ical
s
p
ac
es.
T
h
en
:
(
,
₁
)
→
(
,
₂
)
is
co
n
ti
n
u
o
u
s
i
f
f
:
(
,
(
₁
)
)
→
(
,
(
₂
)
)
is
f
u
zz
y
co
n
ti
n
u
o
u
s
.
P
r
o
p
o
s
itio
n
2
.
1
4
.
[
9
]
L
et
:
(
,
ℑ
₁
)
→
(
,
ℑ
₂
)
b
e
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
m
ap
an
d
b
e
a
co
llect
io
n
o
f
f
u
zz
y
p
o
in
ts
o
f
.
T
h
en
w
e
h
av
e
t
h
e
f
o
llo
w
in
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
4
,
Au
g
u
s
t 2
0
1
9
:
3
2
5
6
-
3261
3258
(
(
)
)
=
(
(
)
)
.
I
f
is
d
en
s
e
(
I
)
in
(
,
ℑ
₁
)
,
th
en
(
)
is
d
en
s
e(
I
)
in
(
,
ℑ
₂
)
.
P
r
o
p
o
s
itio
n
2
.
1
5
.
[
9
]
L
et
(
X
,
ℑ
)
b
e
a
s
e
m
i
-
d
is
cr
ete
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
T
h
en
w
e
h
a
v
e
th
e
f
o
llo
w
in
g
I
f
is
co
u
n
tab
le
d
en
s
e
(
I
)
in
(
,
ℑ
)
,
th
en
(
)
=
.
(
X
,
ℑ
)
is
s
ep
ar
ab
le
if
f
is
co
u
n
tab
le.
P
r
o
p
o
s
itio
n
2
.
1
6
.
[
9
]
L
et
b
e
a
co
u
n
tab
le
s
et
an
d
let
(
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
T
h
en
(
,
ℑ
)
is
C
DH
i
f
f
(
,
ℑ
)
is
a
s
e
m
i
-
d
i
s
cr
ete
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
P
r
o
p
o
s
itio
n
2
.
1
7
.
[
3
4
]
L
et
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
a
n
d
let
:
(
,
ℑ
)
→
(
,
ℑ
)
b
e
a
f
u
zz
y
co
n
tin
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
i
s
m
)
m
ap
.
T
h
en
:
(
,
ℑ
)
→
(
,
ℑ
)
is
co
n
ti
n
u
o
u
s
(
h
o
m
eo
m
o
r
p
h
is
m
)
f
o
r
all
∈
[
0
,
1
)
.
P
r
o
p
o
s
itio
n
2
.
1
8
.
[
9
]
L
et
(
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
L
et
b
e
a
s
u
b
s
et
o
f
an
d
let
b
e
a
co
llectio
n
o
f
f
u
zz
y
p
o
in
t
s
o
f
.
T
h
en
w
e
h
av
e
t
h
e
f
o
llo
w
in
g
I
f
is
d
en
s
e
i
n
(
,
ℑ
₀
)
,
th
en
ℚ(
)
is
d
en
s
e(
I
)
in
(
,
ℑ
)
.
I
f
is
d
en
s
e(
I
)
in
(
,
ℑ
)
,
th
en
(
)
is
d
en
s
e
in
(
,
ℑ
₀
)
.
3.
DH
F
U
Z
Z
Y
T
O
P
O
L
O
G
I
CA
L
SPAC
E
S
I
n
th
is
s
ec
tio
n
,
w
e
w
ill
d
ef
i
n
e
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
s
.
W
e
w
il
l
p
r
o
v
e
t
h
at
o
u
r
n
e
w
co
n
ce
p
t
is
a
f
u
zz
y
to
p
o
lo
g
ical
p
r
o
p
er
ty
a
n
d
a
g
o
o
d
ex
te
n
s
io
n
o
f
DH
to
p
o
lo
g
ical
p
r
o
p
e
r
ty
i
n
t
h
e
s
en
s
e
o
f
L
o
w
en
.
Def
i
n
itio
n
3
.
1
.
A
co
llecti
o
n
o
f
f
u
zz
y
p
o
i
n
ts
o
f
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
,
ℑ
)
is
s
aid
to
b
e
-
s
e
m
i
-
d
is
cr
ete
i
f
f
(
)
=
⋃
∞
=
1
w
ith
(
,
ℑ
)
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
∈
ℕ
.
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
)
if
f
is
σ
-
s
e
m
i
-
d
is
cr
ete
an
d
is
d
en
s
e
(
I
)
.
-
s
e
m
i
-
di
s
cr
ete
d
en
s
e
(
I
I
)
if
f
is
σ
-
s
e
m
i
-
d
is
cr
ete
an
d
is
d
en
s
e
(I
I).
Def
i
n
itio
n
3
.
2
.
A
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
)
is
s
aid
to
b
e
d
en
s
el
y
h
o
m
o
g
e
n
eo
u
s
(
DH)
if
f
(
,
ℑ
)
h
as a
-
s
e
m
i
-
d
i
s
cr
ete
d
en
s
e(
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
.
I
f
an
d
ar
e
tw
o
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
ℑ
)
,
th
en
t
h
er
e
is
a
f
u
z
z
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
s
u
ch
t
h
at
ℎ
(
(
)
)
=
(
)
.
L
e
m
m
a
3
.
3
.
L
et
(
,
ℑ
)
b
e
a
f
u
zz
y
t
o
p
o
lo
g
ical
s
p
ac
e
an
d
b
e
a
-
s
e
m
i
-
d
is
cr
ete
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
.
T
h
en
ℚ(
(
)
)
is
a
-
s
e
m
i
-
d
is
cr
ete
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
ℑ
)
.
P
r
o
o
f
.
I
t
is
ea
s
y
to
s
ee
th
at
(
)
=
(
ℚ
(
(
)
)
)
an
d
h
en
ce
th
e
r
esu
l
t
is
o
b
v
io
u
s
.
T
h
eo
r
em
3
.
4
.
A
f
u
z
z
y
to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
)
is
DH
i
f
f
(
,
ℑ
)
h
as a
σ
-
s
e
m
i
-
d
i
s
cr
ete
d
en
s
e(
I
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
.
I
f
an
d
ar
e
t
w
o
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
t
s
o
f
(
,
ℑ
)
,
th
en
th
er
e
i
s
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
s
u
ch
t
h
at
ℎ
(
(
)
)
=
(
)
.
P
r
o
o
f
.
I
f
(
,
ℑ
)
is
DH,
th
en
(
,
ℑ
)
h
as
a
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(I)
co
llectio
n
o
f
f
u
zz
y
p
o
in
t
s
.
B
y
P
r
o
p
o
s
itio
n
2
.
1
1
(
i)
,
ℚ
(
(
)
)
is
d
e
n
s
e
(
I
I
)
an
d
b
y
L
e
m
m
a
3
.
3
,
ℚ
(
(
)
)
is
-
s
e
m
i
-
d
is
cr
ete.
L
et
an
d
b
e
an
y
t
w
o
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
ℑ
)
.
T
h
en
b
y
P
r
o
p
o
s
itio
n
2
.
1
1
(
ii)
an
d
L
e
m
m
a
3
.
3
,
ℚ
(
(
)
)
an
d
ℚ
(
(
)
)
ar
e
b
o
th
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
i
n
ts
o
f
(
,
ℑ
)
.
T
h
en
th
er
e
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
i
s
m
ℎ
:
(
,
ℑ
)
→
(
,
ℑ
)
s
u
c
h
th
a
t
ℎ
(
(
ℚ
(
(
)
)
)
)
=
(
ℚ
(
(
)
)
)
.
T
h
u
s
,
ℎ
(
(
)
)
=
(
)
.
T
h
e
p
r
o
o
f
o
f
th
e
o
t
h
er
d
ir
ec
tio
n
o
f
t
h
i
s
th
eo
r
e
m
is
s
i
m
ilar
to
t
h
e
ab
o
v
e
o
n
e.
L
e
m
m
a
3
.
5
.
L
et
(
X
,
τ
)
b
e
a
to
p
o
lo
g
ical
s
p
ac
e.
L
e
t
A
b
e
a
n
o
n
-
e
m
p
t
y
s
u
b
s
et
o
f
an
d
b
e
a
c
o
llectio
n
o
f
f
u
zz
y
p
o
in
t
s
o
f
.
T
h
en
τ
is
th
e
d
is
cr
ete
to
p
o
lo
g
y
i
f
f
(
,
(
)
)
is
s
e
m
i
-
d
is
cr
ete.
If
is
-
d
is
cr
ete
in
(
,
)
,
th
en
ℚ
(
)
is
-
s
e
m
i
-
d
is
cr
ete
i
n
(
,
(
)
)
.
I
f
is
-
s
e
m
i
-
d
is
cr
ete
in
(
,
(
)
)
,
th
en
(
)
is
-
d
is
cr
ete
in
(
,
)
.
P
r
o
o
f
.
(
i)
Su
p
p
o
s
e
th
at
τ
is
th
e
d
is
cr
ete
to
p
o
l
o
g
y
an
d
let
∈
.
T
h
en
t
h
er
e
ex
is
ts
∈
su
ch
th
a
t
{
}
=
∩
.
So
,
∩
=
∩
=
{
}
∈
(
)
.
B
u
t
clea
r
ly
(
)
is
th
e
cr
is
p
p
o
in
t
w
it
h
s
u
p
p
o
r
t
.
C
o
n
v
er
s
el
y
,
s
u
p
p
o
s
e
th
at
(
,
(
)
)
is
a
s
e
m
i
-
d
is
cr
ete
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
an
d
let
∈
.
T
h
en
th
e
r
e
ex
is
t
s
a
f
u
zz
y
p
o
i
n
t
o
r
a
f
u
zz
y
cr
is
p
p
o
in
t
s
u
c
h
t
h
at
∈
(
)
.
C
h
o
o
s
e
∈
(
)
s
u
c
h
t
h
at
=
∩
.
T
h
u
s
,
{
}
=
⁻
¹
(
0
,
1
]
∩
an
d
h
e
n
ce
{
}
∈
τ
.
(
ii)
Sin
ce
is
-
d
is
cr
ete
in
(
,
)
,
th
e
n
=
⋃
∞
=
1
w
i
th
is
th
e
d
i
s
cr
ete
to
p
o
lo
g
y
f
o
r
all
∈
ℕ
.
So
,
b
y
p
ar
t
(
i)
(
,
(
)
)
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
.
Sin
ce
(
ℚ
(
)
)
=
,
th
en
ℚ
(
)
is
-
s
e
m
i
-
d
is
cr
et
e
in
(
,
(
)
)
.
(
iii)
Si
n
ce
is
-
s
e
m
i
-
d
is
cr
et
e
in
(
,
(
)
)
,
th
e
n
(
)
=
⋃
∞
=
1
wi
th
(
,
(
)
)
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
∈
ℕ
.
So
,
b
y
p
ar
t
(
i)
is
th
e
d
is
cr
ete
to
p
o
lo
g
y
f
o
r
all
.
T
h
er
ef
o
r
e,
(
)
is
-
d
i
s
cr
ete
in
(
,
)
.
T
h
eo
r
em
3
.
6
.
L
et
(
,
)
b
e
a
t
o
p
o
lo
g
ical
s
p
ac
e.
T
h
en
(
,
)
is
DH
if
f
(
,
(
)
)
is
DH.
P
r
o
o
f
.
Su
p
p
o
s
e
t
h
at
(
,
)
is
D
H.
T
h
en
(
,
)
h
as
a
-
d
is
cr
ete
d
en
s
e
s
u
b
s
e
t
.
B
y
L
e
m
m
a
3
.
5
(
ii)
an
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Den
s
ely
h
o
mo
g
en
e
o
u
s
fu
z
z
y
s
p
a
ce
s
(
S
a
mer A
l G
h
o
u
r
)
3259
P
r
o
p
o
s
itio
n
2
.
1
2
(
i)
,
ℚ
(
)
is
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
)
in
(
,
(
)
)
.
L
et
an
d
b
e
t
w
o
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
(
)
)
.
T
h
en
b
y
L
e
m
m
a
3
.
5
(
iii)
an
d
Pro
p
o
s
itio
n
2
.
1
2
(
ii),
(
)
an
d
(
)
ar
e
b
o
th
-
d
is
cr
ete
d
e
n
s
e
s
u
b
s
et
s
o
f
(
,
)
.
T
h
u
s
,
th
er
e
is
a
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
)
→
(
,
)
s
u
c
h
th
a
t
ℎ
(
(
)
)
=
(
)
.
P
r
o
p
o
s
itio
n
2
.
1
3
en
d
s
th
e
p
r
o
o
f
o
f
th
i
s
d
ir
ec
tio
n
.
C
o
n
v
er
s
el
y
i
f
(
,
(
)
)
is
DH,
th
en
(
,
(
)
)
h
as
a
-
d
is
cr
ete
d
en
s
e
(
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
.
B
y
L
e
m
m
a
3
.
5
(
iii)
an
d
P
r
o
p
o
s
itio
n
2
.
1
2
(
ii),
(
)
is
-
d
is
cr
ete
d
en
s
e
in
(
,
)
.
L
et
an
d
b
e
tw
o
-
d
is
cr
ete
d
en
s
e
s
u
b
s
ets
o
f
(
,
)
.
T
h
en
b
y
L
e
m
m
a
3
.
5
(
ii)
an
d
P
r
o
p
o
s
itio
n
2
.
1
2
(
i)
,
ℚ
(
)
an
d
ℚ
(
)
ar
e
b
o
t
h
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
(
)
)
.
T
h
u
s
,
th
er
e
i
s
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
(
)
)
→
(
,
(
)
)
s
u
c
h
th
at
ℎ
(
(
ℚ
(
)
)
)
=
(
ℚ
(
)
)
.
So
,
ℎ
(
)
=
.
P
r
o
p
o
s
itio
n
2
.
1
3
en
d
s
th
e
p
r
o
o
f
o
f
th
i
s
d
ir
ec
tio
n
.
C
o
r
o
llar
y
3
.
7
.
DH
in
f
u
zz
y
t
o
p
o
lo
g
ical
s
p
ac
es
is
a
g
o
o
d
ex
ten
s
io
n
o
f
DH
i
n
to
p
o
lo
g
ical
s
p
ac
es.
R
ec
all
t
h
at
a
p
r
o
p
er
ty
Ƥ
o
f
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
s
is
ca
lled
a
f
u
zz
y
to
p
o
lo
g
ical
p
r
o
p
e
r
t
y
if
w
h
e
n
ev
er
(
,
ℑ
₁
)
p
o
s
s
ess
es
Ƥ
an
d
ℎ
:
(
,
ℑ
₁
)
→
(
,
ℑ
₂
)
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
,
th
e
n
(
,
ℑ
₂
)
p
o
s
s
ess
es
Ƥ
.
L
e
m
m
a
3
.
8
.
L
et
:
→
b
e
a
b
i
j
ec
tiv
e
m
ap
.
T
h
en
Fo
r
an
y
t
w
o
f
u
zz
y
s
et
s
,
in
,
(
∩
)
=
(
)
∩
(
)
.
Fo
r
an
y
⊆
,
(
)
=
(
)
.
P
r
o
o
f
.
Stra
ig
h
t
f
o
r
w
ar
d
.
L
e
m
m
a
3
.
9
.
L
et
:
(
,
ℑ
₁
)
→
(
,
ℑ
₂
)
b
e
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
.
L
et
b
e
a
n
o
n
-
e
m
p
t
y
s
u
b
s
et
o
f
an
d
b
e
a
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
.
T
h
en
I
f
(
,
(
ℑ
₁
)
)
is
s
e
m
i
-
d
is
cr
ete,
th
e
n
(
(
)
,
(
ℑ
₂
)
(
)
)
is
s
e
mi
-
d
is
cr
ete.
I
f
is
-
s
e
m
i
-
d
is
cr
ete,
th
e
n
(
)
is
-
s
e
m
i
-
d
is
cr
ete.
P
r
o
o
f
.
(
i)
L
et
∈
(
)
,
s
ay
=
(
)
f
o
r
s
o
m
e
∈
.
Sin
ce
(
,
(
ℑ
₁
)
)
is
s
e
m
i
-
d
is
cr
ete,
t
h
er
e
ex
is
t
s
∈
(
0
,
1
]
s
u
c
h
th
a
t
∈
(
ℑ
₁
)
.
C
h
o
o
s
e
∈
ℑ
₁
s
u
ch
t
h
at
=
∩
.
T
h
en
b
y
L
e
m
m
a
3
.
8
=
(
(
)
)
=
(
)
=
(
∩
)
=
(
)
∩
(
)
=
(
)
∩
(
)
.
Sin
ce
is
f
u
zz
y
o
p
en
,
it
f
o
llo
w
s
t
h
at
∈
(
ℑ
₂
)
(
)
.
ii)
Sin
ce
is
-
s
e
m
i
-
d
is
cr
ete,
(
)
=
⋃
∞
=
1
w
it
h
(
,
ℑ
)
is
s
e
m
i
-
d
i
s
cr
ete
f
o
r
all
∈
ℕ
.
B
y
P
r
o
p
o
s
itio
n
2
.
1
4
(
i)
,
(
(
)
)
=
(
(
)
)
=
(
⋃
∞
=
1
)
=
⋃
(
∞
=
1
)
.
A
ls
o
,
b
y
(
i)
w
e
h
av
e
(
(
)
,
(
ℑ
₂
)
(
)
)
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
∈
ℕ
.
I
t f
o
llo
w
s
th
a
t
(
)
is
-
s
e
m
i
-
d
is
cr
et
e.
T
h
eo
r
em
3
.
1
0
.
I
n
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es,
"
B
ein
g
"
DH"
is
a
f
u
zz
y
to
p
o
lo
g
ical
p
r
o
p
er
t
y
.
P
r
o
o
f
.
Ass
u
m
e
(
,
ℑ
₁
)
is
a
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
a
n
d
let
:
(
,
ℑ
₁
)
→
(
,
ℑ
₂
)
b
e
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
i
s
m
w
h
er
e
(
,
ℑ
₂
)
is
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
C
h
o
o
s
e
a
-
s
e
m
i
-
d
i
s
cr
et
e
d
en
s
e(
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
ℑ
₁
)
.
A
cc
o
r
d
in
g
to
L
e
m
m
a
3
.
9
(
i
i)
an
d
P
r
o
p
o
s
itio
n
2
.
1
4
(
ii),
(
)
w
i
ll
b
e
σ
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e(
I
)
in
(
,
ℑ
₂
)
.
L
et
an
d
b
e
an
y
t
w
o
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
ts
o
f
(
,
ℑ
₂
)
.
T
h
en
b
y
L
e
m
m
a
3
.
9
(
ii)
an
d
P
r
o
p
o
s
itio
n
2
.
1
4
(
ii),
⁻
¹
(
)
an
d
⁻
¹
(
)
ar
e
tw
o
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
i
n
ts
o
f
(
,
ℑ
₁
)
.
Sin
ce
(
,
ℑ
₁
)
is
DH,
th
er
e
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
ℎ
:
(
,
ℑ
₁
)
→
(
,
ℑ
₁
)
s
u
ch
th
at
ℎ
(
(
⁻
¹
(
)
)
)
=
(
⁻
¹
(
)
)
.
Def
in
e
:
(
,
ℑ
₂
)
→
(
,
ℑ
₂
)
b
y
=
∘
ℎ
∘
⁻
¹
.
T
h
en
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
.
Usi
n
g
P
r
o
p
o
s
itio
n
2
.
1
4
(
i)
,
w
e
ca
n
s
ee
th
a
t
(
(
)
)
=
(
)
.
4.
RE
L
AT
I
O
NSH
I
P
S B
E
T
W
E
E
N
DH
AND
CDH
F
U
Z
Z
Y
T
O
P
O
L
O
G
I
CA
L
SPAC
E
S
I
n
th
i
s
s
ec
tio
n
,
w
e
w
i
ll g
iv
e
s
o
m
e
r
elatio
n
s
h
ip
s
b
et
w
ee
n
DH
an
d
C
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es.
T
h
e
f
o
llo
w
in
g
u
s
e
f
u
l
le
m
m
a
f
o
llo
w
s
ea
s
il
y
:
L
e
m
m
a
4
.
1
.
L
e
t
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
an
d
P
b
e
a
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
X
w
ith
S
(
P
)
is
co
u
n
tab
le
an
d
n
o
n
-
e
m
p
t
y
.
T
h
en
P
is
σ
-
s
e
m
i
-
d
is
cr
ete.
T
h
eo
r
em
4
.
2
.
L
e
t
(
X
,
ℑ
)
b
e
a
f
u
zz
y
t
o
p
o
lo
g
ical
s
p
ac
e
f
o
r
w
h
ic
h
X
is
co
u
n
tab
le.
T
h
en
(
X
,
ℑ
)
is
DH
i
f
f
(
X
,
ℑ
)
is
s
e
m
i
-
d
is
cr
ete.
P
r
o
o
f
.
Sin
ce
th
e
r
esu
lt
is
o
b
v
io
u
s
w
h
en
|
X
|
=
1
,
w
e
w
ill
ass
u
m
e
t
h
at
|
X
|
>1
.
Su
p
p
o
s
e
th
at
(
X
,
ℑ
)
is
DH
an
d
ass
u
m
e
o
n
t
h
e
co
n
tr
ar
y
t
h
at
(
X
,
ℑ
)
is
n
o
t
s
e
m
i
-
d
is
cr
ete.
T
h
en
th
er
e
ex
i
s
ts
y
∈
X
s
u
c
h
th
at
y
a
∉
ℑ
f
o
r
all
0
<
a
≤
1
.
Set
P
=
ℚ
(
X
)
an
d
W
=
ℚ
(
X
∖
{
y
}
)
.
I
t
is
n
o
t
d
i
f
f
icu
l
t
to
s
ee
t
h
at
P
an
d
W
ar
e
d
en
s
e
(
I
)
.
A
l
s
o
,
b
y
L
e
m
m
a
4
.
1
,
P
an
d
W
ar
e
σ
-
s
e
m
i
-
d
i
s
cr
ete.
So
th
er
e
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
i
s
m
h
:
(
X
,
ℑ
)
→
(
X
,
ℑ
)
s
u
c
h
t
h
a
t
h
(
S
(
P
)
)
=
S
(
W
)
,
th
er
ef
o
r
e,
h
(
X
)
=
X
∖
{
y
}
w
h
ic
h
is
a
co
n
tr
ad
ictio
n
s
in
ce
h
is
an
o
n
to
m
ap
.
C
o
n
v
er
s
el
y
,
s
u
p
p
o
s
e
th
at
(
X
,
ℑ
)
is
s
e
m
i
-
d
is
cr
ete.
T
h
en
b
y
P
r
o
p
o
s
i
tio
n
2
.
1
5
(
ii),
(
X
,
ℑ
)
is
s
ep
ar
ab
le.
C
h
o
o
s
e
a
co
u
n
tab
le
d
en
s
e
(
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
i
n
ts
P
.
T
h
en
S
(
P
)
is
co
u
n
tab
le
a
n
d
b
y
L
e
m
m
a
4
.
1
,
P
is
σ
-
s
e
m
i
-
d
is
cr
ete.
L
et
P
an
d
W
b
e
an
y
t
w
o
σ
-
s
e
m
i
-
d
is
cr
ete
d
e
n
s
e
(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
t
s
.
T
h
en
b
y
P
r
o
p
o
s
itio
n
2
.
1
5
(
i)
,
S
(
P
)
=
S
(
W
)
=
X
an
d
th
e
i
d
en
tit
y
f
u
zz
y
m
ap
co
m
p
letes
th
e
p
r
o
o
f
.
C
o
r
o
llar
y
4
.
3
.
L
et
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
f
o
r
w
h
ic
h
X
is
co
u
n
tab
le.
T
h
en
(
X
,
ℑ
)
is
C
DH
if
f
(
X
,
ℑ
)
is
DH.
P
r
o
o
f
.
Fo
llo
w
s
f
r
o
m
P
r
o
p
o
s
itio
n
2
.
1
6
an
d
T
h
eo
r
em
4
.
2
.
T
h
eo
r
e
m
4
.
4
.
I
f
(
X
,
ℑ
)
is
s
ep
ar
ab
le
an
d
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e,
th
en
(
X
,
ℑ
)
is
C
D
H.
P
r
o
o
f
.
Fo
llo
w
s
f
r
o
m
t
h
e
d
ef
i
n
it
io
n
s
an
d
L
e
m
m
a
4
.
1
.
R
ec
all
t
h
at
a
f
u
zz
y
to
p
o
lo
g
ic
al
s
p
ac
e
(
X
,
ℑ
)
is
h
er
ed
itar
il
y
s
ep
ar
ab
le
if
ev
er
y
s
u
b
s
p
ac
e
o
f
(
X
,
ℑ
)
is
s
ep
ar
ab
le.
R
ec
all
th
at
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
is
s
ec
o
n
d
co
u
n
tab
le
if
it
h
as
a
co
u
n
t
ab
le
b
ase.
I
t
is
w
e
ll
k
n
o
w
n
t
h
at
s
ec
o
n
d
co
u
n
tab
le
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
es
ar
e
h
er
ed
itar
il
y
s
ep
ar
ab
le
.
L
e
m
m
a
4
.
5
.
I
f
(
X
,
ℑ
)
is
a
h
er
ed
itar
il
y
s
ep
ar
ab
le
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e
an
d
P
is
a
σ
-
s
e
m
i
-
d
is
cr
ete
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
o
f
(
X
,
ℑ
)
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
4
,
Au
g
u
s
t 2
0
1
9
:
3
2
5
6
-
3261
3260
th
en
S
(
P
)
is
co
u
n
tab
le.
P
r
o
o
f
.
Si
n
ce
P
is
σ
-
s
e
m
i
-
d
is
cr
ete,
th
e
n
S
(
P
)
=
⋃
A
n
∞
n
=
1
w
it
h
(
A
n
,
ℑ
A
n
)
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
n
∈
ℕ
.
Sin
ce
(
X
,
ℑ
)
is
h
er
ed
itar
il
y
s
ep
ar
ab
le,
th
en
f
o
r
e
ac
h
n
∈
ℕ
,
(
A
n
,
ℑ
A
n
)
is
s
ep
ar
ab
le
an
d
b
y
P
r
o
p
o
s
itio
n
2
.
1
5
(
ii)
it f
o
llo
w
s
th
at
A
n
is
co
u
n
tab
le.
T
h
u
s
,
S
(
P
)
is
co
u
n
tab
le.
T
h
eo
r
em
4
.
6
.
I
f
(
X,
ℑ)
is
h
er
ed
itar
il
y
s
ep
ar
ab
le
an
d
C
DH
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e,
th
e
n
(
X
,
ℑ
)
is
DH.
P
r
o
o
f
.
Si
n
ce
(
X
,
ℑ
)
is
h
er
ed
itar
il
y
s
ep
ar
ab
le,
th
en
it
i
s
s
ep
ar
ab
le.
So
,
th
er
e
e
x
is
ts
a
co
u
n
tab
le
d
en
s
e
(I)
co
llectio
n
o
f
f
u
zz
y
p
o
in
t
s
P
an
d
b
y
L
e
m
m
a
4
.
1
,
P
is
σ
-
se
mi
-
d
i
s
cr
ete.
L
et
P
an
d
W
b
e
t
w
o
σ
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e(
I
)
co
llectio
n
s
o
f
f
u
zz
y
p
o
in
ts
.
T
h
en
b
y
L
e
m
m
a
4
.
5
,
S
(
P
)
an
d
S
(
W
)
ar
e
co
u
n
tab
le.
B
y
P
r
o
p
o
s
itio
n
2
.
1
1
,
ℚ
(
S
(
P
)
)
an
d
ℚ
(
S
(
W
)
)
ar
e
co
u
n
tab
le
d
en
s
e(
I
)
.
Sin
ce
(
X
,
ℑ
)
is
C
DH,
t
h
er
e
is
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
h
:
(
X
,
ℑ
)
→
(
X
,
ℑ
)
s
u
ch
t
h
at
h
(
S
(
P
)
)
=
h
(
S
(
ℚ
(
S
(
P
)
)
)
)
=
S
(
ℚ
(
S
(
W
)
)
)
=
S
(
W
)
.
C
o
r
o
llar
y
4
.
7
.
L
et
(
X
,
ℑ
)
b
e
a
h
er
ed
itar
il
y
s
ep
ar
ab
le
f
u
z
z
y
to
p
o
lo
g
ical
s
p
ac
e.
T
h
en
(
X
,
ℑ
)
is
C
DH
i
f
f
(
X
,
ℑ
)
is
D
H.
P
r
o
o
f
.
Fo
llo
w
s
f
r
o
m
T
h
eo
r
em
s
4
.
4
an
d
4
.
6
.
C
o
r
o
lla
r
y
4
.
8
.
L
et
(
X
,
ℑ
)
b
e
a
s
ec
o
n
d
co
u
n
ta
b
le
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
T
h
en
(
X
,
ℑ
)
is
C
DH
i
f
f
(
X
,
ℑ
)
is
DH.
5.
CUT T
O
P
O
L
O
G
I
CA
L
SPA
CE
S
I
n
th
is
s
ec
tio
n
w
e
w
il
l
m
ain
l
y
s
h
o
w
t
h
at
a
-
c
u
t to
p
o
lo
g
ical
s
p
ac
e
(
X
,
ℑ
a
)
o
f
a
f
u
zz
y
to
p
o
lo
g
ical
(
X
,
ℑ
)
is
DH
in
g
en
er
al
o
n
l
y
i
f
a
=
0
.
L
em
m
a
5
.
1
.
L
et
(
X
,
ℑ
)
b
e
a
f
u
zz
y
to
p
o
lo
g
ical
s
p
ac
e.
L
et
B
b
e
n
o
n
-
e
m
p
t
y
s
u
b
s
et
o
f
X
an
d
let
P
b
e
a
c
o
llectio
n
o
f
f
u
zz
y
p
o
i
n
ts
o
f
X
.
T
h
en
(
,
ℑ
)
is
s
e
m
i
-
d
i
s
cr
ete
if
f
(
ℑ
₀
)
is
th
e
d
is
c
r
ete
to
p
o
lo
g
y
o
n
.
I
f
is
-
d
is
cr
ete
in
(
,
ℑ
₀
)
,
th
en
ℚ
(
)
is
-
s
e
m
i
-
d
is
cr
ete
i
n
(
,
ℑ
)
.
If
is
-
s
e
m
i
-
d
is
cr
ete
in
(
,
ℑ
)
,
th
en
(
)
is
-
d
is
cr
ete
i
n
(
,
ℑ
₀
)
.
P
r
o
o
f
.
(
i)
Su
p
p
o
s
e
t
h
at
(
B
,
ℑ
B
)
is
s
e
m
i
-
d
is
cr
ete
a
n
d
let
x
∈
B
.
T
h
en
th
e
r
e
ex
is
t
s
a
f
u
zz
y
p
o
i
n
t
o
r
a
f
u
zz
y
cr
i
s
p
p
o
in
t
x
a
f
o
r
s
o
m
e
a
w
ith
x
a
∈
ℑ
B
.
C
h
o
o
s
e
λ
∈
ℑ
s
u
c
h
t
h
at
x
a
=
λ
∩
B
.
T
h
en
(
λ
∩
B
)
(
x
)
=
min
{
λ
(
x
)
,
B
(
x
)
}
>
0
an
d
s
o
,
{
x
}
=
λ
⁻
¹
(
0
,
1
]
∩
B
∈
(
ℑ
₀
)
B
.
C
o
n
v
er
s
el
y
,
s
u
p
p
o
s
e
th
at
(
ℑ
₀
)
B
is
th
e
d
is
cr
ete
to
p
o
lo
g
y
o
n
B
an
d
let
x
∈
B
.
T
h
en
t
h
er
e
ex
is
t
s
λ
∈
ℑ
s
u
ch
th
a
t
{
x
}
=
λ
⁻
¹
(
0
,
1
]
∩
B
.
No
w
,
λ
∩
B
is
th
e
f
u
zz
y
o
r
cr
is
p
p
o
in
t
x
λ
(
x
)
,
o
n
th
e
o
th
er
h
an
d
,
λ
∩
B
∈
ℑ
B
.
ii)
Sin
ce
B
is
σ
-
d
is
cr
ete
in
(
X
,
ℑ
₀
)
,
th
en
⋃
B
n
∞
n
=
1
w
it
h
(
ℑ
₀
)
B
n
is
th
e
d
i
s
cr
ete
to
p
o
lo
g
y
f
o
r
all
n
∈
ℕ
.
B
y
(
i)
,
ℑ
B
n
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
n
∈
ℕ
.
Sin
ce
S
(
ℚ
(
B
)
)
=
B
,
th
en
ℚ
(
B
)
is
σ
-
d
is
cr
ete
in
(
X
,
ℑ
)
.
iii)
Si
n
ce
P
is
σ
-
d
is
cr
ete
i
n
(
X
,
ℑ
)
,
th
en
S
(
P
)
=
⋃
A
n
∞
n
=
1
w
it
h
ℑ
A
n
is
s
e
m
i
-
d
is
cr
ete
f
o
r
all
n
∈
ℕ
.
B
y
(
i)
,
(
ℑ
₀
)
A
n
is
th
e
d
is
cr
ete
to
p
o
lo
g
y
o
n
A
n
f
o
r
all
n
∈
ℕ
.
I
t
f
o
llo
w
s
th
a
t
S
(
P
)
is
σ
-
d
is
cr
ete
in
(
X
,
ℑ
₀
)
.
T
h
eo
r
em
5
.
2
.
I
f
(
X,
ℑ)
is
a
DH
f
u
zz
y
to
p
o
lo
g
ic
al
s
p
ac
e,
th
en
(
X,
ℑ₀)
is
DH.
P
r
o
o
f
.
Su
p
p
o
s
e
t
h
at
(
X
,
ℑ
)
is
D
H.
T
h
en
(
X
,
ℑ
)
h
a
s
a
σ
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e(
I
)
co
llectio
n
o
f
f
u
zz
y
p
o
in
ts
P
.
B
y
L
e
m
m
a
5
.
1
(
iii)
an
d
P
r
o
p
o
s
itio
n
2
.
1
8
(
ii),
S
(
P
)
is
σ
-
d
is
cr
ete
d
en
s
e
i
n
(
X
,
ℑ
₀
)
.
L
et
A
an
d
B
b
e
an
y
t
w
o
σ
-
d
is
cr
ete
d
en
s
e
s
et
s
i
n
(
X
,
ℑ
₀
)
.
T
h
en
b
y
L
e
m
m
a
5
.
1
(
ii)
an
d
P
r
o
p
o
s
itio
n
2
.
1
8
(
i)
,
ℚ
(
A
)
an
d
ℚ
(
B
)
ar
e
σ
-
s
e
m
i
-
d
is
cr
ete
d
en
s
e
(
I
)
in
(
X
,
ℑ
)
.
Sin
ce
(
X
,
ℑ
)
is
DH,
t
h
er
e
ex
is
t
s
a
f
u
zz
y
h
o
m
eo
m
o
r
p
h
is
m
h
:
(
X
,
ℑ
)
→
(
X
,
ℑ
)
s
u
c
h
t
h
at
h
(
S
(
ℚ
(
A
)
)
)
=
S
(
ℚ
(
(
B
)
)
)
.
B
y
P
r
o
p
o
s
itio
n
2
.
1
7
,
h
:
(
X
,
ℑ
₀
)
→
(
X
,
ℑ
₀
)
is
a
h
o
m
eo
m
o
r
p
h
is
m
.
O
n
t
h
e
o
th
er
h
an
d
,
S
(
ℚ
(
A
)
)
=
A
an
d
S
(
ℚ
(
B
)
)
=
B
.
T
h
er
ef
o
r
e,
(
X
,
ℑ
₀
)
is
DH.
T
h
e
f
o
llo
w
i
n
g
p
r
o
p
o
s
itio
n
is
w
ell
k
n
o
w
n
:
P
r
o
p
o
s
itio
n
5
.
3
.
L
et
(
,
)
b
e
a
to
p
o
lo
g
ical
s
p
ac
e
w
it
h
X
is
co
u
n
ta
b
le.
T
h
en
th
e
f
o
llo
w
i
n
g
ar
e
eq
u
iv
ale
n
t
:
(
,
)
is
C
DH.
is
th
e
d
is
cr
ete
to
p
o
lo
g
y
o
n
.
(
,
)
is
DH.
T
h
eo
r
em
5
.
4
.
L
et
X
b
e
a
co
u
n
tab
le
s
et
an
d
le
t
(
X
,
ℑ
)
b
e
a
f
u
z
z
y
to
p
o
lo
g
ical
s
p
ac
e.
T
h
en
t
h
e
f
o
llo
w
in
g
ar
e
eq
u
i
v
alen
t:
(
,
ℑ
)
is
DH.
(
,
ℑ
)
is
C
DH.
(
,
ℑ
₀
)
is
DH.
(
,
ℑ
₀
)
is
C
DH.
P
r
o
o
f
.
Fo
llo
w
s
f
r
o
m
T
h
eo
r
e
m
4
.
2
,
L
e
m
m
a
5
.
1
(
i)
an
d
P
r
o
p
o
s
itio
n
5
.
3
.
I
n
f
ac
t
if
a
>
0
,
th
en
(
X
,
ℑ
)
b
ein
g
DH
d
o
es
n
o
t
i
m
p
l
y
,
in
g
en
er
al,
t
h
at
(
X
,
ℑ
a
)
is
DH.
T
h
is
w
ill
b
e
e
x
p
lai
n
ed
in
th
e
f
o
llo
w
i
n
g
co
u
n
ter
e
x
a
m
p
le:
E
x
a
m
p
le
5
.
5
.
Fo
r
f
ix
ed
0
<
a
<
1
,
let
X
=
{
x
,
y
}
an
d
d
ef
in
e
ℑ
=
{
0
,
1
,
x
a
/
2
,
y
a
/
4
,
x
a
/
2
∪
y
a
/
4
}
.
I
t
is
clea
r
th
at
(
X
,
ℑ
)
is
s
e
m
i
-
d
is
cr
ete
an
d
s
o
b
y
T
h
eo
r
e
m
4
.
2
,
it
is
DH.
On
t
h
e
o
th
er
h
an
d
,
s
i
n
ce
ℑ
a
=
{
∅
,
X
}
,
th
en
(
X
,
ℑ
a
)
is
n
o
t D
H.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Den
s
ely
h
o
mo
g
en
e
o
u
s
fu
z
z
y
s
p
a
ce
s
(
S
a
mer A
l G
h
o
u
r
)
3261
RE
F
E
R
E
NC
E
S
[1
]
L
.
A
.
Zad
e
h
,
"
F
u
z
z
y
S
e
ts,"
In
fo
rm
a
n
d
c
o
n
tro
l
,
v
o
l.
8
,
p
p
.
3
3
8
-
3
5
3
,
1
9
6
5
.
[2
]
C.
L
.
Ch
a
n
g
,
"
F
u
z
z
y
T
o
p
o
lo
g
i
c
a
l
S
p
a
c
e
s,"
J
o
u
rn
a
l
o
f
M
a
th
e
ma
ti
c
a
l
An
a
lys
is
a
n
d
Ap
p
li
c
a
ti
o
n
s
,
v
o
l.
2
4
,
p
p
.
1
8
2
-
1
9
0
,
1
9
6
8
.
[3
]
S
.
A
l
G
h
o
u
r,
"
Ho
m
o
g
e
n
e
it
y
in
f
u
z
z
y
sp
a
c
e
s
a
n
d
th
e
ir
in
d
u
c
e
d
sp
a
c
e
s,"
Qu
e
stio
n
s
a
n
d
A
n
sw
e
rs
in
Ge
n
e
ra
l
T
o
p
o
lo
g
y
,
v
o
l
.
2
1
,
p
p
.
1
8
5
-
1
9
5
,
2
0
0
3
.
[4
]
S
.
A
l
G
h
o
u
r,
"
S
L
H f
u
z
z
y
sp
a
c
e
s,
"
Af
ric
a
n
Dia
s
p
o
r
a
J
o
u
r
n
a
l
o
f
M
a
th
e
ma
t
ics
,
v
o
l.
2
,
p
p
.
6
1
-
6
7
,
2
0
0
4
.
[5
]
S
.
A
l
G
h
o
u
r,
A
.
F
o
ra
,
"
M
i
n
im
a
l
it
y
a
n
d
Ho
m
o
g
e
n
e
it
y
in
F
u
z
z
y
S
p
a
c
e
s,"
J
o
u
rn
a
l
o
f
F
u
zz
y
M
a
th
e
ma
ti
c
s,
v
o
l.
1
2
,
p
p
.
7
2
5
--
7
3
7
,
2
0
0
4
.
[6
]
S
.
A
l
G
h
o
u
r,
"
L
o
c
a
l
h
o
m
o
g
e
n
e
it
y
in
f
u
z
z
y
to
p
o
lo
g
ica
l
sp
a
c
e
s,"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
M
a
th
e
ma
t
ics
a
n
d
M
a
th
e
ma
ti
c
a
l
S
c
ien
c
e
s
,
A
rt.
ID 8
1
4
9
7
,
1
4
p
p
,
2
0
0
6
.
[7
]
S
.
A
l
G
h
o
u
r,
"
S
o
m
e
G
e
n
e
ra
li
z
a
ti
o
n
s
o
f
M
in
im
a
l
F
u
z
z
y
Op
e
n
S
e
ts,"
Acta
M
a
t
h
e
ma
ti
c
a
Un
ive
rs
it
a
ti
s
Co
me
n
ia
n
a
e
,
v
o
l.
7
5
,
p
p
.
1
0
7
-
1
1
7
,
2
0
0
6
.
[8
]
S
.
A
l
G
h
o
u
r,
K.
A
l
-
Zo
u
b
i,
"
On
so
m
e
o
rd
in
a
ry
a
n
d
f
u
z
z
y
h
o
m
o
g
e
n
e
it
y
t
y
p
e
s,
"
Acta
M
a
th
e
ma
ti
c
a
Un
ive
rs
it
a
ti
s
Co
me
n
ia
n
a
e
,
v
o
l.
7
7
,
p
p
.
1
9
9
-
2
0
8
,
2
0
0
8
.
[9
]
S
.
A
l
G
h
o
u
r,
A
.
F
o
ra
,
"
On
CDH
f
u
z
z
y
sp
a
c
e
s,"
J
o
u
rn
a
l
o
f
In
telli
g
e
n
t
&
Fu
zz
y
S
y
ste
ms
,
v
o
l.
3
0
,
p
p
.
9
3
5
-
9
4
1
,
2
0
1
6
.
[1
0
]
A
.
F
o
ra
,
S
.
A
l
G
h
o
u
r,
"
Ho
m
o
g
e
n
e
it
y
in
F
u
z
z
y
S
p
a
c
e
s,"
Qu
e
stio
n
s
a
n
d
An
swe
rs
in
Ge
n
e
ra
l
T
o
p
o
lo
g
y
,
v
o
l.
1
9
,
pp.
1
5
9
-
1
6
4
,
2
0
0
1
.
[1
1
]
S
.
A
l
G
h
o
u
r,
A
.
A
z
a
iz
e
h
,
"
F
u
z
z
y
Ho
m
o
g
e
n
e
o
u
s
Bit
o
p
o
lo
g
ica
l
S
p
a
c
e
s,"
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
El
e
c
trica
l
a
n
d
Co
mp
u
ter
E
n
g
in
e
e
rin
g
,
v
o
l.
8
,
p
p
.
2
0
8
8
-
8
7
0
8
,
2
0
1
8
.
[1
2
]
R.
Be
n
n
e
tt
,
"
Co
u
n
tab
le d
e
n
se
h
o
m
o
g
e
n
e
o
u
s sp
a
c
e
s,"
F
u
n
d
a
m
e
n
ta
M
a
th
e
m
a
ti
c
a
e
,
v
o
l.
7
4
,
p
p
.
1
8
9
-
1
9
4
,
1
9
7
2
.
[1
3
]
A
.
V
.
A
rh
a
n
g
e
l'
s
k
ii
,
J.
v
a
n
M
i
ll
,
"
On
th
e
c
a
rd
in
a
li
ty
o
f
c
o
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s,"
Pro
c
e
e
d
in
g
s
o
f
th
e
Ame
ric
a
n
M
a
th
e
ma
ti
c
a
l
S
o
c
iety
,
v
o
l.
1
4
1
,
p
p
.
4
0
3
1
-
4
0
3
8
,
2
0
1
3
.
[1
4
]
R.
He
rn
a
n
d
e
z
-
G
u
ti
é
rre
z
,
"
Co
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
it
y
a
n
d
th
e
d
o
u
b
le
a
rro
w
sp
a
c
e
,
"
T
o
p
o
lo
g
y
a
n
d
I
ts
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
1
6
0
,
p
p
.
1
1
2
3
-
1
1
2
8
,
2
0
1
3
.
[1
5
]
R.
He
rn
a
n
d
e
z
-
G
u
ti
é
rre
z
,
M
.
Hr
u
ša
k
,
"
No
n
-
m
e
a
g
e
r
P
-
f
il
t
e
rs
a
re
c
o
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
o
u
s,"
Co
ll
o
q
u
iu
m
M
a
th
e
ma
ti
c
u
m
,
v
o
l.
1
3
0
,
p
p
.
2
8
1
-
289
,
2
0
1
3
.
[1
6
]
R.
He
rn
a
n
d
e
z
-
G
u
ti
é
rre
z
,
M
.
Hru
ša
k
,
J.
v
a
n
M
il
l,
"
Co
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
it
y
a
n
d
λ
-
se
ts,"
Fu
n
d
a
me
n
t
a
M
a
th
e
ma
ti
c
a
e
,
v
o
l.
2
2
6
,
p
p
.
1
5
7
-
172
,
2
0
1
4
.
[1
7
]
M
.
Hru
sa
k
,
J.
v
a
n
M
il
l,
"
Ne
a
rly
c
o
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s,"
Ca
n
a
d
i
a
n
J
o
u
rn
a
l
o
f
M
a
th
e
ma
ti
c
s
,
v
o
l.
6
6
,
p
p
.
7
4
3
-
758
,
2
0
1
4
.
[1
8
]
J.
v
a
n
M
il
l,
"
On
c
o
u
n
tab
le
d
e
n
s
e
a
n
d
n
-
h
o
m
o
g
e
n
e
it
y
,
"
Ca
n
a
d
ia
n
M
a
th
e
ma
ti
c
a
l
Bu
ll
e
ti
n
,
v
o
l.
5
6
,
p
p
.
8
6
0
-
8
6
9
,
2
0
1
3
.
[1
9
]
J.
v
a
n
M
il
l,
"
Co
u
n
ta
b
le
d
e
n
se
h
o
m
o
g
e
n
e
o
u
s
rim
c
o
m
p
a
c
t
sp
a
c
e
s
a
n
d
lo
c
a
l
c
o
n
n
e
c
ti
v
it
y
,
"
Fi
lo
ma
t
,
v
o
l
.
2
9
,
p
p
.
1
7
9
-
182
,
2
0
1
5
.
[2
0
]
D.
Re
p
o
v
š,
L
.
Zd
o
m
sk
y
y
,
S
.
Zh
a
n
g
,
"
Co
u
n
tab
le
d
e
n
se
h
o
m
o
g
e
n
e
o
u
s
f
il
ters
a
n
d
th
e
M
e
n
g
e
r
c
o
v
e
rin
g
p
ro
p
e
rty
,
"
Fu
n
d
a
me
n
t
a
M
a
t
h
e
ma
ti
c
a
e
,
v
o
l.
2
2
4
,
p
p
.
2
3
3
-
2
4
0
,
2
0
1
4
.
[2
1
]
B.
F
it
z
p
a
tri
c
k
,
N.
F
.
L
a
u
e
r,
"
De
n
se
l
y
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s
(I),
"
Ho
u
sto
n
J
o
u
rn
a
l
o
f
M
a
th
e
m
a
ti
c
s
,
v
o
l.
1
3
,
p
p
.
19
-
2
5
,
1
9
8
7
.
[2
2
]
S
.
K.
Ch
o
,
"
S
o
m
e
re
su
lt
s
re
late
d
to
d
e
n
se
ly
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s,
"
Co
mm
u
n
ica
ti
o
n
s
o
f
th
e
Ko
re
a
n
M
a
th
e
ma
ti
c
a
l
S
o
c
iety
,
v
o
l.
1
1
,
p
p
.
1
0
6
1
-
1
0
6
6
,
1
9
9
6
.
[2
3
]
D.
L
.
F
e
a
rn
le
y
,
"
A
M
o
o
re
sp
a
c
e
w
it
h
a
-
d
isc
re
te
-
b
a
se
w
h
ich
c
a
n
n
o
t
b
e
d
e
n
se
ly
e
m
b
e
d
d
e
d
in
a
n
y
M
o
o
re
sp
a
c
e
w
it
h
th
e
Ba
ire p
ro
p
e
rty
,
"
Pro
c
e
e
d
in
g
s
o
f
t
h
e
Ame
ric
a
n
M
a
th
e
ma
ti
c
a
l
S
o
c
iety
,
v
o
l.
1
2
7
,
p
p
.
3
0
9
5
-
3
1
0
0
,
1
9
9
9
.
[2
4
]
B.
F
it
z
p
a
tri
c
k
,
H.
X.
Zh
o
u
,
"
D
e
n
se
l
y
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s.
(II
),
"
Ho
u
sto
n
J
o
u
rn
a
l
o
f
M
a
t
h
e
ma
ti
c
s
,
v
o
l.
1
4
,
p
p
.
57
-
6
8
,
1
9
8
8
.
[2
5
]
B.
F
it
z
p
a
tri
c
k
,
H.
X
.
Z
h
o
u
,
"
S
o
m
e
o
p
e
n
p
ro
b
lem
s
in
d
e
n
se
l
y
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s,"
Op
e
n
p
ro
b
le
ms
in
to
p
o
lo
g
y
,
No
rth
-
Ho
l
la
n
d
,
Amste
rd
a
m
,
p
p
.
2
5
1
-
2
5
9
,
1
9
9
0
.
[2
6
]
B.
F
it
z
p
a
tr
ick
,
J.
W
h
it
e
,
H.
X
.
Z
h
o
u
,
"
Ho
m
o
g
e
n
e
it
y
a
n
d
σ
-
d
isc
re
te
se
ts,"
T
o
p
o
lo
g
y
a
n
d
I
ts
Ap
p
li
c
a
ti
o
n
s
,
v
o
l.
4
4
,
p
p
.
1
4
3
-
147
,
1
9
9
2
.
[2
7
]
S
.
V
.
M
e
d
v
e
d
e
v
,
"
M
e
tri
z
a
b
le
DH
-
sp
a
c
e
s
o
f
th
e
f
irst
c
a
teg
o
r
y
,
"
T
o
p
o
lo
g
y
a
n
d
Its
Ap
p
l
ica
ti
o
n
s
,
v
o
l.
1
7
9
,
p
p
.
1
7
1
-
178
,
2
0
1
5
.
[2
8
]
W
.
L
.
S
a
lt
s
m
a
n
,
"
Co
m
p
o
n
e
n
ts
o
f
d
e
n
se
l
y
h
o
m
o
g
e
n
e
o
u
s
sp
a
c
e
s
,
"
Ho
u
sto
n
J
o
u
r
n
a
l
o
f
M
a
t
h
e
ma
ti
c
s
,
v
o
l.
1
8
,
p
p
.
4
1
7
-
4
2
2
,
1
9
9
2
.
[2
9
]
C.
K.
W
o
n
g
,
"
F
u
z
z
y
p
o
in
ts
a
n
d
lo
c
a
l
p
r
o
p
e
rti
e
s
o
f
f
u
z
z
y
to
p
o
lo
g
y
,
"
J
o
u
rn
a
l
o
f
M
a
t
h
e
ma
ti
c
a
l
An
a
lys
is
a
n
d
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
4
6
,
p
p
.
3
1
6
-
3
2
8
,
1
9
7
4
.
[3
0
]
R.
S
riv
a
sta
v
a
,
S
.
N.
L
a
l,
A
.
K.
S
ri
v
a
sta
v
a
,
"
F
u
z
z
y
Ha
u
sd
o
rff
to
p
o
lo
g
ica
l
sp
a
c
e
s,"
J
o
u
rn
a
l
o
f
M
a
th
e
m
a
ti
c
a
l
An
a
lys
is
a
n
d
Ap
p
li
c
a
ti
o
n
s
,
v
o
l.
8
1
,
p
p
.
4
9
7
-
5
0
6
,
1
9
8
1
.
[3
1
]
M
.
H.G
h
a
n
im
,
E.
E.
Ke
rre
,
A
.
S
.
M
a
sh
h
o
u
r,
"
S
e
p
a
ra
ti
o
n
a
x
io
m
s,
su
b
sp
a
c
e
s
a
n
d
su
m
s
in
f
u
z
z
y
to
p
o
lo
g
y
,
"
J
o
u
rn
a
l
o
f
M
a
t
h
e
ma
ti
c
a
l
An
a
lys
is a
n
d
I
ts
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
1
0
2
,
p
p
.
1
8
9
-
2
0
2
,
1
9
8
4
.
[3
2
]
A
.
F
o
ra
,
"
S
e
p
a
ra
ti
o
n
a
x
io
m
s f
o
r
fu
z
z
y
sp
a
c
e
s,"
Fu
zz
y
S
e
ts a
n
d
S
y
ste
ms
,
v
o
l.
3
3
,
p
p
.
5
9
-
7
5
,
1
9
8
9
.
[3
3
]
R.
L
o
w
e
n
,
"
A
c
o
m
p
a
riso
n
o
f
d
iffere
n
t
c
o
m
p
a
c
tn
e
ss
n
o
ti
o
n
s
in
f
u
z
z
y
to
p
o
l
o
g
ica
l
sp
a
c
e
s,"
J
o
u
rn
a
l
o
f
M
a
th
e
ma
ti
c
a
l
An
a
lys
is
a
n
d
Its
A
p
p
l
ica
ti
o
n
s
,
v
o
l
.
6
4
,
p
p
.
4
4
6
-
4
5
4
,
1
9
7
8
.
[3
4
]
G
.
J.
W
a
n
g
,
"
T
h
e
o
r
y
o
f
L
-
f
u
z
z
y
t
o
p
o
lo
g
ica
l
sp
a
c
e
,
"
S
h
a
n
x
i
No
rm
a
l
Un
iv
e
rsity
P
re
ss
,
X
ian
,
(i
n
Ch
i
n
e
se
),
1
9
8
8
.
[3
5
]
A
.
F
o
ra
,
"
S
e
p
a
ra
ti
o
n
a
x
io
m
s,
su
b
sp
a
c
e
s
a
n
d
p
ro
d
u
c
t
sp
a
c
e
s
in
fu
z
z
y
to
p
o
lo
g
y
,
"
Ara
b
Gu
lf
J
o
u
rn
a
l
fo
r
S
c
ien
ti
fi
c
Res
e
a
rc
h
,
v
o
l.
8
,
p
p
.
1
-
1
6
,
1
9
9
0
.
Evaluation Warning : The document was created with Spire.PDF for Python.