I
nte
rna
t
io
na
l J
o
urna
l o
f
Adv
a
nces in Applie
d Science
s
(
I
J
AAS)
Vo
l.
6
,
No
.
2
,
J
u
n
e
201
7
,
p
p
.
1
1
7
~
1
2
5
I
SS
N:
2252
-
8814
117
J
o
ur
na
l ho
m
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a
g
e
:
h
ttp
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e
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jo
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.
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h
p
/I
JA
A
S
Federe
r Mea
sure
s, G
o
o
d and
No
np
la
na
r F
unctions
of Metri
c
Dio
pha
ntine
App
ro
x
i
m
a
tion
F
a
iza
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ra
m
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c
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n
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2
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0
0
9
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,
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.
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.
Ch
in
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9
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1
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cc
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1
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li
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e
m
a
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re
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lt
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[1
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a
n
d
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u
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q
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n
t
p
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rs
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n
m
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y
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o
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a
r
f
o
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m
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I
n
p
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rti
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u
lar,
w
e
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li
sh
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jo
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t
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a
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ly
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ial
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lo
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o
f
latti
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s.
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ey
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m
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p
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t
©
201
7
In
s
t
it
u
te o
f
A
d
v
a
n
c
e
d
E
n
g
i
n
e
e
rin
g
a
n
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S
c
ien
c
e
.
Al
l
rig
h
ts
re
se
rv
e
d
.
C
o
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r
e
s
p
o
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A
uth
o
r
:
Faiza
Ak
r
a
m
,
Dep
ar
t
m
en
t o
f
Ma
th
e
m
at
ics,
S
ch
o
o
l o
f
Scien
ce
,
Nan
j
in
g
Un
i
v
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it
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o
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Scien
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d
T
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h
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y
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Nan
j
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g
,
J
ian
g
s
u
,
2
1
0
0
9
7
,
P
.
R
.
C
h
i
n
a
.
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m
ail:
f
aiza
.
ak
r
a
m
7
8
6
@
y
a
h
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o
.
co
.
u
k
1.
I
NT
RO
D
UCT
I
O
N
T
h
e
th
eo
r
y
o
f
s
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m
u
lta
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e
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p
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m
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s
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ce
r
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ed
w
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th
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q
u
esti
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:
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f
Y
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a
n
×
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ea
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m
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tr
ix
(
i
n
ter
p
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s
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te
m
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r
m
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m
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m
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ize
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n
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g
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f
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|
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−
(
1
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δ
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.6
,
No
.
2,
J
u
n
e
201
7
:
117
–
1
2
5
118
d
o
es
n
o
t h
av
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i
n
f
in
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p
r
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ties
o
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to
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s
∕
f
o
r
m
s
h
ap
p
en
to
b
e
g
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w
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h
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s
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e
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s
u
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e.
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h
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c
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cle
o
f
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o
b
le
m
s
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ates
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ac
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to
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1930s
,
n
am
e
l
y
,
to
K.
Ma
h
l
er
’
s
w
o
r
k
o
n
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f
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n
o
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tr
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ce
n
d
en
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l
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u
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s
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et
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o
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tio
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s
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s
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tate
th
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lt
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o
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y
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e
s
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ac
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s
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m
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ties
:
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+
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a
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d
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‖
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≤
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n
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d
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s
u
c
h
t
h
at
‖
+
‖
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<
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‖
∞
−
∕
.
(
1
.
1
.
3
)
Nex
t,
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e
s
a
y
th
a
t Y
is
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e
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b
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e
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∈
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∖
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n
d
∈
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th
‖
Y
+
‖
∞
<
‖
‖
∞
−
n
m
⁄
−
δ
.
(
1
.
1
.
4
)
On
e
ca
n
s
h
o
w
th
a
t
L
eb
e
s
g
u
e
-
a.
e
Y
is
n
o
t
VW
A
b
y
B
o
r
el
-
C
a
n
tell
i
le
m
m
a
(
w
e
w
ill
i
n
tr
o
d
u
ce
th
is
le
m
m
a
i
n
o
u
r
n
e
x
t
s
ec
tio
n
)
.
Als
o
n
o
te
t
h
at
b
y
K
h
in
tch
i
n
e
’
s
T
r
an
s
f
er
en
ce
P
r
in
cip
le,
s
ee
e.
g
.
[
8
,
C
h
ap
ter
V]
,
Y
is
VW
M
if
f
t
h
e
tr
an
s
p
o
s
e
o
f
Y
is
.
W
ith
th
e
s
e
d
ef
in
itio
n
s
an
d
n
o
tatio
n
s
,
let
u
s
g
o
b
ac
k
to
o
n
e
th
eo
r
e
m
co
n
j
ec
tu
r
ed
b
y
Ma
h
ler
[
9
]
in
1932
an
d
p
r
o
v
ed
th
r
ee
d
ec
a
d
es
la
ter
b
y
V
.
Sp
r
i
n
d
z
̌
uk
,
s
ee
[
7
,
6
]
,
w
h
ic
h
s
tate
s
th
at
f
o
r
λ
–
a.
e.
.
∈
ℝ
,
th
e
r
o
w
v
ec
to
r
.
(
)
=
(
,
2
,
⋯
,
)
(
1
.
1
.
5
)
is
n
o
t V
W
M.
B
y
e
x
te
n
d
in
g
th
is
p
r
o
b
le
m
in
t
o
a
m
o
r
e
g
e
n
er
al
s
etti
n
g
,
th
at
i
s
,
f
o
r
∈
ℝ
d
,
b
y
d
e
f
in
itio
n
:
(
x
)
=
(
f
1
(
x
)
,
⋯
,
f
n
(
x
)
)
,
Wi
th
’
s
co
n
t
in
u
o
u
s
m
ap
s
f
r
o
m
ℝ
to
ℝ
,
o
n
e
as
k
w
h
et
h
er
o
r
n
o
t
f
o
r
al
m
o
s
t
ev
er
y
x
∈
ℝ
d
,
f
(
x
)
is
n
o
t
VW
A
,
w
it
h
r
esp
ec
t
to
L
eb
esg
u
e
m
ea
s
u
r
e
o
r
s
o
m
e
o
th
er
m
ea
s
u
r
es.
I
n
th
i
s
s
et
tin
g
,
[
4
]
p
r
o
v
ed
th
e
r
esu
lt
f
o
r
L
eb
es
g
u
e
Me
as
u
r
e
an
d
n
o
n
d
eg
en
er
ate
m
ap
f
an
d
[
1
]
p
r
o
v
ed
th
e
r
esu
lt
f
o
r
m
o
r
e
g
en
e
r
al
ass
u
m
p
tio
n
s
o
n
m
ea
s
u
r
es
a
n
d
m
ap
s
.
B
ef
o
r
e
w
e
s
tate
r
es
u
lt
o
f
[
4
]
,
r
ec
all
t
h
at
a
s
m
o
o
t
h
m
ap
f
f
r
o
m
⊂
ℝ
to
ℝ
is
ca
lled
n
o
n
d
eg
e
n
er
ate
at
∈
U
if
p
ar
tial
d
e
r
iv
ati
v
es
o
f
f
at
x
s
p
an
ℝ
an
d
f
i
s
n
o
n
d
eg
e
n
er
ate
i
f
it
is
n
o
n
d
eg
en
er
ate
at
λ
-
a.
e.
x
∈
U.
T
h
eo
r
em
1
.
1
.
1
.
L
et
b
e
a
Fed
er
er
m
ea
s
u
r
e
o
n
ℝ
d
,
U
an
o
p
en
s
u
b
s
et
o
f
ℝ
d
‚
an
d
F:U
→
,
a
co
n
tin
u
o
u
s
m
ap
s
u
c
h
th
a
t
(
D
F
‚
)
is
g
o
o
d
an
d
n
o
n
p
lan
ar
.
T
h
en
f
o
r
-
a.
e
0
U
th
er
e
ex
is
t
a
b
all
B
⊂
U
ce
n
ter
ed
at
0
an
d
C
,
α
>
0
s
u
ch
th
at
f
o
r
an
y
=
(
t
1
,
∙
∙
∙
‚
t
m
,
t
m
+
1
,
⋅
⋅
⋅
,
t
m
+
n
)
∈
+
an
d
an
y
ε
>
0
,
({
x
ϵ
B
:
̅
(
F(x
)
)
∉
})
<
C
.
W
e
w
ill p
r
o
v
e
th
is
t
h
eo
r
e
m
lat
er
.
Nex
t,
let
u
s
i
n
tr
o
d
u
ce
B
o
r
el
-
C
a
n
tell
i
le
m
m
a:
L
e
m
m
a
1
.
1
.
2
(
B
o
r
el
-
C
an
telli).
L
et
b
e
a
f
in
ite
m
ea
s
u
r
eo
n
B
,
i.e
.
(
B
)
<∞.
I
f
E
i
i
1
is
a
s
eq
u
en
ce
o
f
s
ets i
n
B
s
u
c
h
t
h
at
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S
I
SS
N:
2252
-
8814
F
ed
erer
Mea
s
u
r
e
s
,
Go
o
d
a
n
d
N
o
n
p
la
n
a
r
F
u
n
ctio
n
s
o
f Ma
tr
ic
Dio
p
h
a
n
ti
n
e
…
(
F
a
iz
a
A
kra
m
)
119
1
i
v
(
)
<
∞,
T
h
en
it f
o
llo
w
s
t
h
at:
(
s
u
p
l
i
m
i
E
i
)
=
0
Re
c
a
ll
th
a
t
E
i
i
s
u
p
lim
=
1
1
i
j
i
E
,
i.e
.
ea
ch
ele
m
e
n
t o
f
t
h
e
E
i
i
s
u
p
l
i
m
1
b
elo
n
g
to
in
f
i
n
itel
y
m
a
n
y
’
s
.
2.
Q
UAN
T
I
T
A
T
I
V
E
NO
N
DI
VE
R
G
E
NCE
I
n
th
e
s
ec
t
io
n
w
e
w
ill
s
tate
T
h
eo
r
em
2
.
2
.
2
th
at
h
as
b
ee
n
p
r
o
v
ed
in
[
1
]
an
d
ap
p
ly
i
t
to
p
r
o
v
e
T
h
eo
r
em
1
.
1
.
1
.
I
n
o
r
d
er
to
s
tat
e
T
h
eo
r
em
2
.
2
.
2
w
e
n
ee
d
to
i
n
tr
o
d
u
ce
d
s
o
m
e
n
o
tat
io
n
s
a
n
d
d
ef
in
i
tio
n
s
.
R
ec
all
t
h
at
{
1
,
⋯
,
,
1
,
⋯
,
}
is
a
s
ta
n
d
ar
d
b
asis
o
f
ℝ
+
.
Def
i
n
e
≝
{1
,
⋯
,
}
an
d
≝
{1
,
⋯
,
}.
T
h
e
f
o
llo
w
i
n
g
n
o
tatio
n
s
w
il
l
b
e
in
tr
o
d
u
ce
d
o
n
th
e
e
x
ter
io
r
alg
eb
r
a
o
f
ℝ
+
(
⋀
(
ℝ
+
)).
T
ak
e
I
=
{
1
,
⋯
,
}
⊂
M
w
it
h
1
<
⋯
<
an
d
J
=
{
1
,
⋯
,
}
⊂
N
w
it
h
1
<
…<
(
2
.
2
.
1
)
a
n
d
de
n
o
te
≝
1
∧
,
⋯
,
∧
an
d
≝
1
∧
,
⋯
,
∧
,
w
it
h
th
e
co
n
v
en
t
io
n
∅
=
∅
=1
.
Den
o
t
e
by
|
|
th
e
ca
r
d
in
al
it
y
o
f
s
et
I
,
s
o
th
at
∧
’
s
ar
e
th
e
b
asis
e
le
m
e
n
t
s
o
f
⋀
|
|
+
|
|
(
ℝ
+
)
.
W
e
s
a
y
th
a
t
a
s
u
b
s
p
ac
e
of
ℝ
+
is
r
atio
n
al
i
f
it
is
s
p
a
n
n
ed
b
y
v
ec
to
r
s
w
it
h
r
atio
n
al
co
o
r
d
in
ates
o
r
eq
u
i
v
alen
tl
y
i
n
te
g
er
co
o
r
d
in
ates.
Def
i
n
e
:
≝
th
e
s
et
o
f
n
o
n
ze
r
o
r
atio
n
al
s
u
b
s
p
ac
es
o
f
ℝ
+
.
Fo
r
∈
G
(
r
ec
all
th
at
G=
S
+
(
ℝ
)
)
an
d
W
∈
,
let
{
1
,
⋯
,
}
b
e
a
g
en
er
atin
g
s
et
f
o
r
ℤ
+
⋂
W
,
i
.
e
.
ℤ
+
⋂
W
=
ℤ
(
1
,
⋯
,
)
,
an
d
d
ef
in
e
th
e
g
ac
tio
n
o
n
1
∧
. . . .
∧
by
(
1
∧
. . . .
∧
)
≝
(
1
)
∧
.
.
.
.
∧
(
)
W
e
w
ill
w
r
ite
w
=
1
∧
.
.
.
∧
an
d
w
e
w
il
l
s
a
y
th
at
w
c
o
r
r
esp
o
n
d
s
to
th
e
n
o
n
ze
r
o
r
atio
n
al
s
u
b
s
p
ac
es
W
.
T
h
en
d
ef
i
n
e
ℓ
(
)
as th
e
co
v
o
l
u
m
e
o
f
∩
ℤ
+
in
,
i.e
.
ℓ
(
g
)
≝
‖
(
)
‖
=
‖
(
1
⋀
⋯
⋀
)
‖
.
(
2
.
2
.
2
)
T
h
e
n
o
r
m
‖
⋅
‖
is
an
e
x
te
n
s
io
n
o
f
E
u
clid
ea
n
n
o
r
m
o
f
ℝ
+
,
i
.
e.
if
ˊ
ϵ
⋀
(
ℝ
+
)
ca
n
b
e
w
r
itte
n
as
w
′
=
l
J
I
N
J
M
I
|
|
|
:|
,
,
⋀
,
w
h
er
e
,
’
s
ar
e
co
ef
f
icie
n
t a
n
d
I
,
J
ar
e
d
ef
in
e
in
(
1
.
2
.
1
)
,
th
en
‖
ˊ
‖
=
l
J
I
J
I
a
|
|
|
|
2
,
.
No
te
th
at
ℓ
(
)
in
d
ep
en
d
en
t
o
f
t
h
e
ch
o
ice
o
f
a
g
en
er
ati
n
g
s
et
s
.
No
w
,
let
u
s
s
tate
T
h
eo
r
e
m
2
.
2
.
2
as
f
o
llo
w
s
:
T
h
eo
r
e
m
2
.
2
.
2
(
T
h
eo
r
e
m
4
.
3
o
f
[
1
]
)
.
L
et
′′′
,
D,
α
b
e
p
o
s
itiv
e
co
n
s
ta
n
ts
.
S
u
p
p
o
s
e
U
⊂
ℝ
is
o
p
en
,
is
a
m
ea
s
u
r
e
w
h
ic
h
is
D
-
Fed
er
er
o
n
U
,
h
is
a
co
n
tin
u
o
u
s
m
ap
U
→
G
,
0
<
ϱ
≤
1
,
0
∈
U
∩
s
u
p
p
,
an
d
B
=B
(
0
,
r
)
is
a
b
all
s
u
ch
th
a
t
̃
≝
B(
0
,
3
+
r
)
is
co
n
tin
u
o
u
s
in
U
,
f
o
r
ea
ch
W
∈
,
(
1
)
T
h
e
f
u
n
ct
io
n
ℓ
∘
ℎ
is
(
′′′
,
α
)
-
g
o
o
d
o
n
̃
w
it
h
r
esp
ec
t to
,
an
d
(
2
)
‖
ℓ
∘
ℎ
‖
,
≥
,
T
h
en
th
er
e
ex
i
s
t
′
′′
>0
s
u
ch
t
h
at
f
o
r
an
y
0
<ε
≤
ϱ,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.6
,
No
.
2,
J
u
n
e
201
7
:
117
–
1
2
5
120
(
{
∈
B
: π
(
ℎ
(
x
)
)
∉
})
≤
′
(
)
(
B
)
.
T
h
is
th
eo
r
e
m
is
k
n
o
w
n
as
n
o
n
d
iv
er
g
e
n
ce
th
eo
r
e
m
.
I
n
[
4
,
T
h
eo
r
em
5
.
2
]
,
D.
Klein
b
o
ck
an
d
G.
Ma
r
g
u
lis
p
r
o
v
ed
th
is
r
es
u
lt
f
o
r
L
eb
esg
u
e
m
ea
s
u
r
e
λ
,
f
o
llo
w
in
g
th
e
id
ea
o
f
[
5
,
Ma
in
L
e
m
m
a]
.
A
n
o
t
h
er
v
er
s
io
n
w
h
ic
h
r
ep
lace
s
co
n
d
itio
n
(
2
)
b
y
w
ea
k
er
co
n
d
itio
n
s
ap
p
ea
r
ed
in
[
3
]
.
T
h
e
p
r
o
o
f
o
f
T
h
eo
r
e
m
2
.
2
.
2
w
h
ich
a
s
s
u
m
e
is
D
-
Fed
er
er
m
ea
s
u
r
e
i
s
in
t
h
e
p
ap
er
[
1
]
.
B
ef
o
r
e
ap
p
ly
i
n
g
T
h
e
-
o
r
e
m
2
.
2
.
2
to
p
r
o
v
e
th
is
th
eo
r
e
m
1
.
1
.
1
,
let
u
s
d
ef
in
e
ex
p
an
d
i
n
g
b
asis
ele
m
e
n
ts
:
Fix
t
∈
+
,
an
d
s
a
y
t
h
at
b
asis
ele
m
e
n
t o
f
⋀
(
ℝ
+
)
,
∧
is
ex
p
an
d
ed
b
y
(
is
d
ef
in
e
as )
=
d
iag
(
1
,
⋯
,
,
−
+
1
,
⋯
,
−
+
)
,
w
h
er
e
t =
(
1
,
. . . ,
+
)
∈
+
I
f
‖
(
∧
)
‖
≥
‖
∧
‖
.
I
n
th
is
ca
s
e
w
e
s
a
y
t
h
at
∧
i
s
an
ex
p
an
d
i
n
g
b
asis
ele
m
e
n
t.M
o
r
eo
v
er
,
w
e
s
a
y
th
a
t
s
tr
i
ctl
y
ex
p
an
d
∧
if
‖
(
∧
)
‖
>
‖
∧
‖
.
On
o
t
h
er
h
a
n
d
,
w
e
s
a
y
th
a
t
s
tr
ictl
y
co
n
tr
ac
t
s
∧
if
it
d
o
es
n
o
t
ex
p
a
n
d
∧
.
L
et
=
∑
∈
an
d
=
∑
+
∈
.
C
lear
l
y
,
s
tr
ictl
y
e
x
p
an
d
s
∧
if
−
>0
s
in
ce
‖
(
∧
)
‖
=
−
‖
∧
‖
.
I
n
th
is
ca
s
e,
w
e
d
ef
in
e
t
h
e
s
u
b
s
p
ac
e
o
f
⋀
(
ℝ
+
)
g
en
er
ated
b
y
∧
’
s
w
it
h
−
>
0
as
s
tr
ictly
e
x
p
a
n
d
in
g
s
u
b
s
p
ac
e
,
an
d
d
en
o
te
b
y
+
.
Si
m
i
lar
l
y
,
th
e
s
u
b
s
p
ac
e
g
e
n
er
ated
b
y
∧
’
s
w
it
h
−
=
0
is
d
en
o
ted
b
y
0
an
d
th
e
s
u
b
s
p
ac
e
g
en
er
ated
b
y
∧
’
s
w
it
h
−
<
0
is
d
en
o
ted
b
y
−
.
So
w
e
ca
n
d
ec
o
m
p
o
s
e
th
e
s
p
ac
e
⋀
(
ℝ
+
)
=
+
⊕
0
⊕
−
,
W
h
er
e
s
tr
ictl
y
e
x
p
an
d
s
t
h
e
n
o
r
m
o
f
th
e
e
le
m
en
ts
i
n
+
,
d
o
es
n
o
t
ch
a
n
g
e
th
e
n
o
r
m
o
f
ele
m
e
n
ts
i
n
0
an
d
c
o
n
tr
ac
ts
t
h
e
n
o
r
m
o
f
ele
m
e
n
t
s
i
n
−
.
On
e
f
ac
t
i
s
t
h
at
+
,
0
an
d
−
a
r
e
d
ep
en
d
en
t
o
n
t
s
o
t
h
at
d
i
f
f
er
en
t
t
give
d
if
f
er
e
n
t
d
ec
o
m
p
o
s
iti
o
n
o
f
⋀
(
ℝ
+
)
.
I
f
w
∈
⋀
(
ℝ
+
)
ca
n
b
e
w
r
itte
n
as
w
=
1
+
2
w
h
er
e
1
∈
+
⊕
0
a
n
d
2
∈
−
,
th
en
w
e
s
a
y
1
is
th
e
ex
p
an
d
in
g
p
ar
t
o
f
w
an
d
2
is
t
h
e
co
n
tr
ac
ti
n
g
p
ar
t
o
f
w
.
Si
m
i
lar
l
y
,
e
x
p
an
d
i
n
g
p
ar
t
a
n
d
co
n
tr
ac
tin
g
p
ar
t
ar
e
also
d
ep
en
d
en
t
o
n
t
.
W
e
w
il
l u
s
e
th
e
d
ef
in
i
tio
n
s
an
d
n
o
tatio
n
s
ab
o
v
e
to
s
o
lv
e
t
h
eo
r
e
m
in
th
i
s
p
ap
er
.
3.
P
RO
O
F
O
F
M
AIN
RE
SUL
T
1
.
1
.
1
.
L
et
h
(
x
)
=
(
τ
(
F(x
)
)
an
d
f
o
r
–
a.
e.
0
,
tak
e
̃
=
B
(
0
,
3
+
r)
⊂
U
s
uc
h
tha
t
(
,
)
is
(
C
,
α
)
-
g
o
o
d
o
n
̃
f
o
r
s
o
m
e
C
,
α
>
0
a
n
d
n
o
n
p
la
n
ar
o
n
B
=
B
(
0
,
r
)
.
To
ap
p
ly
T
h
eo
r
em
2
.
2
.
2
,
w
e
n
ee
d
to
s
h
o
w
t
h
at
the
r
e
e
xist
′′′
,
α
>
0
an
d
s
o
m
e
0
<
ϱ
≤
1
s
u
ch
th
at
f
o
r
an
y
W
∈
w
h
er
e
i
s
th
e
s
et
o
f
n
o
n
ze
r
o
r
atio
n
al
s
u
b
s
p
ac
es
:
ℓ
∘
h
is
(
′′′
,
α
)
-
g
o
o
d
o
n
̃
w
ith
r
e
s
p
ec
t
to
(
co
n
d
itio
n
(
1
)
)
an
d
‖
ℓ
∘
ℎ
‖
,
≥
ϱ
(
c
on
diti
on
(
2
)
)
.
First,
w
e
w
a
n
t
to
s
h
o
w
th
a
t
th
er
e
ex
is
t
′′′
,
α
>
0
s
u
ch
th
at
f
o
r
an
y
W
∈
,
ℓ
°h
is
(
′′′
,
)
-
g
o
o
d
o
n
̃
w
i
th
r
esp
ec
t
to
.
T
h
at
is
,
w
e
w
a
n
t
to
s
h
o
w
th
at
f
o
r
{
1
,
.
.
.
,
}
a
g
en
er
atin
g
s
et
o
f
r
atio
n
al
s
u
b
s
p
ac
e
W
(
w
e
as
s
u
m
e
it
is
d
im
e
n
s
io
n
al
r
atio
n
al
s
u
b
s
p
ac
e
)
an
d
f
o
r
w
=
1
∧
.
.
.
∧
a
h
o
m
o
g
e
n
eo
u
s
ele
m
e
n
t
o
f
∧
(
ℝ
+
)
c
or
r
e
s
pon
din
g
to
W
,
‖
(
(
(
)
)
)
(
)
‖
is
(
′′′
,
α
)
-
g
o
o
d
o
n
̃
w
it
h
r
esp
ec
t
to
.
T
h
e
s
tr
a
te
gy
is
as
fol
l
ow
:
fir
s
t
,
w
e
w
a
n
t
to
ap
p
ly
τ
(
F
(
)
)
to
w
an
d
ca
lcu
late
th
e
r
es
u
lt.
Af
ter
ca
lcu
lati
n
g
τ
(
F(x
)
)
(
w
)
,
w
e
w
ill
u
s
e
it to
s
h
o
w
(
‖
(
(
(
)
)
)
(
)
‖
,
)
is
(
′′′
,
α
)
-
g
o
o
d
o
n
̃
.
T
o
ca
lcu
late
τ
(
F
(
)
)
(
)
,
let
u
s
f
ir
s
t
ap
p
ly
τ
(
F
(
)
)
to
b
asis
ele
m
e
n
ts
o
f
⋀
(
ℝ
+
)
.
Fo
r
r
ea
d
e
r
’s
co
n
v
e
n
ien
ce
,
w
e
s
tar
t b
y
ap
p
ly
in
g
τ
(
F(
x
)
)
to
b
a
s
is
ele
m
en
t
s
o
f
⋀
(
ℝ
+
)
of
dime
n
s
ion
1
an
d
d
im
e
n
s
io
n
2
:
Fo
r
d
im
e
n
s
io
n
1
,
ap
p
ly
τ
(
F (
x
)
)
to
b
a
s
is
ele
m
en
ts
an
d
f
o
r
=
1
,
⋯
,
an
d
j
=
1
,
⋯
,
:
τ
(
F
(
)
)
(
)
=
for
=
1
,
⋯
,
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S
I
SS
N:
2252
-
8814
F
ed
erer
Mea
s
u
r
e
s
,
Go
o
d
a
n
d
N
o
n
p
la
n
a
r
F
u
n
ctio
n
s
o
f Ma
tr
ic
Dio
p
h
a
n
ti
n
e
…
(
F
a
iz
a
A
kra
m
)
121
(
F(x
)
)
(
)
=
=
1
(
x
)
1
+ . .
. +
(
x
)
for
j
=
1
,
⋯
,
.
So
τ
(
F
(
x
)
)
f
ix
e
s
i
an
d
w
h
e
n
a
p
p
ly
i
n
g
τ
(
F(
x
)
)
to
,
o
n
e
g
et
s
a
lin
ea
r
co
m
b
i
n
atio
n
o
f
ele
m
e
n
ts
o
f
d
i
m
en
s
io
n
1
an
d
t
h
e
co
ef
f
icie
n
ts
i
n
f
r
o
n
t o
f
ea
ch
ele
m
e
n
t a
r
e
eith
er
1
o
r
s
o
m
e
co
m
p
o
n
e
n
t
s
(
x
)
’
s
i
n
F(x
)
.
Fo
r
d
i
m
en
s
io
n
2
,
th
er
e
ar
e
t
h
r
ee
d
if
f
er
en
t
t
y
p
e
s
o
f
b
a
s
is
elem
en
ts
,
t
h
e
y
ar
e
∧
j
,
∧
a
n
d
i
∧
.
W
e
w
a
n
t to
ap
p
l
y
τ
(
F(
x
)
)
to
all
ty
p
e
s
o
f
b
a
s
is
ele
m
e
n
ts
:
(
F(
x
))(
i
∧
j
)
=
∧
j
for
a
n
y
1
≤
,
≤
.
(
F(
x
))(
∧
)
=
∧
+
m
l
1
(
)
∧
−
m
l
1
(
x
)
∧
+
m
l
k
1
|
(
)
(
)
(
)
(
)
|
∧
for
a
n
y
1
≤
,
≤
.
(
F(
x
))(
∧
)
=
∧
+
i
l
m
l
1
(
)
⋀
for
a
n
y
1
≤
≤
,
1
≤
≤
.
Si
m
i
lar
l
y
a
s
f
o
r
d
i
m
e
n
s
io
n
1
,
w
e
ca
n
co
n
cl
u
d
e
th
at
t
h
e
r
es
u
l
t
o
f
ap
p
l
y
in
g
(
F(x
)
to
an
y
t
y
p
e
o
f
b
asis
ele
m
e
n
t
is
a
lin
ea
r
co
m
b
i
n
at
io
n
o
f
b
asis
ele
m
e
n
ts
o
f
d
im
en
s
io
n
2
w
i
th
co
ef
f
icie
n
t
ar
e
1
,
(
)
s
o
r
th
e
d
eter
m
in
a
n
ts
o
f
2
b
y
2
s
u
b
m
a
tr
ices
o
f
F(x
)
.
T
h
ese
o
b
s
er
v
atio
n
s
lead
u
s
to
th
e
g
e
n
er
a
l
l
r
esu
lt
s
:
b
y
ap
p
l
y
i
n
g
(
F(x
)
)
to
an
y
b
asis
e
le
m
e
n
t
s
o
f
d
i
m
en
s
io
n
⋀
(
ℝ
+
)
,
th
e
r
e
s
u
lt
i
s
a
l
in
ea
r
co
m
b
i
n
atio
n
o
f
b
asi
s
ele
m
en
ts
o
f
d
i
m
en
s
io
n
⋀
(
ℝ
+
)
w
ith
t
h
e
co
ef
f
ici
en
ts
ar
e
1
o
r
d
eter
m
i
n
an
t
s
o
f
s
u
b
m
atr
ice
s
o
f
(
)
.
T
o
p
r
o
v
e
th
is
,
let
|
I
|
b
e
th
e
ca
r
d
in
alit
y
o
f
s
et
I
an
d
I
∧
b
e
a
b
asis
ele
m
e
n
t
o
f
d
i
m
en
s
io
n
=
|
|
+
|
|
.
A
p
p
l
y
i
n
g
(
(
)
)
to
∧
,
o
n
e
h
as:
(
(
)
)
(
∧
)
=
J
S
S
J
L
S
K
I
M
K
\
|,
|
|
|
,
\
(
−
1
)
(
,
)
,
(
x
)
∪
∧
(
2
.
2
.
3
)
W
h
er
e
,
(
x
)
’s
d
e
f
in
ed
as:
,
≝
|
1
,
1
⋯
1
,
,
1
⋯
,
|
,
An
d
m
(
⋅
)
d
eter
m
in
e
s
th
e
s
i
g
n
o
f
th
e
co
ef
f
icie
n
ts
.
C
lear
l
y
,
(
(
x
))
(
I
∧
J
)
is
a
lin
ea
r
co
m
b
i
n
atio
n
o
f
b
asis
ele
m
en
t
s
o
f
⋀
(
ℝ
+
)
an
d
th
e
co
ef
f
icie
n
t
s
o
f
t
h
ese
b
asi
s
ele
m
e
n
ts
ar
e
d
eter
m
i
n
a
n
ts
o
f
s
q
u
ar
e
s
u
b
m
atr
ices o
f
F
(
)
.
No
w
,
let
u
s
ap
p
l
y
τ
(
F
(
)
)
(
∧
)
is
a
li
n
ea
r
co
m
b
i
n
atio
n
o
f
b
asis
ele
m
e
n
t
s
o
f
⋀
(
ℝ
+
)
an
d
t
h
e
co
ef
f
icie
n
t
o
f
t
h
e
s
e
b
asi
s
ele
m
en
ts
ar
e
d
eter
m
i
n
an
ts
o
f
s
q
u
ar
e
s
u
b
m
atr
ices
o
f
F(x
)
.
No
w
,
le
t
u
s
ap
p
l
y
(
(
)
)
to
w
w
h
er
e
w
is
a
h
o
m
o
g
en
o
u
s
ele
m
e
n
t
o
f
⋀
(
ℝ
+
)
co
r
r
esp
o
n
d
in
g
to
d
im
e
n
s
io
n
r
atio
n
a
l
s
u
b
s
p
ac
e
W
.
Sin
ce
{
1
,
.
.
.
,
}
is
a
g
en
er
ati
n
g
s
et
o
f
n
o
n
ze
r
o
r
atio
n
al
s
u
b
s
p
ac
e
W
,
w
e
ca
n
w
r
ite
w
=
1
∧
.
.
.
∧
=
∑
,
,
I
∧
∈
⋀
(
ℝ
+
)
,
w
h
er
e
t
h
e
s
u
m
m
a
tio
n
i
s
o
v
er
all
I
,
J
s
a
tis
f
y
in
g
(
2
.
2
.
1
)
w
it
h
=
|
I
|
+
|
J
|
i
s
f
ix
e
d
an
d
,
’
s
ar
e
in
teg
er
s
co
ef
f
icie
n
ts
.
B
y
eq
u
atio
n
(
2
.
2
.
3
)
s
in
ce
w
is
a
lin
ea
r
co
m
b
in
a
ti
o
n
o
f
∧
’
s
τ
(
F
(
)
)
(
)
is
also
a
lin
ea
r
co
m
b
in
at
io
n
o
f
b
asis
ele
m
e
n
ts
⋀
(
ℝ
+
)
w
it
h
t
h
e
co
ef
f
icie
n
t
s
o
f
ea
ch
I
∧
i
n
τ
(
F
(
)
)
(
)
a
co
m
b
in
at
io
n
o
f
,
(x
)
’
s
.
L
et
ˊ
,
(
)
b
e
th
e
co
ef
f
icie
n
t
o
f
∧
in
τ
(
F
(
)
)
(
)
,
th
en
w
e
ca
n
w
r
ite:
(
F(
x
))(
w
)
=
J
I
,
ˊ
,
(
x
)
∧
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.6
,
No
.
2,
J
u
n
e
201
7
:
117
–
1
2
5
122
No
w
,
let
u
s
ap
p
l
y
,
f
o
r
t
in
+
.
T
ak
e
=
I
i
an
d
=
J
j
+
.
W
e
h
av
e:
(
(
F(
x
)))
=
J
I
,
−
ˊ
,
(
)
I
∧
.
L
et
,
(
x
)
=
−
ˊ
,
(x
)
.
By
th
e
as
s
u
m
p
tio
n
th
at
(
,
)
is
(
,
)
-
g
o
o
d
on
̃
,
th
o
s
e
,
(x
)
’
s
w
h
ic
h
ar
e
lin
ea
r
co
m
b
i
n
atio
n
o
f
,
’
s
ar
e
also
(
,
)
-
g
o
o
d
o
n
̃
w
ith
r
esp
ec
t
to
.
Fu
r
th
er
m
o
r
e,
th
i
s
i
m
p
lie
s
th
at
Q(
x
)
≝
‖
(
(
(
)
)
)
‖
is
(
′′′
,
)
-
g
o
o
d
o
n
̃
w
it
h
r
esp
ec
t
t
o
f
o
r
an
y
h
o
m
o
g
e
n
o
u
s
w
an
d
s
o
m
e
ˊˊˊ
,
α
>
0.
T
o
s
h
o
w
th
is
,
let
∗
∧
∗
∈
⋀
|
|
+
|
|
(
ℝ
+
)
b
e
s
u
c
h
th
at
t
h
e
co
ef
f
icie
n
t
∗
,
∗
(
x
)
s
atis
f
ies t
h
e
f
o
llo
w
i
n
g
:
‖
∗
,
∗
(
)
‖
,
̃
=
m
a
x
|
|
|
|
l
J
I
{
‖
,
(
)
‖
,
̃
}
,
w
h
er
e
th
e
m
a
x
i
m
u
m
is
tak
e
n
a
m
o
n
g
all
th
e
n
o
r
m
s
o
f
c
o
ef
f
icie
n
t
s
o
f
(
F
(
)
(
)
.
Fo
llo
w
i
n
g
th
e
d
ef
in
i
tio
n
o
f
t
h
e
n
o
r
m
‖
⋅
‖
o
n
th
e
ex
ter
io
r
alg
eb
r
a
⋀
(
ℝ
+
),
‖
(
)
‖
,
̃
=
∥
√
l
J
I
|
|
|
|
,
2
(
)
∥
,
̃
≤
√
l
J
I
|
|
|
|
‖
,
2
(
)
‖
,
̃
.
T
h
en
it
is
clea
r
th
at
th
er
e
ex
i
s
t
s
o
m
e
c
o
n
s
tan
t
ˊˊˊ
≥
0
(
′′′
d
ep
en
d
s
o
n
,
b
u
t
n
o
t
o
n
)
s
u
ch
th
a
t
th
e
f
o
llo
w
i
n
g
is
s
a
tis
f
ied
:
‖
∗
,
∗
(
)
‖
,
̃
≥
ˊˊˊ
‖
(
)
‖
,
̃
.
I
t
f
o
llo
w
t
h
at
t
h
er
e
ex
i
s
t so
m
e
ˊˊˊ
>
0
s
u
ch
t
h
at:
({
x
∈
̃
:
|
Q(
x
)
|
<
ε}
)
≤
(
{
∈
̃
:
|
∗
,
∗
(
x
)
|
<
ε
}
)
≤
C
(
‖
∗
,
∗
(
)
‖
,
̃
)
(
̃
)
=
ˊˊˊ
(
‖
(
)
‖
,
̃
)
(
̃
).
T
h
e
s
ec
o
n
d
in
eq
u
ali
t
y
f
o
llo
w
i
n
g
f
r
o
m
(
C
,
α
)
-
g
o
o
d
p
r
o
p
er
ty
o
f
∗
,
∗
(
x
)
o
n
̃
w
it
h
r
esp
ec
t
to
.
t
h
i
s
p
r
o
v
es
th
at
(
Q
(
x
),
)
is
(
ˊˊˊ
,
α
)
-
g
o
o
d
o
n
̃
f
o
r
an
y
w
an
d
co
n
s
eq
u
en
tl
y
,
ℓ
∘
ℎ
(
′′′
,
)
-
g
o
o
d
o
n
̃
w
it
h
r
esp
ec
t to
f
o
r
an
y
n
o
n
ze
r
o
r
ati
o
n
al
s
u
b
s
p
ac
e
W
a
n
d
h
(
)
=
(
τ
(
)
)
.
No
w
,
let
u
s
s
h
o
w
t
h
at
t
h
e
s
ec
o
n
d
co
n
d
itio
n
is
s
ati
s
f
ied
;
t
h
at
is
:
t
h
er
e
ex
i
s
t
s
o
m
e
ϱ
w
i
th
0
<
≤
1
s
u
ch
t
h
at
f
o
r
an
y
n
o
n
ze
r
o
r
ati
o
n
al
s
u
b
s
p
ac
e
W
o
f
ℝ
+
an
d
an
y
t
,
‖
ℓ
∘
ℎ
‖
,
̃
≥
ϱ,
w
h
e
n
ℎ
(
x
)
=
(
τ
(
F (
x
)
)
)
.
R
ec
all
f
r
o
m
t
h
e
p
ar
ag
r
ap
h
b
ef
o
r
e
t
h
e
p
r
o
o
f
o
f
T
h
eo
r
em
1
.
1
.
1
th
at,
an
ex
p
an
d
i
n
g
b
a
s
is
ele
m
en
t
is
a
b
asis
ele
m
e
n
t
∧
s
u
c
h
t
h
at
‖
(
⋀
)
‖
≥
‖
⋀
‖
,
o
r
eq
u
iv
alen
t
l
y
,
∧
∈
+
⊕
0
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S
I
SS
N:
2252
-
8814
F
ed
erer
Mea
s
u
r
e
s
,
Go
o
d
a
n
d
N
o
n
p
la
n
a
r
F
u
n
ctio
n
s
o
f Ma
tr
ic
Dio
p
h
a
n
ti
n
e
…
(
F
a
iz
a
A
kra
m
)
123
Ou
r
s
tr
ate
g
y
to
s
h
o
w
co
n
d
iti
o
n
(
2
)
o
f
T
h
eo
r
em
2
.
2
.
2
is
as
f
o
llo
w
.
First,
ap
p
l
y
(
F
(
x
)
)
to
w
=
∑
,
a
,
I
∧
an
d
ca
lcu
late
t
h
e
r
es
u
lt;
n
ex
t
tak
e
th
e
e
x
p
an
d
i
n
g
P
ar
t o
f
(
F(
x
)
)
(
w
)
an
d
s
h
o
w
‖
ℓ
∘
ℎ
‖
,
≥
ϱ f
o
r
an
y
n
o
n
ze
r
o
r
atio
n
al
s
u
b
s
p
ac
e
W
w
h
e
n
ℎ
(
x
)
=
(
τ
(
F (
x
)
)
)
.
L
et
u
s
co
n
s
id
er
t
w
o
ca
s
e
s
:
Ca
s
e1
.
W
h
en
≤
d
en
o
te
b
y
E
th
e
s
p
ac
e
g
en
er
ated
b
y
{
1
,
⋯
,
}
an
d
b
y
V
th
e
s
p
ac
e
g
en
er
ated
b
y
{
1
,
⋯
,
},
an
d
co
n
s
id
er
a
p
r
o
j
ec
tio
n
P
∶
⋀
(
ℝ
+
)
⟶
⋀
(
E
)
.
Fro
m
th
e
d
e
f
in
itio
n
o
f
P
,
th
e
i
m
a
g
e
o
f
P
co
n
s
is
t
s
o
f
li
n
ea
r
co
m
b
i
n
atio
n
o
f
ex
p
a
n
d
in
g
b
asis
ele
m
e
n
ts
(
i
.
e
.
e
l
e
me
n
t
s
in
+
⊕
0
)
s
in
ce
⋀
(
E
)
=
s
p
an
(
)
⊂
,
an
d
f
o
r
an
y
L
⊂
M,
‖
‖
=
‖
‖
≥
‖
‖
w
h
er
e
=
∑
∈
.
Usi
n
g
eq
u
atio
n
(
2
.
2
.
3
)
,
it is
ea
s
y
to
s
ee
t
h
at
P
(
τ
(
F(
x
)
)
(
∧
)
)
=
∧
|
|
|
|
,
\
J
K
I
M
K
(
−
1
)
(
,
)
,
(
x
)
,
(
2
.
2
.
4
)
W
h
er
e
K
= {
1
,
⋯
,
}
⊂
M
.
No
te
th
at
|
I
|
ca
n
ta
k
e
v
alu
e
s
b
et
w
ee
n
m
a
x
(
0
,
−
)
a
n
d
;
eq
u
iv
ale
n
tl
y
,
−
|
|
r
an
g
e
s
b
et
w
ee
n
0
an
d
–
ma
x
(
0
,
−
)
=
min
(
,
)
.
T
h
en
w
e
h
a
v
e:
P
(
τ
(
F
(
)
)
(
)
)
=
l
I
n
l
M
I
|
|
)
,
0
m
a
x
(
∧
|
|
|
|
I
l
J
N
J
,
|
|
|
|
|
|
\
J
K
I
M
K
(
−
1
)
(
,
)
,
(
x
)
.
R
ea
r
r
an
g
in
g
ter
m
s
an
d
s
u
b
s
ti
t
u
ti
n
g
L
=
I
⋃
K
,
w
e
g
et
P
(
τ
(
F
(
)
)
(
)
)
=
l
L
M
L
|
|
(
l
I
n
l
L
I
|
|
)
,
0
m
a
x
(
|
|
|
|
I
l
J
N
J
(
−
1
)
(
,
)
,
\
,
(
)
)
,
O
r
eq
u
iv
ale
n
tl
y
,
P
(
τ
(
F
(
)
)
(
)
)
=
l
L
M
L
|
|
(
l
I
n
l
K
M
K
|
|
)
,
m
i
n
(
|
|
0
|
|
|
|
K
J
N
J
(
−
1
)
(
,
)
\
,
(
)
)
.
Fro
m
th
e
d
ef
in
i
tio
n
o
f
ℓ
(
ℎ
(
x
)
)
f
o
r
an
y
W
an
d
f
o
r
ℎ
(
x
)
=
(
τ
(
F
(
)
)
)
,
w
e
k
n
o
w
t
h
at
‖
ℓ
(
ℎ
(
)
)
‖
,
=
‖
‖
(
(
(
)
)
(
)
)
‖
‖
,
,
W
h
er
e
in
t
h
e
r
ig
h
t
h
a
n
d
s
id
e
o
f
eq
u
atio
n
,
t
h
e
i
n
n
er
‖
⋅
‖
is
e
x
te
r
io
r
alg
eb
r
a
o
f
ℝ
+
an
d
‖
⋅
‖
,
th
e
n
o
r
m
d
ef
in
e
i
n
f
o
llo
w
i
n
g
eq
u
atio
n
,
‖
‖
,
=
s
u
p
{c
:
({
x
∈
B
:
|
(
)
|
>
c}
)
>0
}.
Fro
m
th
e
d
ef
i
n
it
io
n
o
f
‖
⋅
‖
n
o
r
m
an
d
th
e
f
ac
t
t
h
at
is
an
ex
p
an
d
in
g
b
asi
s
ele
m
e
n
t,
‖
‖
(
)
(
)
‖
‖
,
is
g
r
ea
ter
th
a
n
o
r
eq
u
al
to
th
e
‖
⋅
‖
,
n
o
r
m
o
f
th
e
co
e
f
f
icien
ts
o
f
a
n
y
,
i.e
.
‖
‖
(
(
(
)
)
(
)
)
‖
‖
,
≥
m
a
x
|
|
l
L
M
L
‖
‖
(
(
l
I
n
l
K
L
K
|
|
)
,
m
i
n
(
|
|
0
|
|
|
|
K
J
N
J
(
−
1
)
(
,
)
\
,
,
(
)
L
‖
‖
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.6
,
No
.
2,
J
u
n
e
201
7
:
117
–
1
2
5
124
≥
m
a
x
|
|
l
L
M
L
‖
l
I
n
l
K
L
K
|
|
)
,
m
i
n
(
|
|
0
|
|
|
|
K
J
N
J
(
−
1
)
(
,
)
\
,
,
(
)
‖
,
.
I
t
r
em
ai
n
s
to
s
h
o
w
t
h
at
th
er
e
ex
is
t
0
≤
ϱ
≤
1
s
u
ch
th
at
f
o
r
an
y
w
=
∑
,
,
∧
co
r
r
esp
o
n
d
in
g
to
a
n
o
n
ze
r
o
r
atio
n
al
s
u
b
s
p
ac
e
W
,
m
a
x
|
|
l
L
M
L
‖
l
I
n
l
K
L
K
|
|
)
.
m
i
n
(
|
|
0
|
|
|
|
K
J
N
J
(
−
1
)
(
,
)
\
,
,
(
)
‖
,
≥
ϱ
.
(
2
.
2
.
5
)
Fro
m
th
e
a
s
s
u
m
p
tio
n
t
h
at
(
,
)
is
n
o
n
p
lan
ar
,
,
(
x
)
’
s
li
n
ea
r
l
y
i
n
d
ep
en
d
en
t o
n
B
.
An
d
th
i
s
i
m
p
l
i
es
th
at
t
h
er
e
ex
is
t
0
<
ϱ
≤
1
s
u
c
h
t
h
at
f
o
r
an
y
,
’
s
w
h
er
e
I
⊂
M
,
J
⊂
N,
|
I
|
=
|
J
|
≤
m
i
n
(
,
)
w
it
h
ma
x
,
{
|
,
|
}
≥
1
,
w
e
h
a
v
e:
‖
J
I
,
,
,
(
)
‖
,
≥
ϱ
.
Fro
m
th
is
f
ac
t
u
n
less
,
=
0
f
o
r
all
I
⊂
M
,
J
⊂
N
an
d
|
I
|
=
|
J
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,
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u
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n
(
2
.
2
.
5
)
is
s
atis
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ied
.
I
f
,
=
0
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o
r
all
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⊂
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n
d
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,
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h
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m
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at
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ze
r
o
.
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h
is
s
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h
at
f
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th
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1
,
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n
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itio
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2
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in
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h
eo
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m
2
.
2
.
1
is
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ati
s
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ied
.
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e2
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f
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h
en
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r
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m
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at
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2
.
2
.
3
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(
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m
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K
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|
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|
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m
K
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K
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−
1
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,
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\
,
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2
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6
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m
u
s
t
h
a
v
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ma
x
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r
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eq
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iv
alen
tl
y
,
0
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\
I
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x
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0
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,
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T
h
er
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r
e:
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(
τ
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F
(
)
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)
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m
I
n
l
M
I
|
|
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,
0
m
a
x
(
|
|
|
|
I
l
J
N
J
,
|
|
|
|
I
m
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K
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−
1
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,
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x
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.
R
ea
r
r
an
g
in
g
ter
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s
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d
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b
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ti
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L
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J
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w
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g
et
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F
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m
l
L
M
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m
I
n
l
M
I
|
|
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,
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m
a
x
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I
l
J
L
J
(
−
1
)
(
,
)
,
\
,
\
(
)
)
∧
L
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S
I
SS
N:
2252
-
8814
F
ed
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≤
ϱ
≤
1
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ch
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at
f
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y
w
=
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∧
co
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r
esp
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in
g
to
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n
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n
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)
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m
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n
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|
0
l
n
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I
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m
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(
2
.
2
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7
)
Usi
n
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a
m
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m
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t
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n
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an
d
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,
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e
in
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t
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(
2
.
2
.
7
)
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f
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h
is
s
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th
a
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2
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n
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itio
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f
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h
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2
.
1
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s
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f
ied
.
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e
h
av
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s
h
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w
n
t
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at
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1
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d
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(
2
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f
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h
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r
e
m
2
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.
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s
atis
f
ied
in
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h
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r
e
m
1
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,
s
o
w
e
ca
n
p
r
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v
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h
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r
em
1
.
1
.
1
b
y
ap
p
l
y
in
g
T
h
eo
r
e
m
2
.
2
.
1
wi
th
=
ˊ
(
1
)
(
)
.
4.
CO
NCLU
SI
O
N
I
n
th
is
p
ap
er
,
w
e
s
tu
d
ied
lin
e
ar
co
m
b
in
at
io
n
,
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n
p
la
n
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itio
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(
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f
u
n
ctio
n
.
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e
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p
ly
τ
(
F
(
)
)
to
w
an
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lcu
late
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e
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d
af
ter
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lcu
lati
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(
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p
ly
it
to
ca
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latin
g
(
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(
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is
(
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.
W
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co
n
clu
d
e
th
at
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s
u
lt o
f
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p
l
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n
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τ
(
F
(
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to
an
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f
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im
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s
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2
w
it
h
th
e
co
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f
icie
n
ts
ar
e
1
,
f
(
x
)
s
o
r
th
e
d
eter
m
i
n
a
n
ts
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f
2
b
y
2
s
u
b
m
atr
ice
s
o
f
F(x
)
.
W
e
h
a
v
e
s
h
o
w
n
t
h
at
b
o
th
co
n
d
itio
n
s
ar
e
s
atis
f
ied
.
RE
F
E
R
E
NC
E
S
[1
]
D.
Kle
in
b
o
c
k
,
E.
L
in
d
e
n
stra
u
ss
a
n
d
B.
W
e
iss,
“
o
n
f
ra
c
tal
m
e
a
su
re
s
a
n
d
Dio
p
h
a
n
ti
n
e
a
p
p
ro
x
i
m
a
ti
o
n
,
”
S
e
lec
ta
M
a
th
.
V
o
l
.
1
0
,
p
p
.
4
9
7
-
5
2
3
,
2
0
0
4
.
[2
]
D.
Kle
in
b
o
c
k
,
“
Ba
d
ly
a
p
p
ro
x
ima
b
le sy
ste
m
s o
f
a
ff
in
e
f
o
rm
s,
J
.
Nu
mb
e
r T
h
e
o
ry
,
”
V
o
l
.
7
9
,
p
p
.
8
3
-
1
0
2
,
1
9
9
9
.
[3
]
D.
Kle
in
b
o
c
k
,
“
A
n
e
x
ten
sio
n
o
f
q
u
a
n
ti
tativ
e
n
o
n
d
iv
e
rg
e
n
c
e
a
n
d
a
p
p
l
ica
ti
o
n
s
t
o
Dio
p
h
a
n
t
in
e
e
x
p
o
n
e
n
ts,
”
T
ra
n
s.
Am
e
r.
M
a
th
.
S
o
c
.
V
o
l.
3
6
0
,
p
p
.
6
4
9
7
-
6
5
2
3
,
2
0
0
8
.
[4
]
D.
Kle
in
b
o
c
k
a
n
d
G
.
A
.
M
a
r
g
u
li
s,
“
F
lo
w
s
o
n
h
o
m
o
g
e
n
o
u
s
sp
a
c
e
s
a
n
d
Dio
p
h
a
n
ti
n
e
a
p
p
r
o
x
im
a
ti
o
n
o
n
m
a
n
if
o
ld
s,
”
An
n
.
M
a
th
.
Vo
l.
1
4
8
,
p
p
.
3
3
9
-
3
6
0
,
1
9
9
8
.
[5
]
G
.
M
a
rg
u
li
s,
“
On
th
e
a
c
ti
o
n
o
f
u
n
i
p
o
te
n
t
g
ro
u
p
in
th
e
s
p
a
c
e
o
f
latti
c
e
s,
”
P
r
o
c
e
e
d
in
g
s
o
f
th
e
S
u
m
m
e
r
S
c
h
o
o
l
o
n
g
ro
u
p
re
p
re
se
n
tatio
n
s,
(Bu
d
a
p
e
st
1
9
7
1
),
A
c
a
d
e
m
i
a
i
Kia
d
o
,
B
u
d
a
p
e
st,
p
p
.
3
6
5
-
3
7
0
,
1
9
7
5
.
[6
]
V
.
G
.
S
p
ri
n
d
ž
u
k
.
“
M
a
h
ler
’s
p
ro
b
lem
in
me
tric
n
u
m
b
e
r
th
e
o
ry
,
”
T
ra
n
sla
ted
f
ro
m
th
e
Ru
ss
ian
b
y
B.
Vo
lk
m
a
n
n
.
T
ra
n
sla
ti
o
n
s o
f
M
a
th
e
m
a
ti
c
a
l
M
o
n
o
-
g
ra
p
h
s,
Am
e
r.
M
a
th
.
S
o
c
.
,
P
ro
v
id
e
n
c
e
,
R.
I.
,
1
9
6
9
.
[7
]
V
.
G
.
S
p
ri
n
d
ž
u
k
.
“
M
o
re
o
n
M
a
h
le
r’s co
n
jec
tu
re
.
D
o
k
l.
Aka
d
.
N
a
u
k
S
S
S
R
,
”
p
p
.
1
5
5
:5
4
-
5
6
,
1
9
6
4
.
[8
]
J.W
.
S
.
Ca
ss
e
ls,
“
An
i
n
tro
d
u
c
ti
o
n
to
Dio
p
h
a
n
ti
n
e
a
p
p
ro
x
ima
ti
o
n
,
”
Ca
m
b
rid
g
e
Un
iv
e
rsit
y
p
re
ss
,
Ne
w Yo
rk
,
1
9
5
7
.
[9
]
K.
M
a
h
ler,
“
Üb
e
r
d
a
s M
a
ss
d
e
r
M
e
n
g
e
a
ll
e
r
S
-
Zah
len
,
”
M
a
t
h
.
A
n
n
.
Vo
l.
1
0
6
,
p
p
.
1
3
1
-
1
3
9
,
1
9
3
2
.
Evaluation Warning : The document was created with Spire.PDF for Python.