Geoscience Reference
In-Depth Information
and the function
u
ðÞ
¼
m
g
ðÞw
ðÞ:
1
[ X
2
, where
Lemma 4.1 The
φ
(h) function is nondecreasing at
8
h
2 X
=
;
1
¼
h
1
h
1
¼ inf H
þ
X
h
0
:
h
2
0
;
;
F
ð
0
Þ=
N
ð
0
Þ
\
m
n
h
i
\
H
¼ 0
o
2
¼
h
i
h
j
h
j
2
H
h
j
2
H
þ
h
i
h
j
X
h
0
:
h
2
;
=
;
;
;
1
[ X
2
where
Lemma 4.2 The
φ
(h) function is nonincreasing at
8
h
2 X
=
;
1
¼
h
1
h
1
¼ inf H
X
h
0
:
h
2
0
;
;
F
ð
0
Þ=
N
ð
0
Þ
\
m
n
h
i
\
H
þ
¼ 0
o
2
¼
h
i
h
j
h
i
2
H
þ
h
j
2
H
h
i
h
j
X
h
0
:
h
2
;
=
;
;
;
The validity of the Lemmas
4.1
and
4.2
follows from the elementary properties
of the function
; u
0
ðÞ
¼
m
φ
(h). In particular, we have:
u
ðÞ
¼0
N
ðÞ
F
ðÞ; ...;
ðÞ
ðÞ
¼vN
i
ð Þ
ðÞ
F
ð
i
1
Þ
ðÞ:
Note 1
. The conclusion of the lemma
4.1
remains also valid at F(0)/N(0) =
u
ʽ
,
provided
h
i
9ðÞ
F
ð
s
Þ
ðÞ=
N
ð
s
Þ
ðÞ
\
m;
F
ð
j
Þ
ðÞ=
N
ð
j
Þ
ðÞ
¼
m;
j
¼
0
;
; ...;
:
1
s1
Note 2
. The conclusion of the lemma
4.2
remains also valid at F(0)/N(0) =
ʽ
,
provided
h
i
:
9ðÞ
F
ð
s
Þ
ðÞ=
N
ð
s
Þ
ðÞ
[
m;
F
ð
j
Þ
ðÞ=
N
ð
j
Þ
ðÞ
¼
m;
j ¼ 0
;
1
; ...;
s1
According to Eq. (
4.42
) the function p(h) is the function of
φ
(h):
8
<
9
=
Z
h
p
ð
h
Þ
¼ p
ð
0
Þ
w
ð
y
Þ
exp
½
uð
y
Þ
dy
exp
uð
h
Þ
½
ð
4
:
43
Þ
:
;
0
Based on the results of the lemmas
4.1
and
4.2
, from (
4.43
) one can easily
formulate the monotonicity properties of the function p(h). Since
φ
(h) does not
þ
¼
X
1
[ X
2
, the function exp[
decrease in the set
X
−φ
(h)] will be nonincreasing
At w
ðÞ
0 the function p
ðÞ
R
w
ðÞ
exp½
u
þ
at
8
h
2 X
:
ðÞ
dy is nonincreasing at
þ
.
any values of h
≥
0. Therefore p(h) is the nonincreasing function of
8
h
2 X
Search WWH ::
Custom Search