Biology Reference
In-Depth Information
concentration at a location
y
and expanding it into a Taylor series
84
around
x
:
r
+
1
1
È
Í
(
)
˘
˙
i
Â
uu
()
y
=
()
x
+
(
y
-
x
)
◊ —
u
(
x
¢
)
x
¢
i
!
i
=
1
xx
¢=
(
•
)
.
r
+
2
+
O
yx
-
u
(19)
2
Subtracting
u
(
x
) on both sides, multiplying the whole equation by a
scaled kernel function
η
(
x
)
=
−
d
η
(
x
/
) of core size
>
0, and integrat-
ing over
y
yields
Ú
(()
uu
y
-
()) (
x
h
x
-
y
)
d
y
=
d
IR
r
+
1
1
[
]
Â
(
)
i
Ú
(
yx
-◊ —
)
u
(
x
¢
)
h
(
xyy
-
)
d
x
¢
d
i
!
IR
xx
¢¢ =
i
=
1
(
)
r
+
2
Ú
yx
-
(
xyy
-
)
d
+
u
O
h
.
(20)
•
d
2
IR
For the approximation to be consistent, we have to ask the following
requirement for the kernel function
η
103
:
Ï
Ô
Ô
d
Â
α
d
α
d
0
,
"Œ
IN
,
π
2
e
,
1
£
a
£ +
r
1
'
a
h
i
i
Ú
x
()
xx
d
=
(21)
i
i
=
1
d
IR
i
=
1
α
2
,
if
=
2
e
,
i
Œ
{ , ,
1 2
K
,,}.
d
i
r
is the order of the approximation and
x
=
(
x
1
,
x
2
,…,
x
d
)
∈
IR
d
.
α =
(
∈ IN
d
is a
d
-dimensional index and (
e
1
,
e
2
,…,
e
d
) is the
canonical basis of IR
d
. In the 3D case, the above requirement can be
expressed as
α
1
,
α
2
,…,
α
d
)
Ú
xx
h
()
xx
d
=
2
d
for
i j
,
=
1 2 3
, ,
(22)
ij
j
IR
3
