... by
ω
W
1,p
,
l
,w
α
ω
W
1,p
,
l
,w
α
diam
Ω
−1
ω
p ,Ω, w
α
ω
p ,Ω, w
α
, 1.6
where α is a real number.
We denote the exterior derivative by d : D
,
l
... Lip
k
,
l
for loc Lip
k
,
l
,w
α
. Similarly, for u ∈ L
1
loc
,
l
,w
α
, l
0, 1, ,n, we write u ∈ BMO ,
l
,w
α
if
u
, ,w...
... the performance of watermar k retrieval
process, normalized correlation (NC) is used. Here, W
1
and W
2
are the original and recovered watermark signals,
respectively. The normalized correlation ... watermarks
and popular host images such as Lena, Barbara, B aboon,
Cameraman, City, Couple, Man, Boat, Elaine, Peppers,
and Zelda of size 512
×512. The watermark is a grayscale
fingerprint...
... x
1
, ,x
n
, w w
1
, ,w
n
, we define x
i
x
1
, ,x
i
, w
i
w
1
, ,w
i
,W
i
i
j1
w
j
, M
i ,r
M
i ,r
x
i
M
i ,r
w
i
/W
i
, x
i
, and M
i ,r
M
1 ,r
, ,M
i ,r
. Then we have the ... symmetric means.
1. Introduction
Let M
n ,r
x be the generalized weighted power means: M
n ,r
q, x
n
i1
q
i
x
r
i
1 /r
,...
... Mathematics, Marcel Dekker, New York, NY, USA, 1992.
2 E. F. Beckenbach and R. Bellman, Inequalities, Springer, New York, NY, USA, 1965.
3 D. S. Mitrinovi
´
c, Analytic Inequalities, Springer, New ... r
A
r
dr
× exp
−
s
−s
k
r, r
B
r
dr
t
s
r
r
∂
r
k
r, y
B
y
dy dr
ds
t
0
k
s, s
A
s
exp
−
s
−s
k
r, r
B
r
dr...
... 73860 3, 11 pages, 2008.
4 L C. Jang, S D. Kim, D W. Park, and Y S. Ro, “A note on Euler number and polynomials,” Journal
of Inequalities and Applications, vol. 200 6, Article ID 3460 2, 5 pages, ... higher order,” Advanced Studies in Contemporary Mathematics,
vol. 1 9, no. 1, pp. 25–2 9, 2009.
22 T. Kim, M S. Kim, L C. Jang, and S H. Rim, “New q-Euler number...
... explained.
2. Regularization and Error Estimate
First, let u be the Fourier transform of u:
u
x, ξ
1
√
2π
∞
−∞
u
x, y
e
−iξy
dy, ξ ∈ R.
2.1
Taking Fourier transformation for 1.2 , we ... 1. Of course, since g and h are assumed to be measured, there must be
measurement errors, and we would actually have noisy data function g
δ
,h
δ
∈ L
2
R ,for
which the measureme...
... York, NY, USA, 1989.
6 Y. Du and Y. Lou, “Proof of a conjecture for the perturbed Gelfand equation from combustion theory,”
Journal of Differential Equations, vol. 17 3, no. 2, pp. 213–23 0, 2001.
7 ... boundary value problems of the form −Δu λfu; ,
αx, u∂u/∂η1 − αx, uu 0; Ω, whereΩ is a bounded domain in
R
n
with n ≥ 1, Δ is
the Laplace operator, λ is a...
... Taking
Geoinformation Science One Step Further Se ries,B.Lars,F.
Anders, and P. Hardy, Eds ., Lecture Notes in Geoinformation
and Cartography, Springer, Heidelberg, Germany, 2008.
[7]A.Demiriz,K.P.Bennet,andM.J.Embrechts,“Semi-
supervised ... homogeneous
regions,” in Proceedings of 3rd International Conference on Geo-
graphical Information Science (GIScience ’04),M.J.Egenhofer,
C. Fres...
... Mstep:
µ
new
k ,r
=
1
Q
k ,r
N
n
l:( r
k,n,l
=r)
q
k,n
x
n,l
,
Σ
new
k ,r
=
1
Q
k ,r
N
n
l:( r
k,n,l
=r)
q
k,n
(x
n,l
− µ
new
k ,r
)(x
n,l
− µ
new
k ,r
)
t
,
π
new
k
=
N
k
N
,
where Q
k ,r
=
N
n
l:( r
k,n,l
=r)
q
k,n
, N
k
=
N
n
q
k,n
.
(4) If parameter updating converges, the ... Estep:
q
k,n
=
π
k
p(I
n
|θ
k
)
K
k
π
k
p(I
n
|θ
k
)
,wherep(I
n...
... and the lth lighting condition.
These training images are sorted in a specific order so as to
construct a data matrix X
∈ R
m×n
s
n
v
n
l
:
X
=
x
1,1 ,1
, x
2,1 ,1
, , x
n
s
,1 ,1
, x
1,2 ,1
, , ... Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Linear D...