We estimate the
L
p
L^p
(
p
>
0
p>0
) local distance between piecewise constant solutions to the Cauchy problem of conservation laws and propose a shock admissibility condition for having an
L
p
L^p
...local contraction of such solutions. Moreover, as an application, we prove that there exist
L
p
L^p
locally contractive solutions on some set of initial functions, to the Cauchy problem of conservation laws with convex or concave flux functions. As a result, for conservation laws with convex or concave flux functions, we see that rarefaction waves have an
L
q
L^q
(
q
≥
1
q\geq 1
) local contraction and shock waves have an
L
r
L^r
(
0
>
r
≤
1
0>r\leq 1
) local contraction.
Introducing the notion of extended Schrödinger spaces, we define the criticality and subcriticality of Schrödinger forms in the manner similar to the recurrence and transience of Dirichlet forms. We ...show that a Schrödinger form is critical (resp. subcritical) if and only if there exists an excessive function of the associated Schrödinger semigroup and the Dirichlet form defined by
h
h
-transform of the excessive function is recurrent (resp. transient). We give an analytical condition for the subcriticality of Schrödinger forms in terms of the bottom of spectrum.
We introduce a subclass
K
H
{\mathcal {K}}_H
of the local Kato class and show a Schrödinger form with potential in
K
H
{\mathcal {K}}_H
is critical. Critical Schrödinger forms lead us to critical Hardy-type inequalities. As an example, we treat fractional Schrödinger operators with potential in
K
H
{\mathcal {K}}_H
and reconsider the classical Hardy inequality by our approach.
In this short note, we prove that a generalized Kähler-Einstein metric
g
g
on a Fano manifold is actually a Kähler-Einstein metric if and only if one of the following conditions is satisfied: (i)
g
g
...is a Kähler-Ricci soliton; (ii)
g
g
is an extremal Kähler metric.
We will show the following three theorems on the diffeomorphism and homeomorphism groups of a
K
3
K3
surface. The first theorem is that the natural map
π
0
(
D
i
f
f
(
K
3
)
)
→
A
u
t
(
H
2
(
K
3
;
Z
...)
)
\pi _{0}(Diff(K3)) \to Aut(H^{2}(K3;\mathbb {Z}))
has a section over its image. The second is that there exists a subgroup
G
G
of
π
0
(
D
i
f
f
(
K
3
)
)
\pi _{0}(Diff(K3))
of order two over which there is no splitting of the map
D
i
f
f
(
K
3
)
→
π
0
(
D
i
f
f
(
K
3
)
)
Diff(K3) \to \pi _{0}(Diff(K3))
, but there is a splitting of
H
o
m
e
o
(
K
3
)
→
π
0
(
H
o
m
e
o
(
K
3
)
)
Homeo(K3) \to \pi _{0}(Homeo(K3))
over the image of
G
G
in
π
0
(
H
o
m
e
o
(
K
3
)
)
\pi _{0}(Homeo(K3))
, which is non-trivial. The third is that the map
π
1
(
D
i
f
f
(
K
3
)
)
→
π
1
(
H
o
m
e
o
(
K
3
)
)
\pi _{1}(Diff(K3)) \to \pi _{1}(Homeo(K3))
is not surjective. Our proof of these results is based on Seiberg-Witten theory and the global Torelli theorem for
K
3
K3
surfaces.
In this article, we show that the Riemann hypothesis for an
L
L
-function
F
F
belonging to the Selberg class implies that all the derivatives of
F
F
can have at most finitely many zeros on the left ...of the critical line with imaginary part greater than a certain constant. This was shown for the Riemann zeta function by Levinson and Montgomery in 1974 Acta Math. 133 (1974), pp. 49–65.
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