bluuebunny/crossref_metadata_2025_split huggingface dataset:
Existence and uniqueness of the Riemann problem for a nonlinear system of conservation laws of mixed type
Authors: Hsiao L. , de Mottoni P. 
Published: June 2012

We study the system of conservation laws given by
\[

{

υ
t

+

[
υ
(
a
+
u
)
]

x

=
0

(
a
>
1

is

a

constant

)
,

u
t

+

[
u
(
1
−
υ
)
]

x

=
0
,

\left \{ {_{{\upsilon _t} + {{[\upsilon (a + u)]}_x} = 0\quad (a > 1{\text {is}}{\text {a}}{\text {constant}}),}^{{u_t} + {{[u(1 - \upsilon )]}_x} = 0,}} \right .

\]
with any Riemann initial data

(

u
∓

,

υ
∓

)

({u_ \mp },{\upsilon _ \mp })

. The system is elliptic in the domain where

(
υ
−
u
+
a
−
1

)
2

+
4
(
a
−
1
)
u
>
0

{(\upsilon - u + a - 1)^2} + 4(a - 1)u > 0

and strictly hyperbolic when

(
υ
−
u
+
a
−
1

)
2

+
4
(
a
−
1
)
u
>
0

{(\upsilon - u + a - 1)^2} + 4(a - 1)u > 0

. We combine and generalize Lax criterion and Oleinik-Liu criterion to introduce the generalized entropy condition (G.E.C.) by which we can show that the Riemann problem always has a weak solution (any discontinuity satisfies the G.E.C.) for any initial data, however not necessarily unique. We introduce the minimum principle then in the definition of an admissible weak solution for the Riemann problem and the existence and uniqueness of the solution for any Riemann data.
Immersions of positively curved manifolds into manifolds with curvature bounded above
Authors: Menninga Nadine L. 
Published: June 2012

Let

M
M

be a compact, connected, orientable Riemannian manifold of dimension

n
−
1
⩾
2

n - 1 \geqslant 2

, and let

x
x

be an isometric immersion of

M
M

into an

n
n

-dimensional Riemannian manifold

N
N

. Let

K
K

denote sectional curvature and

i
i

denote the injectivity radius. Assume, for some constant positive constant

c
c

, that

K
(
N
)
⩽
1

/

(
4

c
2

)
,

1

/

c
2

⩽
K
(
M
)

K(N) \leqslant 1/(4{c^2}),\quad 1/{c^2} \leqslant K(M)

, and

π
c
⩽
i
(
N
)

\pi c \leqslant i(N)

. Then the radius of the smallest

N
N

-ball containing

x
(
M
)

x(M)

is less than

1
2

π
c

\tfrac {1} {2}\pi c

and

x
x

is in fact an imbedding of

M
M

into

N
N

, whose image bounds a convex body.
Two differential-difference equations arising in number theory
Authors: Wheeler Ferrell S. 
Published: June 2012

We survey many old and new results on solutions of the following pair of adjoint differential-difference equations: (1)
\[

u

p
′

(
u
)
=
−
a
p
(
u
)
−
b
p
(
u
−
1
)
,

up’(u) = - ap(u) - bp(u - 1),

\]
(2) (
\[

(
u
q
(
u
)

)
′

=
a
q
(
u
)
+
b
q
(
u
+
1
)
.

(uq(u))’ = aq(u) + bq(u + 1).

\]
We bring together scattered results usually proved only for specific

(
a
,
b
)

(a,b)

pairs, while emphasizing the connections between the two equations. We also point out some of the ways these two equations are used in number theory. We giv s several new integral relationships between (1) and (2) and use them to prove a new application of (2) in number theory, namy el
\[

∑

1
>
n
⩽
x

P
2

(
n
)
⩽

P
1

(
n
)

1

/

u

(
log
⁡

P
1

(
n
)
)

α

∼

u
α

f
(
u
)
x

(
log
⁡
x
)

α

(
x
→
∞
,

u
⩾
1
,

α
∈

R

)

\sum \limits _{\begin {array}{*{20}{c}} {1 > n \leqslant x} \\ {{P_2}(n) \leqslant {P_1}{{(n)}^{1/u}}} \\ \end {array} } {{{(\log {P_1}(n))}^\alpha } \sim {u^\alpha }f(u)x{{(\log x)}^\alpha }} \qquad (x \to \infty ,\;u \geqslant 1,\;\alpha \in {\mathbf {R}})

\]
where

P
1

(
n
)

{P_1}(n)

and

P
2

(
n
)

{P_2}(n)

are the first and second largest prime divisors of

n
n

and

f
(
u
)

f(u)

satisfies (2) with

(
a
,
b
)
=
(
1
−
α
,
−
1
)

(a,b) = (1 - \alpha , - 1)

.
Bounds for projection constants and 1-summing norms
Authors: König Hermann , Tomczak-Jaegermann Nicole 
Published: June 2012

It is shown that projection constants

λ
(

X
n

)

\lambda ({X_n})

of

n
n

-dimensional normed spaces

X
n

{X_n}

satisfy

λ
(

X
n

)
⩽

n

−
c

/

n

\lambda ({X_n}) \leqslant \sqrt n - c/\sqrt n

where

c
>
0

c > 0

is a numerical constant. Similarly, the

1
1

-summing norms of (the identity of)

X
n

{X_n}

can be estimated by

π
1

(

X
n

)
⩾

n

+
c

/

n

{\pi _1}({X_n}) \geqslant \sqrt n + c/\sqrt n

. These estimates are the best possible: for prime

n
n

, translation-invariant

n
n

-dimensional spaces

X
n

{X_n}

such that

λ
(

X
n

)
⩾

n

−
2

/

n

\lambda ({X_n}) \geqslant \sqrt n - 2/\sqrt n

and

π
1

(

X
n

)
⩽

n

+
2

/

n

{\pi _1}({X_n}) \leqslant \sqrt n + 2/\sqrt n

can be constructed. For these spaces Gordon-Lewis constants and distances to Hilbert spaces are large as well:

gl
⁡
(

X
n

)
⩾

1
3

n

,
d
(

X
n

,

l
2
n

)
=

n

\operatorname {gl} ({X_n}) \geqslant \tfrac {1} {3}\sqrt n ,d({X_n},l_2^n) = \sqrt n

.
Mean summability methods for Laguerre series
Authors: Stempak Krzysztof 
Published: June 2012

We apply a construction of generalized convolution in
\[

L
1

(

R

+

×

R

,

x

2
α
−
1

d
x
d
t
)
,

α
⩾
1
,

{L^1}({\mathbb {R}_ + } \times \mathbb {R},{x^{2\alpha - 1}}dxdt),\qquad \alpha \geqslant 1,

\]
cf. [8], to investigate the mean convergence of expansions in Laguerre series. Following ideas of [4, 5] we construct a functional calculus for the operator
\[

L
=
−

(

∂
2

∂

x
2

+

2
α
−
1

x

∂

∂
x

+

x
2

∂
2

∂

t
2

)

,

x
>
0
,

t
∈

R

,

α
⩾
1.

L = - \left ( {\frac {{{\partial ^2}}} {{\partial {x^2}}} + \frac {{2\alpha - 1}} {x}\frac {\partial } {{\partial x}} + {x^2}\frac {{{\partial ^2}}} {{\partial {t^2}}}} \right ),\qquad x > 0,\quad t \in \mathbb {R},\quad \alpha \geqslant 1.

\]
Then, arguing as in [3], we prove results concerning the mean convergence of some summability methods for Laguerre series. In particular, the classical Abel-Poisson and Bochner-Riesz summability methods are included.
Hölder domains and Poincaré domains
Authors: Smith Wayne , Stegenga David A. 
Published: June 2012

A domain

D
⊂

R
d

D \subset {R^d}

of finite volume is said to be a

p
p

-Poincaré domain if there is a constant

M
p

(
D
)

{M_p}(D)

so that
\[

∫
D

|

u
−

u
D

|

p

d
x
≤

M
p
p

(
D
)

∫
D

|

∇
u

|

p

d
x

{\int \limits _D {|u - {u_D}|} ^p}dx \leq M_p^p(D){\int \limits _D {|\nabla u|} ^p}dx

\]
for all functions

u
∈

C
1

(
D
)

u \in {C^1}(D)

. Here

u
D

{u_D}

denotes the mean value of

u
u

over

D
D

. Techniques involving the quasi-hyperbolic metric on

D
D

are used to establish that various geometric conditions on

D
D

are sufficient for

D
D

to be a

p
p

-Poincaré domain. Domains considered include starshaped domains, generalizations of John domains and Hàlder domains.

D
D

is a Hàlder domain provided that the quasi-hyperbolic distance from a fixed point

x
0

∈
D

{x_0} \in D

to

x
x

is bounded by a constant multiple of the logarithm of the euclidean distance of

x
x

to the boundary of

D
D

. The terminology is derived from the fact that in the plane, a simply connected Hàlder domain has a Hàlder continuous Riemann mapping function from the unit disk onto

D
D

. We prove that if

D
D

is a Hàlder domain and

p
≥
d

p \ge d

, then

D
D

is a

p
p

-Poincaré domain. This answers a question of Axler and Shields regarding the image of the unit disk under a Hàlder continuous conformal mapping. We also consider geometric conditions which imply that the imbedding of the Sobolev space

W

1
,
p

(
D
)
→

L
p

(
D
)

{W^{1,p}}(D) \to {L^p}(D)

is compact, and prove that this is the case for a Hàlder domain

D
D

.
The 𝑣₁-periodic homotopy groups of an unstable sphere at odd primes
Authors: Thompson Robert D. 
Published: June 2012

The

mod

p

\bmod \;p

v
1

{v_1}

-periodic homotopy groups of a space

X
X

are defined by considering the homotopy classes of maps of a Moore space into

X
X

and then inverting the Adams self map. In this paper we compute the

p
p

v
1

{v_1}

-periodic homotopy groups of an odd dimensional sphere, localized at an odd prime. This is done by showing that these groups are isomorphic to the stable

mod

p

\bmod \;p

v
1

{v_1}

-periodic homotopy groups of

B

Σ
p

2
(
p
−
1
)
n

B\Sigma _p^{2(p - 1)n}

, the

2
(
p
−
1
)
n

2(p - 1)n

skeleton of the classifying space for the symmetric group

Σ
p

{\Sigma _p}

. There is a map

Ω

2
n
+
1

S

2
n
+
1

→

Ω
∞

(
J
∧
B

Σ
p

2
(
p
−
1
)
n

)

{\Omega ^{2n + 1}}{S^{2n + 1}} \to {\Omega ^\infty }(J \wedge B\Sigma _p^{2(p - 1)n})

, where

J
J

is a spectrum constructed from connective

K
K

-theory, and the image in homotopy is studied.
Linear series with an 𝑁-fold point on a general curve
Authors: Schubert David 
Published: June 2012

A linear series

(
V
,

L

)

(V,\mathcal {L})

on a curve

X
X

has an

N
N

-fold point along a divisor

D
D

of degree

N
N

if

dim
⁡
(
V
∩

H
0

(
X
,

L

(
−
D
)
)
)
≥
dim

V
−
1

\dim (V \cap {H^0}\;(X,\mathcal {L}\,(- D))) \geq \dim \;V - 1

. The dimensions of the families of linear series with an

N
N

-fold point are determined for general curves.
A cubic counterpart of Jacobi’s identity and the AGM
Authors: Borwein J. M. , Borwein P. B. 
Published: June 2012

We produce exact cubic analogues of Jacobi’s celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is
\[

a

n
+
1

:=

a
n

+
2

b
n

3

and

b

n
+
1

:=

b
n

(

a
n
2

+

a
n

b
n

+

b
n
2

3

)

.

3

{a_{n + 1}}: = \frac {{{a_n} + 2{b_n}}} {3}\quad {\text {and}}\quad {b_{n + 1}}: = \sqrt [3]{{{b_n}\left ( {\frac {{a_n^2 + {a_n}{b_n} + b_n^2}} {3}} \right ).}}

\]
The limit of this iteration is identified in terms of the hypergeometric function

2

F
1

(
1

/

3
,
2

/

3
;
1
;
⋅
)

{}_2{F_1}(1/3,2/3;1; \cdot )

, which supports a particularly simple cubic transformation.
On the distance between unitary orbits of weighted shifts
Authors: Marcoux Laurent 
Published: June 2012

In this paper, we consider invertible bilateral weighted shift operators acting on a complex separable Hilbert space

H

\mathcal {H}

. They have the property that there exist a constant

τ
>
0

\tau > 0

and an orthonormal basis

{

e
i

}

i
∈

Z

{\{ {{e_i}} \}_{i \in \mathbb {Z}}}

for

H

\mathcal {H}

with respect to which a shift

V
V

acts by

W

e
i

=

w
i

e

i
+
1

,
i
∈

Z

W{e_i}= {w_i}{e_{i + 1}},i \in \mathbb {Z}

and

|

w
i

|

≥
τ

{\mathbf {|}}{w_i}{\mathbf {|}} \geq \tau

. The equivalence class

U

(
W
)
=
{

U

∗

W
U
:
U
∈

B

(

H

)
,
U

unitary

}

\mathcal {U}(W)= \{ {U^{\ast }}\;WU:U \in \mathcal {B}(\mathcal {H}),U\;{\text {unitary}}\}

of weighted shifts with weight sequence (with respect to the basis

{

U

∗

e
i

}

i
∈

Z

{\{ {U^{\ast }}{e_i}\} _{i \in \mathbb {Z}}}

for

H

)

\mathcal {H})

identical to that of

W
W

forms the unitary orbit of

W
W

. Given two shifts

W
W

and

V
V

, one can define a distance

d
(

U

(
V
)
,

U

(
W
)
)
=
inf
{
∥

X
−
Y
∥:
X
∈

U

(
V
)
,
Y
∈

U

(
W
)
}

d(\mathcal {U}(V),\mathcal {U}(W))= \inf \{\parallel \,X - Y\parallel :X \in \mathcal {U}(V),Y \in \mathcal {U}(W)\}

between the unitary orbits of

W
W

and

V
V

. We establish numerical estimates for upper and lower bounds on this distance which depend upon information drawn from finite dimensional restrictions of these operators.
A multidimensional Wiener-Wintner theorem and spectrum estimation
Authors: Benedetto John J. 
Published: June 2012

Sufficient conditions are given for a bounded positive measure

μ
\mu

on

R

d

{\mathbb {R}^d}

to be the power spectrum of a function

φ
\varphi

. Applications to spectrum estimation are made for the cases in which a signal

φ
\varphi

is known or its autocorrelation

P
ϕ

{P_\phi }

is known. In the first case, it is shown that
\[

∫

|

f
^

(
γ
)

|

2

d

μ
ϕ

(
γ
)
=

lim

R
→
∞

1

|

B
(
R
)

|

∫

B
(
R
)

|

f
∗
φ
(
t
)

|

2

d
t
,

\int {|\hat f(\gamma )|^2}d{\mu _\phi }(\gamma )= \lim \limits _{R \to \infty }\,\frac {1}{|B(R )|}\,\int _{B(R)} |f \ast \varphi (t)|^2\;dt,

\]
where

P
^

φ

=

μ
φ

{\hat P}_{\varphi }= {\mu _\varphi }

,

B
(
R
)

B(R)

is the

d
d

-dimensional ball of radius

R
R

, and

f
f

ranges through a prescribed function space. In the second case, an example, which is a variant of the Tomas-Stein restriction theorem, is
\[

∀
f
∈

L
1

(

R

d

)

∩

L
p

(

R

d

)
,

(

∫

∑

d
−
1

|

f
^

(
θ
)

|

2

d

μ

d
−
1

(
θ
)
)

1

/

2

≤

(

1
2

∥

μ

d
−
1

∨

∥

p
′

1

/

2

)

(
∥
f

∥

1

+
∥
f

∥

p

)

,

\forall f \in {L^1}({\mathbb {R}^d})\, \cap \,{L^p}({\mathbb {R}^d}), \\ \left (\int _{\sum _{d - 1}} | \hat f(\theta )|^2 d {\mu _{d - 1}}(\theta ) \right )^{1/2} \leq \left (\frac {1}{2}\parallel \mu _{d - 1}^{\vee } \parallel _{p’}^{1/2} \right )\;\left (\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right ),

\]
where

1
≤
p
>
2
d

/

(
d
+
1
)

1 \leq p > 2d/(d + 1)

and the power spectrum

μ

d
−
1

{\mu _{d - 1}}

is the compactly supported restriction of surface measure to the unit sphere

∑

d
−
1

⊆

R

^

d

\sum \nolimits _{d - 1} { \subseteq } \;{{\hat {\mathbb {R}}}^d}

.
Iteration of a composition of exponential functions
Authors: Dong Xiaoying 
Published: June 2012

We show that for certain complex parameters

λ
1

,
…
,

λ

n
−
1

{\lambda _1},\ldots ,{\lambda _{n - 1}}

and

λ
n

{\lambda _n}

the Julia set of the function
\[

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

{e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}

\]
is the whole plane

C

\mathbb {C}

. We denote by

Λ
\Lambda

the set of

n
n

-tuples

(

λ
1

,
…
,

λ
n

)
,

λ
1

,
…
,

λ
n

∈

R

({\lambda _1},\ldots ,{\lambda _n}),{\lambda _1},\ldots ,{\lambda _n} \in \mathbb {R}

for which the equation
\[

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

−
z
=
0

{e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}} - z= 0

\]
has exact two real solutions. In fact, one of them is an attracting fixed point of
\[

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

,

{e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}},

\]
which is denoted by

q
q

. We also show that when

(

λ
1

,
…
,

λ
n

)

∈
Λ

({\lambda _1},\ldots ,{\lambda _n})\, \in \Lambda

, the Julia set of
\[

e

λ
1

e

.

.

.

λ

n
−
1

e
λ

n
z

{e^{{\lambda _1}{e^{^{{.^{{.^{{.^{{{^{{\lambda _{n - 1}}}}^{{e^\lambda }^{_{{n^z}}}}}}}}}}}}}}}}

\]
is the complement of the basin of attraction of

q
q

. The ideas used in this note may also be applicable to more general functions.
Continuity of translation in the dual of 𝐿^{∞}(𝐺) and related spaces
Authors: Graham Colin C. , Lau Anthony T. M. , Leinert Michael 
Published: June 2012

Let

X
X

be a Banach space and

G
G

a locally compact Hausdorff group that acts as a group of isometric linear operators on

X
X

. The operation of

x
∈
G

x \in G

on

X
X

will be denoted by

L
x

{L_x}

. We study the set

X
c

{X_c}

of elements

μ
∈
X

\mu \in X

such that

x
↦

L
x

μ

x \mapsto {L_x}\mu

is continuous with respect to the topology on

G
G

and the norm-topology on

X
X

. The spaces

X
X

studied include

M

(
G

)

∗

,

LUC

(
G

)

∗

,

L
∞

(
G

)

∗

,

VN

(
G
)

M{(G)^{\ast } },{\text {LUC}}{(G)^{\ast } },{L^\infty }{(G)^{\ast } },{\text {VN}}(G)

, and

VN

(
G

)

∗

{\text {VN}}{(G)^{\ast } }

. In most cases, characterizations of

X
c

{X_c}

do not appear to be possible, and we give constructions that illustrate this. We relate properties of

X
c

{X_c}

to properties of

G
G

. For example, if

X
c

{X_c}

is sufficiently small, then

G
G

is compact, or even finite, depending on the case. We give related results and open problems.
Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of 𝐂ⁿ
Authors: Shi Ji Huai 
Published: June 2012

In this paper, the following two inequalities are proved:
\[

∫
0
1

(
1
−
r
)

a

|

α

|

+
b

M
p
a

(
r
,

D

α

f
)

d
r
≤
K

∫
0
1

(
1
−
r
)

b

M
p
a

(
r
,
f
)

d
r
,

∫
0
1

(
1
−
r
)

b

M
p
a

(
r
,
f
)

d
r

≤
K

{

∑

|

α

|

≤
m
−
1

|

(

D
α

f
)
(
0
)

|

a

+

∑

|

α

|

=
m

∫
0
1

(
1
−
r
)

a
m
+
b

M
p
a

(
r
,

D

α

f
)

d
r
}

\int _0^1 {{(1 - r)}^{a|\alpha | + b}}M_p^a(r,D^{\alpha } f)\,dr \leq K\int _0^1 {{(1 - r)}^b}M_p^a(r,f)\,dr, \int _0^1 {{(1 - r)}^b}M_p^a(r,f)\,dr \\ \leq K\left \{ \sum \limits _{|\alpha | \leq m - 1} |({D^\alpha }f)(0)|^a + \sum \limits _{|\alpha | = m} \int _0^1 {(1 - r)}^{am + b}M_p^a(r,D^{\alpha }f)\,dr \right \}

\]
where

α
=
(

α
1

,
…
,

α
n

)

\alpha = ({\alpha _1}, \ldots ,{\alpha _n})

is multi-index,

0
>
p
>
∞
,
0
>
a
>
∞

0 > p > \infty ,0 > a > \infty

and

−
1
>
b
>
∞

- 1 > b > \infty

. These are a generalization of some classical results of Hardy and Littlewood. Using these two inequalities, we generalize a theorem in

[
9
]

[9]

. The methods used in the proof of Theorem 1 lead us to obtain Theorem 7, which enables us to generalize some theorems about the pluriharmonic conjugates in

[
8
]

[8]

and

[
2
]

[2]

.
A measure of smoothness related to the Laplacian
Authors: Ditzian Z. 
Published: June 2012

A

K
K

-functional on

f
∈
C

(

R
d

)

f \in C\,({R^d})

given by
\[

K
~

(
f
,

t
2

)
=
inf
(

|

|

f
−
g

|

|

+

t
2

|

|

Δ
g

|

|

;
g
∈

C
2

(

R
d

)
)

\tilde K\,(f,{t^2})= \inf (||f - g|| + {t^2}||\Delta g||;g \in {C^2}\,({R^d}))

\]
will be shown to be equivalent to the modulus of smoothness
\[

w
~

(
f
,
t
)
=

sup

0
>
h
≤
t

‖

2

d
f
(
x
)
−

∑

i
=
1

d

[
f
(
x
+
h

e
i

)
+
f
(
x
−
h

e
i

)
]

‖

.

\tilde w\,(f,t)= \sup \limits _{0 > h \leq t} \,\left \| {2\,df(x) - \sum \limits _{i = 1}^d {[f(x + h{e_i}) + f(x - h{e_i})]} } \right \|.

\]
The situation for other Banach spaces of functions on

R
d

{R^d}

will also be resolved.
Stable and uniformly stable unit balls in Banach spaces
Authors: Suárez Granero Antonio 
Published: June 2012

Let

X
X

be a Banach space with closed unit ball

B
X

{B_X}

and, for

x
∈
X

x \in X

,

r
≥
0

r \geq 0

, put

B
(
x
;
r
)
=
{
u
∈
X
:

|

|

u
−
x

|

|

≤
r
}

B(x;r)= \{ u \in X:||u - x|| \leq r\}

and

V
(
x
,
r
)
=

B
X

∩
B
(
x
;
r
)

V(x,r)= {B_X} \cap B(x;r)

. We say that

B
X

{B_X}

(or in general a convex set) is stable if the midpoint map

Φ

1

/

2

:

B
X

×

B
X

→

B
X

{\Phi _{1/2}}:{B_X} \times {B_X} \to {B_X}

, with

Φ

1

/

2

(
u
,
υ
)
=

1
2

(
u
+
υ
)

{\Phi _{1/2}}(u,\upsilon )= \frac {1}{2}(u + \upsilon )

, is open. We say that

B
X

{B_X}

is uniformly stable (US) if there is a map

α
:
(
0
,
2
]
→
(
0
,
2
]

\alpha :(0,2] \to (0,2]

, called a modulus of uniform stability, such that, for each

x
,
y
∈

B
X

x,y \in {B_X}

and

r
∈
(
0
,
2
]
,
V
(

1
2

(
x
+
y
)
;
α
(
r
)
)
⊆

1
2

(
V
(
x
;
r
)
+
V
(
y
;
r
)
)

r \in (0,2],V(\frac {1} {2}(x + y);\alpha (r)) \subseteq \frac {1} {2}(V(x;r) + V(y;r))

. Among other things, we see: (i) if

dim
⁡
X
≥
3

\dim X \geq 3

, then

X
X

admits an equivalent norm such that

B
X

{B_X}

is not stable; (ii) if

dim
⁡
X
>
∞

\dim X > \infty

,

B
X

{B_X}

is stable iff

B
x

{B_x}

is US; (iii) if

X
X

is rotund,

X
X

is uniformly rotund iff

B
X

{B_X}

is US; (iv) if

X
X

is

3.2.

I.P

3.2.{\text {I.P}}

,

B
X

{B_X}

is US and

α
(
r
)
=
r

/

2

\alpha (r)= r/2

is a modulus of US; (v)

B
X

{B_X}

is US iff

B

X

∗
∗

{B_{{X^{ \ast \ast }}}}

is US and

X
X

,

X

∗
∗

{X^{ \ast \ast }}

have (almost) the same modulus of US; (vi)

B
X

{B_X}

is stable (resp. US) iff

B

C
(
K
,
X
)

{B_{C(K,X)}}

is stable (resp. US) for each compact

K
K

iff

B

A
(
K
,
X
)

{B_{A(K,X)}}

is stable (resp. US) for each Choquet simplex

K
K

; (vii)

B
X

{B_X}

is stable iff

B

L
p

(
μ
,
X
)

{B_{{L_p}(\mu ,X)}}

is stable for each measure

μ
\mu

and

1
≤
p
>
∞

1 \leq p > \infty

.
A sharp inequality for martingale transforms and the unconditional basis constant of a monotone basis in 𝐿^{𝑝}(0,1)
Authors: Choi K. P. 
Published: June 2012

Let

1
>
p
>
∞

1 > p > \infty

. Let

d
=
(

d
1

,

d
2

,
…
)

d = ({d_1},{d_2}, \ldots )

be a real-valued martingale difference sequence,

θ
=
(

θ
1

,

θ
2

,
…
)

\theta = ({\theta _1},{\theta _2}, \ldots )

is a predictable sequence taking values in

[
0
,
1
]

[0,1]

. We show that the best constant of the inequality,
\[

‖

∑

k
=
1

n

θ
k

d
k

‖

p

≤

c
p

‖

∑

k
=
1

n

d
k

‖

p

,

n
≥
1
,

{\left \| {\sum \limits _{k = 1}^n {{\theta _k}{d_k}} } \right \|_p} \leq {c_p}{\left \| {\sum \limits _{k = 1}^n {{d_k}} } \right \|_p}, \quad n \geq 1,

\]
satisfies
\[

c
p

=

p
2

+

1
2

log

(

1
+
y

2

)

+

α
2

p

+
⋯
,

{c_p} = \frac {p}{2} + \frac {1}{2}\;\log \;\left ({\frac {{1 + y}}{2}} \right ) + \frac {{{\alpha _2}}}{p} + \cdots ,

\]
where

γ
=

e

−
2

\gamma = {e^{ - 2}}

and

α
2

=

[

1
2

log

1
+
γ

2

]

2

+

1
2

log

1
+
γ

2

−
2

(

γ

1
+
γ

)

2

{\alpha _2} = {\left [ {\frac {1}{2}\;\log \;\frac {{1 + \gamma }}{2}} \right ]^2} + \frac {1}{2}\;\log \;\frac {{1 + \gamma }}{2} - 2{\left ({\frac {\gamma }{{1 + \gamma }}} \right )^2}

. The best constant equals the unconditional basis constant of a monotone basis of

L
p

(
0
,
1
)

{L^p}(0,1)

.
Nonexistence of nodal solutions of elliptic equations with critical growth in ℝ²
Authors: Adimurthi  , Yadava S. L. 
Published: June 2012

Let

f
(
t
)
=
h
(
t
)

e

b

t
2

f(t) = h(t){e^{b{t^2}}}

be a function of critical growth. Under a suitable assumption on

h
h

, we prove that
\[

−
Δ
u
=
f
(
u
)

a
m
p
;

in

B
(
R
)
⊂

R

2

,

u
=
0

a
m
p
;

on

∂
B
(
R
)
,

\begin {array}{*{20}{c}} { - \Delta u = f(u)} \hfill & {{\text {in}}\;B(R) \subset {\mathbb {R}^2},} \hfill \\ {u = 0} \hfill & {{\text {on}}\;\partial B(R),} \hfill \\ \end {array}

\]
does not admit a radial solution which changes sign for sufficiently small

R
R

.
Existence of smooth solutions to the classical moment problems
Authors: Jorgensen Palle E. T. 
Published: June 2012

Let

s
(
0
)
,
s
(
1
)
,
…

s(0),s(1), \ldots

be a given sequence, and define

s
(
n
)
=

s
(
−
n
)

¯

s(n) = \overline {s( - n)}

for

n
>
0

n > 0

. If

Σ
Σ

ξ
¯

n

ξ
m

s
(
m
−
n
)
≥
0

\Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0

holds for all finite sequences

(

ξ
n

)

n
∈

Z

{({\xi _n})_{n \in \mathbb {Z}}}

, then it is known that there is a positive Borel measure

μ
\mu

on the circle

T

\mathbb {T}

such that

s
(
n
)
=

∫

−
π

π

e

i
n
t

d
μ
(
t
)

s(n) = \smallint _{ - \pi }^\pi {{e^{int}}d\mu (t)}

, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence

(
s
(
n
)
)

(s(n))

that the measure

μ
\mu

may be chosen to be smooth. A measure

μ
\mu

is said to be smooth if it has the same spectral type as the operator

i
d

/

d
t

id/dt

acting on

L
2

(

T

)

{L^2}(\mathbb {T})

with respect to Haar measure

d
t

dt

on

T

\mathbb {T}

: Equivalently,

μ
\mu

is a superposition (possibly infinite) of measures of the form

|

w
(
t
)

|

2

d
t

|w(t){|^2}dt

with

w
∈

L
2

(

T

)

w \in {L^2}(\mathbb {T})

such that

d
w

/

d
t
∈

L
2

(

T

)

dw/dt \in {L^2}(\mathbb {T})

. The condition is stated purely in terms of the initially given sequence

(
s
(
n
)
)

(s(n))

: We show that a smooth representation exists if and only if, for some

ε
∈

R

+

\varepsilon \in {\mathbb {R}_ + }

, the a priori estimate
\[

∑

∑

s
(
m
−
n
)

ξ
¯

n

ξ
m

≥
ε

|

∑

n
s
(
n
)

ξ
n

|

2

\sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left | {\sum {ns(n){\xi _n}} } \right |}^2}} }

\]
is valid for all finite double sequences

(

ξ
n

)

({\xi _n})

. An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.