bluuebunny/crossref_metadata_2025_split huggingface dataset:
A comparison theorem for linear difference equations
Authors: Eloe P. W. 
Published: July 2010

Let a be real,

I
=
{
a
,
a
+
1
,
⋯
,
b
}

I = \{ a,a + 1, \cdots ,b\}

where

b
−
a

b - a

is a positive integer or

I
=
{
a
,
a
+
1
,
…
}

I = \{ a,a + 1, \ldots \}

. Let

n
n

be a positive integer and let

I
n

=
{
a
,
a
+
1
,
…
,
b
+
n
}

{I^n} = \{ a,a + 1, \ldots ,b + n\}

if

b
>
∞

or

I
n

=
I

b > \infty \;{\text {or}}\;{I^n} = I

otherwise. Consider the

n
n

th order difference equation

P
u
(
m
)
=

∑

j
=
0

n

α
j

(
m
)
u
(
m
+
j
)
=
0
,

α
n

(
m
)
=
1
,

α
0

(
m
)
≠
0
,

m
∈
I

Pu(m) = \sum \nolimits _{j = 0}^n {{\alpha _j}(m)u(m + j) = 0,\;{\alpha _n}(m) = 1,\;{\alpha _0}(m) \ne 0,\;m \in I}

. It is shown that if

0
≤
r
(
m
)
≤
q
(
m
)
,

m
∈
I

0 \leq r(m) \leq q(m),\;m \in I

and if the equations

P
u
(
m
)
=
0

Pu(m) = 0

and

P
u
(
m
)
+
q
(
m
)
u
(
m
)
=
0

Pu(m) + q(m)u(m) = 0

are disconjugate on

I
n

{I^n}

, then the equation

P
u
(
m
)
+
r
(
m
)
u
(
m
)
=
0

Pu(m) + r(m)u(m) = 0

is disconjugate on

I
n

{I^n}

.
Martin compactifications of the punctured disk with close to rotation free densities
Authors: Tada Toshimasa 
Published: July 2010

The purpose of this paper is to prove that Martin compactifications

Ω
P
∗

\Omega _P^*

and

Ω
Q
∗

\Omega _Q^*

of the punctured unit disk

Ω
:
0
>

|

z

|

>
1

\Omega :0 > |z| > 1

with respect to equations

Δ
u
=
P
u

\Delta u = Pu

and

Δ
u
=
Q
u

\Delta u = Qu

, respectively, are homeomorphic to each other if

|

P
(
z
)
−
Q
(
z
)

|

=
O
(

|

z

|

−
2

)
(
z
→
0
)

|P(z) - Q(z)| = O(|z{|^{ - 2}})(z \to 0)

and

P
(
z
)
=
P
(

|

z

|

)
(
z
∈
Ω
)

P(z) = P(|z|)(z \in \Omega )

.
On the maximal Riesz-transforms along surfaces
Authors: Chen Lung-Kee 
Published: July 2010

Let

b
(
t
)

b(t)

be an arbitrary bounded radial function. For

x
=
(

x
1

,

x
2

)
,
t
=
(

t
1

,

t
2

)

x = ({x_1},{x_2}),t = ({t_1},{t_2})

in

R
2

,

|
t
|

=

(

t
1

+

t
2

)

1

/

2

{R^2},\left | t \right | = {({t_1} + {t_2})^{1/2}}

, we establish that the following maximal Riesz-transforms along the surfaces

(

t
1

,

t
2

,

|

t

|

a

)
,
a
>
0

({t_1},{t_2},|t{|^a}),a > 0

:
\[

T
∗

f
(
x
)
=

sup

ε
>
0

|

∫

|

t

|

>
ε

f
(

x
1

−

t
1

,

x
2

−

t
2

,

x
3

−

|

t

|

a

)
b
(
t
)

t
1

|

t

|

3

d
t

|

{T^*}f(x) = \sup \limits _{\varepsilon > 0} \left | {\int _{|t| > \varepsilon } {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})b(t)\left . {\frac {{{t_1}}}{{|t{|^3}}}dt} \right |} } \right .

\]
are bounded in

L
p

(

R
3

)

{L^p}({R^3})

for all

1
>
p
>
∞

1 > p > \infty

. The

n
n

-dimensional result can be found at the end of this paper.
Continuous functions with everywhere infinite variation with respect to sequences
Authors: Buczolich Z. 
Published: July 2010

We prove that if

{

a
n

}

n
=
1

∞

\{ {a_n}\} _{n = 1}^\infty

is such that

a
n

↘
0

{a_n} \searrow 0

and
\[

lim

n
→
∞

(

a
n

+
1

/

a
n

)
=
1
,

\lim \limits _{n \to \infty } ({a_n}_{ + 1}/{a_n}) = 1,

\]
then for the typical continuous function

f
f

we have
\[

S

n
0

:=

∑

n
=

n
0

∞

|

f
(

x
n

+
1

)
−
f
(

x
n

)

|

=
+
∞

{S_{{n_0}}}: = \sum \limits _{n = {n_0}}^\infty | f({x_n}_{ + 1}) - f({x_n})| = + \infty

\]
whenever

x
∈
[
0
,
1
−

a

n
0

]

x \in [0,1 - {a_{{n_0}}}]

and

x
n

∈
[
x
+

a

n
+
1

,
x
+

a
n

]

{x_n} \in [x + {a_{n + 1}},x + {a_n}]

. Based on our result in a previous paper, we know that the above theorem fails to hold if

a

n
+
1

/

a
n

=
λ
>
1

{a_{n + 1}}/{a_n} = \lambda > 1

. We also prove that if

{

a
n

}

n
=
1

∞

\{ {a_n}\} _{n = 1}^\infty

is such that

a
n

↘
0

{a_n} \searrow 0

, then for the typical continuous function

f
f

we have

S

n
0

=
+
∞

if

x
n

=
x
+

a
n

{S_{{n_0}}} = + \infty {\text { if }}{x_n} = x + {a_n}

and

x
∈
[
0
,
1
−

a

n
0

]

x \in [0,1 - {a_{{n_0}}}]

.
Bounded sequence-to-function Hausdorff transformations
Authors: Georgakis Constantine 
Published: July 2010

Let
\[

(

T
a

)

(
y
)

=

∑

n
=
0

∞

(

−
y

)

n

g

(
n
)

(
y
)

n
!

a

n
,

y
≥
0

\left ( {Ta} \right )\left ( y \right ) = \sum \limits _{n = 0}^\infty {{{\left ( { - y} \right )}^n}} \frac {{{g^{\left ( n \right )}}\left ( y \right )}}{{n!}}{a_{n,\quad }}\quad y \geq 0

\]
be the sequence-to-function Hausdorff transformation generated by the completely monotone function

g
g

or, what is equivalent, the Laplace transform of a finite positive measure

σ
\sigma

on

[
0
,
∞
)

[0,\infty )

. It is shown that for

1
≤
p
≤
∞

1 \leq p \leq \infty

,

T
T

is a bounded transformation of

l
p

{l^p}

with weight

Γ

(

n
+
s
+
1

)

/

n
!

\Gamma \left ( {n + s + 1} \right ) / n!

into

L
p

[
0
,
∞
)

{L^p}[0,\infty )

with weight

y
s

,
s
>
−
1

{y^s},s > - 1

, whose norm

‖
T
‖

=

∫
0
∞

t

−

(

1
+
s

)

/

p

d
σ

(
t
)

=
C

(

p
,
s

)

\left \| T \right \| = \int _0^\infty {{t^{ - \left ( {1 + s} \right ) / p}}} d\sigma \left ( t \right ) = C\left ( {p,s} \right )

if and only if

C

(

p
,
s

)

>
∞

C\left ( {p,s} \right ) > \infty

, and that for

1
>
p
>
∞
,

‖

T
a

‖

p
,
s

>
C

(

p
,
s

)

‖
a
‖

p
,
s

1 > p > \infty ,{\left \| {Ta} \right \|_{p,s}} > C\left ( {p,s} \right ){\left \| a \right \|_{p,s}}

unless

a
n

{a_n}

is a null sequence. Furthermore, if

1
>
p
>
r
>
∞
,

0
>
λ
>
1

1 > p > r > \infty ,\,\;0 > \lambda > 1

and

σ
\sigma

is absolutely continuous with derivatives

ψ
\psi

such that the function

ψ
r

(
t
)

=

t

−
1

/

r

ψ

(
t
)

{\psi _r}\left ( t \right ) = {t^{ - 1 / r}}\psi \left ( t \right )

belongs to

L

1

/

λ

[
0
,
∞
)

{L^{1 / \lambda }}[0,\infty )

, then the transformation

(

T
λ

a

)

(
y
)

=

y

1
−
λ

(

T
a

)

(
y
)

\left ( {{T_\lambda }a} \right )\left ( y \right ) = {y^{1 - \lambda }}\left ( {Ta} \right )\left ( y \right )

is bounded from

l
p

{l^p}

to

L
r

[
0
,
∞
)

{L^r}[0,\infty )

and has norm

‖

T
λ

‖

≤

‖

ψ
r

‖

1

/

λ

\left \| {{T_\lambda }} \right \| \leq {\left \| {{\psi _r}} \right \|_{1 / \lambda }}

. The transformation

T
T

includes in particular the Borel transform and that of generalized Abel means. These results constitute an improved analogue of a theorem of Hardy concerning the discrete Hausdorff transformation on

l
p

{l^p}

which corresponds to a totally monotone sequence, and lead to improved forms of some inequalities of Hardy and Littlewood for power series and moment sequences.
Integrability of double lacunary sine series
Authors: Móricz Ferenc 
Published: July 2010

We consider mainly the series

∑

∑

a

j
k

sin
⁡

2
j

x
sin
⁡

2
k

y

\sum {\sum {{a_{jk}}\sin {2^j}x\sin {2^k}y} }

, which converges to a finite function

f
(
x
,
y
)

f(x,y)

a.e. if

∑

∑

a

j
k

2

>
∞

\sum {\sum {a_{jk}^2 > \infty } }

. We show that the (Lebesgue) integrability of

x

−
1

y

−
1

f
(
x
,
y
)

{x^{ - 1}}{y^{ - 1}}f(x,y)

over the two-dimensional torus is essentially controlled by the quantity

∑

m
=
1

∞

∑

n
=
1

∞

(

∑

j
=
m

∞

∑

k
=
n

∞

a

j
k

2

)

1

/

2

\sum \nolimits _{m = 1}^\infty {\sum \nolimits _{n = 1}^\infty {{{\left ( {\sum \nolimits _{j = m}^\infty {\sum \nolimits _{k = n}^\infty {a_{jk}^2} } } \right )}^{1/2}}} }

. Our result is an extension of the corresponding one by M. C. Weiss [4] from one-dimensional to two-dimensional lacunary sine series.
Projectivity of the Whitehead product in spheres
Authors: Lin Wên Hsiung 
Published: July 2010

The Whitehead square

[

l

2
i

−
1

,

l

2
i

−
1

]

∈

π

2

i
+
1

−
3

(

S

2
i

−
1

)

\left [ {{l_{{2^i} - 1}},{l_{{2^i} - 1}}} \right ] \in {\pi _{{2^{i + 1}} - 3}}({S^{{2^i} - 1}})

is shown to be projective for

i
≤
10

i \leq 10

.
Oscillations in neutral equations with periodic coefficients
Authors: Ladas G. , Philos Ch. G. , Sficas Y. G. 
Published: July 2010

We obtain a necessary and sufficient condition for the oscillation of all solutions of the neutral delay differential equation: (1)
\[

d

d
t

[
x
(
t
)
+
p
x
(
t
−
τ
)
]
+
Q
(
t
)
x
(
t
−
σ
)
=
0
,

\tfrac {d}{{dt}}[x(t) + px(t - \tau )] + Q(t)x(t - \sigma ) = 0,

\]
where

p
∈

R

,
Q
∈
C
[
[
0
,
∞
)
,

R

+

]
,
Q

p \in {\mathbf {R}},Q \in C[[0,\infty ),{{\mathbf {R}}^ + }],Q

is

ω
\omega

-periodic with

ω
>
0
,
Q
(
t
)
[
u
n
k
]
0

\omega > 0,Q(t)[unk]0

for

t
≧
0

t \geqq 0

, and there exist positive integers

n
1

{n_1}

and

n
2

{n_2}

such that

τ
=

n
1

ω

\tau = {n_1}\omega

and

σ
=

n
2

ω

\sigma = {n_2}\omega

. More precisely we show that every solution of (1) oscillates if and only if every solution of an associated neutral equation with constant coefficients oscillates.
Nonintegrability of superharmonic functions
Authors: Suzuki Noriaki 
Published: July 2010

In this article we prove the following: If

u
u

is a nonzero superharmonic function on a proper subdomain

D
D

in

R
n

{R^n}

, then
\[

∫
D

|

u
(
x
)

|

p

δ
D

(
x

)

n
p
−
n
−
2
p

d
x
=
∞
,

{\int _D {\left | {u(x)} \right |} ^p}{\delta _D}{(x)^{np - n - 2p}}dx = \infty ,

\]
where

0
>
p
≤
1

0 > p \leq 1

and

δ
D

(
x
)

{\delta _D}(x)

denotes the distance between

x
x

and the boundary of

D
D

.
“Lebesgue measure” on 𝑅^{∞}
Authors: Baker Richard 
Published: July 2010

We construct a translation invariant Borel measure

λ
\lambda

on

R

∞

=

∏

i
=
1

∞

R

{{\mathbf {R}}^\infty } = \prod _{i = 1}^\infty {\mathbf {R}}

such that for any infinite-dimensional rectangle

R
=

∏

i
=
1

∞

(

a
i

,

b
i

)
,
−
∞
>

a
i

≤

b
i

>
+
∞

R = \prod _{i = 1}^\infty ({a_i},{b_i}), - \infty > {a_i} \leq {b_i} > + \infty

, if

0
≤

∏

i
=
1

∞

(

b
i

−

a
i

)
>
+
∞

0 \leq \prod _{i = 1}^\infty ({b_i} - {a_i}) > + \infty

, then

λ
(
R
)
=

∏

i
=
1

∞

(

b
i

−

a
i

)

\lambda (R) = \prod _{i = 1}^\infty ({b_i} - {a_i})

. Because

R

∞

{{\mathbf {R}}^\infty }

is an infinite-dimensional locally convex topological vector space, the measure

λ
\lambda

can not be

σ
\sigma

-finite.
Homogeneous polynomials and invariant subspaces in the polydisc. II
Authors: Nakazi Takahiko , Takahashi Katsutoshi 
Published: July 2010

We determine the invariant subspaces

M
M

of

L
2

(

T
2

)

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

for which there is a subspace

S
S

of

M
M

and a positive integer

r
r

such that
\[

M
=

∑

n
=
0

∞

⊕

[

∑

j
=
0

n

z
j

w

r
(
n
−
j
)

S

]

,

M = \sum \limits _{n = 0}^\infty { \oplus \left [ {\sum \limits _{j = 0}^n {{z^j}{w^{r(n - j)}}S} } \right ]} ,

\]
where, for a subspace

A
A

of

L
2

,

T
2

,
[
A
]

{L^2},{T^2},[A]

denotes the closure of

A
A

.
On the weighted estimate of the solution associated with the Schrödinger equation
Authors: Wang Si Lei 
Published: July 2010

Let

u
(
x
,
t
)

u(x,t)

be the solution of the Schrödinger equation with initial data

f
f

in the Sobolev space

H

−
1
+
a

/

2

(

R

n

)

{H^{ - 1 + a/2}}({\mathbb {R}^n})

with

a
>
1

a > 1

. This paper shows that the weighted inequality

∫

R

n

∫

R

|

u
(
x
,
t
)

|

2

d
t

(
1
+

|
x
|

)

−
a

d
x
≤
C

‖
f
‖

H

−
1
+
a

/

2

(

R

n

)

\int _{{\mathbb {R}^n}} {\int _\mathbb {R} {{{\left | {u(x,t)} \right |}^2}dt{{(1 + \left | x \right |)}^{ - a}}dx \leq C{{\left \| f \right \|}_{{H^{ - 1 + a/2}}({\mathbb {R}^n})}}} }

is false. Another improved weighted inequality is proved for the general case.
On the integrability and 𝐿¹-convergence of complex trigonometric series
Authors: Móricz Ferenc 
Published: July 2010

We prove that if a weakly even sequence

{

c
k

:
k
=
0
,
±
1
,
…
}

\{ {c_k}:k = 0, \pm 1, \ldots \}

of complex numbers is such that for some

p
>
1

p > 1

we have
\[

∑

m
=
1

∞

2

m

/

q

(

∑

k
=

2

m
−
1

2
m

−
1

|

Δ

(

c
k

+

c

−
k

)

|

p

)

1

/

p

>
∞
,

1
p

+

1
q

=
1
,

\sum \limits _{m = 1}^\infty {{2^{m/q}}} {\left ( {\sum \limits _{k = {2^{m - 1}}}^{{2^m} - 1} {{{\left | {\Delta \left ( {{c_k} + {c_{ - k}}} \right )} \right |}^p}} } \right )^{1/p}} > \infty ,\frac {1}{p} + \frac {1}{q} = 1,

\]
then the symmetric partial sums of the trigonometric series

(
∗
)

∑

k
=
−
∞

∞

c
k

e

i
k
x

( * )\sum \nolimits _{k = - \infty }^\infty {{c_k}{e^{ikx}}}

converge pointwise, except possibly at

x
=
0
(
mod
⁡
2
π
)

x = 0(\operatorname {mod} 2\pi )

, to a Lebesgue integrable function,

(
∗
)

( * )

is the Fourier series of its sum, and series

(
∗
)

( * )

converges in

L
1

(
−
π
,
π
)

{L^1}( - \pi ,\pi )

-norm if and only if

lim

|

k

|

→
∞

c
k

ln
⁡

|

k

|

=
0

{\lim _{|k| \to \infty }}{c_k}\ln |k| = 0

. In addition, we present new proofs of the theorems by J. Fournier and W. Self [6] and by ČC. V. Stanojević and V. B. Stanojević [10].
A few remarks on Riesz summability of orthogonal series
Authors: Szabłowski PawełJ. 
Published: July 2010

We study convergence behavior of some sequences and series related to a given orthogonal series. Following the developed technique we define (in terms of fourth mixed moments only) a class of orthonormal functions

{

X
i

}

i
≥
1

{\left \{ {{X_i}} \right \}_{i \geq 1}}

such that the condition:

∃
k
∈

N

∑

i
≥
1

μ
i
2

(

ln

(
k
)

i

)

2

>
∞

\exists k \in \mathbb {N}\sum \nolimits _{i \geq 1} {\mu _i^2} {\left ( {{{\ln }^{\left ( k \right )}}i} \right )^2} > \infty

implies almost everywhere convergence of the series

∑

i
≥
1

μ
i

X
i

\sum \nolimits _{i \geq 1} {{\mu _i}{X_i}}

, here for every

i
=
1
,
2
,
…
,
j
=
1
,
…
,
k

i = 1,2, \ldots ,j = 1, \ldots ,k

,
\[

ln

(
1
)

i
=

ln
2

i
,

ln

(
j
)

i
=

ln
2

(
max
(
1
,

ln

(
j
−
1
)

i
)
)
.

{\ln ^{(1)}}i = {\ln _2}i,\quad {\ln ^{(j)}}i = {\ln _2}(\max (1,{\ln ^{(j - 1)}}i)).

\]
Commutator approximants
Authors: Maher P. J. 
Published: July 2010

This paper deals with minimizing

|

|

B
−
(
A
X
−
X
A
)

|

|

p

||B - (AX - XA)|{|_p}

, where

A
A

and

B
B

are fixed,

B
∈

C

p

B \in {\mathcal {C}_p}

, and

X
X

varies such that

A
X
−
X
A
∈

C

p

AX - XA \in {\mathcal {C}_p}

. (Here,

C

p

{\mathcal {C}_p}

denotes the von Neumann-Schatten class and

‖
‖

p

{\left \| \right \|_p}

denotes its norm.) The main result (Theorem 3.2) says that if

A
A

is normal and

A
B
=
B
A

AB = BA

then

|

|

B
−
(
A
X
−
X
A
)

|

|

p

,
1
≤
p
>
∞

||B - (AX - XA)|{|_p},1 \leq p > \infty

, is minimized if and for

1
>
p
>
∞

1 > p > \infty

only if,

A
X
−
X
A
=
0

AX - XA = 0

; and that the map

X
→

|

|

B
−
(
A
X
−
X
A
)

|

|

p
p

,
1
>
p
>
∞

X \to ||B - (AX - XA)||_p^p,1 > p > \infty

, has a critical point at

X
=
V

X = V

if and only if

A
V
−
V
A
=
0

AV - VA = 0

.
Imbedding of any vector field in a scalar semilinear parabolic equation
Authors: Poláčik P. 
Published: July 2010

The scalar semilinear parabolic equation
\[

u
t

=
Δ
u
+
f
(
x
,
u
,
∇
u
)
,

x
∈
Ω
,

t
>
0
,

{u_t} = \Delta u + f(x,u,\nabla u),\quad x \in \Omega ,\quad t > 0,

\]
on a smooth bounded convex domain

Ω
⊂

R

N

\Omega \subset {\mathbb {R}^N}

under Neumann boundary condition (2)
\[

∂
u

∂
[
u
n
k
]

=
0

on

∂
Ω

\quad \frac {{\partial u}}{{\partial [unk]}} = 0\quad {\text {on }}\partial \Omega

\]
is considered. For any prescribed vector field

H
H

on

R

N

{\mathbb {R}^N}

, a function

f
f

is found such that the flow of (1), (2) has an invariant

N
N

-dimensional subspace and the vector field generating the flow of (1), (2) on this invariant subspace coincides, in appropriate coordinates, with

H
H

.
A note on Krylov-Tso’s parabolic inequality
Authors: Escauriaza Luis 
Published: July 2010

We show that if

u
u

is a solution to

∑

i
,
j
=
1

n

a

i
j

(
x
,
t
)

D

i
j

u
(
x
,
t
)
−

D
t

u
(
x
,
t
)
=
ϕ
(
x
)

\sum \nolimits _{i,j = 1}^n {{a_{ij}}(x,t){D_{ij}}u(x,t) - {D_t}u(x,t) = \phi (x)}

on a cylinder

Ω
T

=
Ω
×
(
0
,
T
)

{\Omega _T} = \Omega \times (0,T)

, where

Ω
\Omega

is a bounded open set in

R

n

,
T
>
0

{{\mathbf {R}}^n},T > 0

, and

u
u

vanishes continuously on the parabolic boundary of

Ω
T

{\Omega _T}

. Then the maximum of

u
u

on the cylinder is bounded by a constant

C
C

depending on the ellipticity of the coefficient matrix

(

a

i
j

(
x
,
t
)
)

({a_{ij}}(x,t))

, the diameter of

Ω
\Omega

, and the dimension

n
n

times the

L
n

{L^n}

norm of

ϕ
\phi

in

Ω
\Omega

.
The Fourier-Bessel series representation of the pseudo-differential operator (-𝑥⁻¹𝐷)^{𝜈}
Authors: Singh O. P. , Pandey J. N. 
Published: July 2010

For a certain Fréchet spaceFFconsisting of complex-valuedC∞{C^\infty }functions defined onI=(0,∞)I = (0,\infty )and characterized by their asymptotic behaviour near the boundaries, we show that: (I) The pseudo-differential operator(−x−1D)ν,ν∈R,D=d/dx{( - {x^{ - 1}}D)^\nu },\nu \in \mathbb {R},D = d/dx, is an automorphism (in the topological sense) onFF; (II)(−x−1D)ν{( - {x^{ - 1}}D)^\nu }is almost an inverse of the Hankel transformhν{h_\nu }in the sense that\[hν∘(x−1D)ν(φ)=h0(φ),∀φ∈F,∀ν∈R;{h_\nu } \circ {({x^{ - 1}}D)^\nu }(\varphi ) = {h_0}(\varphi ),\quad \forall \varphi \in F,\quad \forall \nu \in \mathbb {R};\](III)(−x−1D)ν{( - {x^{ - 1}}D)^\nu }has a Fourier-Bessel series representation on a subspaceFb⊂F{F_b} \subset Fand also on its dualFb′{F’_b}.
Nonlinear 𝑝-Laplacian problems on unbounded domains
Authors: Yu Lao Sen 
Published: July 2010

We consider the

p
p

-Laplacian problem
\[

−
d
i
v
(
a
(
x
)

|

∇
u

|

p
−
2

∇
u
)
+
b
(
x
)

|

u

|

p
−
2

u
=
f
(
x
,
u
)
,

x
∈
Ω
,

u

|

∂
Ω

=
0
,

lim

|

x

|

→
∞

u
=
0
,

- div(a(x)|\nabla u{|^{p - 2}}\nabla u) + b(x)|u{|^{p - 2}}u = f(x,u),\quad x \in \Omega ,\quad u{|_{\partial \Omega }} = 0,\quad \lim \limits _{|x| \to \infty } u = 0,

\]
where

1
>
p
>
n
,
Ω
(
⊂

R
n

)

1 > p > n,\Omega ( \subset {R^n})

is an exterior domain. Under certain conditions, we show the existence of solutions for this problem via critical point theory.
On products of polynomials
Authors: Masser D. W. , Wolbert J. 
Published: July 2010

For a complex polynomial

P
P

in a single variable, let

H
(
P
)

H(P)

be the maximum of the absolute values of its coefficients. Given nonnegative integers

n
1

{n_1}

and

n
2

{n_2}

, it is well known that

μ
(

n
1

,

n
2

)
=
inf

H
(

P
1

P
2

)

/

(
H
(

P
1

)
H
(

P
2

)
)
>
0

\mu ({n_1},{n_2}) = \inf \,H({P_1}{P_2})/(H({P_1})H({P_2})) > 0

, where the infimum is taken over all such polynomials

P
1

{P_1}

and

P
2

{P_2}

of degrees

n
1

{n_1}

and

n
2

{n_2}

respectively. We determine here the exact values of

μ
(
1
,
n
)

\mu (1,n)

for every

n
n

.
Unital multiplications on a Hilbert space
Authors: Froelich John 
Published: July 2010

We provide a new proof of two theorems of Ingelstam that identify the Hilbert spaces with a unital multiplication satisfying

|

|

x
y

|

|

⩽

|

|

x

|

|

|

|

y

|

|

||xy|| \leqslant ||x||\,||y||

and

|

|

1

|

|

=
1

||1|| = 1

.
Distortion of sets by inner functions
Authors: Hamilton D. H. 
Published: July 2010

For any inner function

f
f

with

f
(
0
)
=
0

f(0) = 0

and any Borel set

E
⊂

D

E \subset {\mathbf {D}}

\[

M
α

(
z
∈

D

:
f
(
z
)
∈
E
)
⩾

M
α

(
E
)
,

0
>
α
⩽
1
,

{M_\alpha }(z \in {\mathbf {D}}:f(z) \in E) \geqslant {M_\alpha }(E),\qquad 0 > \alpha \leqslant 1,

\]
where

M
α

{M_\alpha }

denotes

α
\alpha

-dimensional Hausdorff measure. In the case that

0
>

M
α

(
E
)
>
∞

0 > {M_\alpha }(E) > \infty

we have equality only for rotations of the identity.
Growth conditions for thin sets in Vilenkin groups of bounded order
Authors: Grubb D. J. 
Published: July 2010

Let

G
G

be a Vilenkin group of bounded order and

H
n

{H_n}

a sequence of clopen subgroups of

G
G

forming a base at the identity. If

E
E

is a subset of

G
G

, let

N
n

(
E
)

{N_n}(E)

denote the number of cosets of

H
n

{H_n}

which intersect

E
E

. If
\[

lim
_

N
n

(
E
)

log
⁡
[
G
:

H
n

]

>
∞
,

\underline {\lim } \frac {{{N_n}(E)}} {{\log [G:{H_n}]}} > \infty ,

\]
then

E
E

is a U-set in the group

G
G

. It is also shown that for

G
G

satisfying a growth condition and

φ
(
n
)
→
∞

\varphi (n) \to \infty

, there is an M-set,

E
E

, with
\[

N
n

(
E
)
=
O
(
φ
(
n
)

log
⁡
[
G
:

H
n

]
)
.

{N_n}(E) = O(\varphi (n)\,\log [G:{H_n}]).

\]
Conjugation and excess in the Steenrod algebra
Authors: Silverman Judith H. 
Published: July 2010

In this paper we prove a formula involving the canonical antiautomorphism

χ
\chi

of the

mod

-

2

\bmod {\text {-}}2

Steenrod algebra

A

(
2
)

\mathcal {A}(2)

, namely,
\[

χ
(
S

q

2
j

(

2

i
+
1

−
1
)

S

q

2

j
−
1

(

2

i
+
1

−
1
)

⋯
S

q

(

2

i
+
1

−
1
)

)

=
S

q

2
i

(

2

j
+
1

−
1
)

S

q

2

i
−
1

(

2

j
+
1

−
1
)

⋯
S

q

(

2

j
+
1

−
1
)

,

\begin {array}{*{20}{c}} {\chi (S{q^{{2^j}({2^{i + 1}} - 1)}}S{q^{{2^{j - 1}}({2^{i + 1}} - 1)}} \cdots S{q^{({2^{i + 1}} - 1)}})} \\ {\qquad \qquad = S{q^{{2^i}({2^{j + 1}} - 1)}}S{q^{{2^{i - 1}}({2^{j + 1}} - 1)}} \cdots S{q^{({2^{j + 1}} - 1)}},} \\ \end {array}

\]
and discuss its implications for the study of the image of the

A

(
2
)

\mathcal {A}(2)

-action on

F

2

[

x
1

,
⋯
,

x
s

]

{\mathbb {F}_2}[{x_1}, \cdots ,{x_s}]

.
Weakly convergent sequence coefficient of product space
Authors: Zhang Guang Lu 
Published: July 2010

W. L. Bynum introduced the weakly convergent sequence coefficient

WCS
⁡
(
X
)

\operatorname {WCS} (X)

of the Banach space

X
X

as

WCS
⁡
(
X
)
=

sup

{
M
:

for each weakly convergent sequence

{

x
n

}

in

X
,

there is some

y
∈

co
¯

(
{

x
n

}
)

such that

M
⋅
lim
sup

|

|

x
n

−
y

|

|

⩽
A
(
{

x
n

}
)
}

\operatorname {WCS} (X) = {\text {sup}}\{ M:{\text {for each weakly convergent sequence}}\;\{ {x_n}\} \;{\text {in}}\;X,\;{\text {there is some }}y \in \overline {\operatorname {co} } (\{ {x_n}\} )\;{\text {such that }}M \cdot \lim \sup ||{x_n} - y|| \leqslant A(\{ {x_n}\} )\}

. We consider the weakly convergent sequence coefficient of the

l
p

{l_p}

-product space

Z
=
(

∏

i
=
1

n

X
i

)

l
p

Z = (\prod \nolimits _{i = 1}^n {{X_i}{)_{lp}}}

of the finite non-Schur space

X
1

,
…
,

X
n

{X_1}, \ldots ,{X_n}

and show that

WCS
⁡
(
Z
)
=
min
{
WCS
⁡
(

X
i

)
:
1
⩽
i
⩽
n
}

\operatorname {WCS} (Z) = \min \{ \operatorname {WCS} ({X_i}):1 \leqslant i \leqslant n\}

.
On the spectrum of the Neumann Laplacian of long-range horns: a note on the Davies-Simon theorem
Authors: Jakšić V. 
Published: July 2010

For a class of regions with cusps (e.g.,

Ω
=
{
(
x
,
y
)
:
x
>
1
,

|

y

|

>
exp
⁡
(
−

x
α

)
}
,

0
>
α
>
1
)

\Omega = \{ (x,y):x > 1,|y| > \exp ( - {x^\alpha })\} ,\;0 > \alpha > 1)

) we show that

σ

ac

(
−

Δ
N
Ω

)
=
[
0
,
∞
)

{\sigma _{\operatorname {ac} }}( - \Delta _N^\Omega ) = [0,\infty )

of uniform multiplicity one,

σ

sing

(
−

Δ
N
Ω

)
=
∅

{\sigma _{\operatorname {sing} }}( - \Delta _N^\Omega ) = \emptyset

, and

σ

pp

(
−

Δ
N
Ω

)

{\sigma _{\operatorname {pp} }}( - \Delta _N^\Omega )

consists of a discrete set of embedded eigenvalues of finite multiplicity.
A double Stirling formula
Authors: Quine J. R. , Song Richard R. 
Published: July 2010

We give an asymptotic expansion as

M
→
∞
,

N
→
∞

M \to \infty ,\;N \to \infty

of

∏

(
m
+
n
τ
)

2

\prod {{{(m + n\tau )}^2}}

, where the product is over

1
⩽

|

m

|

⩽
M
,

1
⩽

|

n

|

⩽
N
,
(
m
,
n
)
≠
(
0
,
0
)

1 \leqslant |m| \leqslant M,\;1 \leqslant |n| \leqslant N,(m,n) \ne (0,0)

. The formula is analogous to the classical Stirling expansion on

M
!

M!

. Of special interest is the constant term in the expansion, which involves the Dedekind eta function

η
(
z
)

\eta (z)

. Finding this constant is related to the Kronecker limit formula for the derivative at

0
0

of the zeta function

Z
(
s
)
=

∑

m
,
n

′

|

m
+
n
τ

|

−
2
s

Z(s) = \sum \nolimits _{m,n}’{|m + n\tau {|^{ - 2s}}}

. We consider instead the zeta function without absolute values.