bluuebunny/crossref_metadata_2025 huggingface dataset:
Poisson semigroups and singular integrals
Authors: Dahlberg Björn E. J.
Published: June 2010

Let

D
⊂

R

n

D \subset {{\mathbf {R}}^n}

be a Lipschitz domain and consider the bilinear form

∫
D

u

(

∂
v

/

∂
y

)

d
P

\int _D {u\left ( {\partial v/\partial y} \right )dP}

. We show that the form is bounded if

v
v

is harmonic with boundary values in

L
2

{L^2}

, if

u
u

is smooth with its nontangential maximal function in

L
2

{L^2}

and

∫
D

dist

{

P
,
∂
D

}

|

grad 

u

|

2

d
P
>
∞

\int _D {{\text {dist}}\left \{ {P,\partial D} \right \}{{\left | {{\text {grad }}u} \right |}^2}dP > \infty }

.
Tangential limits of starlike univalent functions
Authors: Twomey J. B.
Published: June 2010

Let

f
f

be starlike univalent in the unit disc, let

γ
>
1

\gamma > 1

and let

K
>
0

K > 0

. Then

f
(
z
)

f(z)

tends to a limit as

z
→

e

i
θ

z \to {e^{i\theta }}

inside
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet z colon StartAbsoluteValue e Superscript i theta Baseline minus z EndAbsoluteValue less-than-or-equal-to upper K left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue right-parenthesis Superscript 1 slash gamma Baseline EndSet">

{
z
:

|

e

i
θ

−
z

|

≤
K

(
1
−

|
z
|

)

1

/

γ

}

\{ z:\left | {{e^{i\theta }} - z} \right | \leq K{(1 - \left | z \right |)^{1/\gamma }}\}

for every

θ
\theta

in

[
0
,
2
π
]

[0,2\pi ]

. This result is sharp.
An isomorphism for the Grothendieck ring of a Hopf algebra order
Authors: Jensen Anna-Lise , Larson Richard G.
Published: June 2010

If

G
G

is a finite abelian group,

R
R

is a principal ideal domain with field of quotients an algebraic number field

K
K

which splits

G
G

, and if

A
A

is a Hopf algebra order in KG, then the Grothendieck ring of the category of finitely generated

A
A

-modules is isomorphic to the Grothendieck ring of the category of finitely generated RG-modules.
On the Clifford index of algebraic curves
Authors: Ballico Edoardo
Published: June 2010

Here we prove (over

C

{\mathbf {C}}

) that a general

(
e
+
2
)

(e + 2)

-gonal algebraic curve of genus

p
p

has no

g
d
r

g_d^r

with

d
≤
p
−
1
,
r
≥
2

d \leq p - 1,r \geq 2

and

d
−
2
r
≤
e

d - 2r \leq e

.
A new fixed point theorem on demicompact 1-set-contraction mappings
Authors: Li Guo Zhen
Published: June 2010

W. V. Petryshyn studied the fixed point theorem of demi-compact mappings. This paper gives a new fixed point theorem of demi-compact

1
1

-set-contraction mappings in the norm form.
A one-one selection theorem
Authors: Sarbadhikari H.
Published: June 2010

Let

X
X

,

Y
Y

be Polish spaces without isolated points and

B
⊆
X
×
Y

B \subseteq X \times Y

a Borel set such that

x
:

B
x

x:{B_x}

is nonmeager is comeager in

X
X

and

y
:

B
y

y:{B^y}

is nonmeager is comeager in

Y
Y

. There is a comeager Borel

E
⊆
X

E \subseteq X

, a comeager Borel

F
⊆
Y

F \subseteq Y

and a Borel isomorphism

f
f

from

E
E

onto

F
F

such that graph of

f
⊆
B

f \subseteq B

.
Extension of continuous functions into uniform spaces
Authors: Hernández Salvador
Published: June 2010

Let

X
X

be a dense subspace of a topological space

T
T

, let

Y
Y

be a uniformizable space, and let

f
:
X
→
Y

f:X \to Y

a continuous map. In this paper we study the problem of the existence of a continuous extension of

f
f

to the space

T
T

. Thus we generalize basic results of Taimanov, Engelking and Blefko-Mrówka on extension of continuous functions. As a consequence, if

D

\mathcal {D}

is a nest generated intersection ring on

X
X

, we obtain a necessary and sufficient condition for the existence of a continuous extension to

v
(
X
,

D

)

v (X,\mathcal {D})

, of a continuous function over

X
X

, when the range of the map is a uniformizable space, and we apply this to realcompact spaces. Finally, we suppose each point of

T
∖
X

T\backslash X

has a countable neighbourhood base, and we obtain a generalization of a theorem by Blair, herewith giving a solution to a question proposed by Blair.
Polynomials with no small prime values
Authors: McCurley Kevin S.
Published: June 2010

Let

f
(
x
)

f(x)

be a polynomial with integer coefficients, and let
\[

D
(
f
)
=

g

.c

.d

{
f
(
x
)
:
x
∈

Z

}
.

D(f) = {\text {g}}{\text {.c}}{\text {.d}}\{ f(x):x \in {\mathbf {Z}}\}.

\]
It was conjectured by Bouniakowsky in 1857 that if

f
(
x
)

f(x)

is nonconstant and irreducible over

Z

{\mathbf {Z}}

, then

|

f
(
x
)

|

/

D
(
f
)

|f(x)|/D(f)

is prime for infinitely many integers

x
x

. It is shown that there exist irreducible polynomials

f
(
x
)

f(x)

with

D
(
f
)
=
1

D(f) = 1

such that the smallest integer

x
x

for which

|

f
(
x
)

|

|f(x)|

is prime is large as a function of the degree of

f
f

and the size of the coefficients of

f
f

.
On perfect 𝐶*-algebras
Authors: Archbold R. J.
Published: June 2010

It is shown that injective

C
∗

{C^ * }

-algebras are perfect, as are certain maximal simple

C
∗

{C^ * }

-algebras. Some properties of the perfect

C
∗

{C^ * }

-algebra

A
c

{A_c}

arising from a UHF algebra

A
A

are obtained by considering a masa with the extension property.
A partition theorem for triples
Authors: Milner E. C. , Prikry K.
Published: June 2010

Consider a partition of triples of enumerable ordinals into two classes. We show that either for each natural number

k
k

, the first class contains all triples from a set of type

ω
+
k

\omega + k

, or the second class contains all triples of a four element set.
On stably extended projective modules over polynomial rings
Authors: Roitman Moshe
Published: June 2010

We prove here that if

A
A

is a commutative noetherian ring of Krull dimension

d
d

and of finite characteristic prime to

d
!

d!

, then stably extended projective

A
[

X
1

,
…
,

X
n

]

A[{X_1}, \ldots ,{X_n}]

-modules of rank

⩾
d

/

2
+
1

\geqslant d/2 + 1

are extended from

A
A

.
Compact derivations of nest algebras
Authors: Peligrad C.
Published: June 2010

In this paper we determine all the weakly compact derivations of a nest algebra. We also obtain necessary and sufficient conditions in order that a nest algebra admit compact derivations. Finally we prove that every compact derivation of a nest algebra

A

\mathcal {A}

is the norm limit of finite-rank derivations.
On a question of Faith in commutative endomorphism rings
Authors: Clark John
Published: June 2010

Given a commutative ring

R
R

, let

Q
(
R
)

Q(R)

denote its maximal ring of quotients and, for any ideal

I
I

of

R
R

, let

End
⁡
(
I
)

\operatorname {End} (I)

denote the ring of

R
R

-endomorphisms of

I
I

. It is known that if

Q
(
R
)

Q(R)

is a self-injective ring then

End
⁡
(
I
)

\operatorname {End} (I)

is commutative for each ideal

I
I

of

R
R

. Carl Faith has asked if the converse holds. It does if

R
R

is either Noetherian or has no nontrivial nilpotent elements but here we produce an example to show that it does not hold in general.
Oscillation theorems for second order linear differential equations with damping
Authors: Yan Ju Rang
Published: June 2010

In this paper, we present some criteria for the oscillation of the differential equation with damping
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis r left-parenthesis t right-parenthesis x prime left-parenthesis t right-parenthesis right-parenthesis prime plus p left-parenthesis t right-parenthesis x prime left-parenthesis t right-parenthesis plus q left-parenthesis t right-parenthesis x left-parenthesis t right-parenthesis equals 0 comma t element-of left-parenthesis t 0 comma normal infinity right-parenthesis comma">

(
r
(
t
)

x
′

(
t
)

)
′

+
p
(
t
)

x
′

(
t
)
+
q
(
t
)
x
(
t
)
=
0
,

t
∈
(

t
0

,
∞
)
,

(r(t)x’(t))’ + p(t)x’(t) + q(t)x(t) = 0,\quad t \in ({t_0},\infty ),

\]
where

p
(
t
)

p(t)

and

q
(
t
)

q(t)

are allowed to change sign on

[

t
0

,
∞
)

[{t_0},\infty )

, and

γ
(
t
)
>
0

\gamma (t) > 0

. One of our results is new even for the differential equations
\[

x
(
t
)
+
q
(
t
)
x
(
t
)
=
0
,

x(t) + q(t)x(t) = 0,

\]
and
\[

x
(
t
)
+
p
(
t
)

x
′

(
t
)
+
q
(
t
)
x
(
t
)
=
0.

x(t) + p(t)x’(t) + q(t)x(t) = 0.

\]
On square roots of the uniform distribution on compact groups
Authors: Diaconis Persi , Shahshahani Mehrdad
Published: June 2010

Let

G
G

be a compact separable topological group. When does there exist a probability

P
P

such that

P
∗
P
=
U

P * P = U

, where

U
U

is Haar measure and

P
≠
U

P \ne U

? We show that such square roots exist if and only if

G
G

is not abelian, nor the product of the quaternions and a product of two element groups. In the course of proving this we classify compact groups with the property that every closed subgroup is normal.
On free subalgebras of certain division algebras
Authors: Lorenz Martin
Published: June 2010

Using ideas of Makar-Limanov, we construct noncommutative free subalgebras in certain division algebras which arise as fields of fractions of skew polynomial rings. Our result applies in particular to the Weyl division algebra

D
1

{D_1}

.
A simple proof of the classification of rational rotation 𝐶*-algebras
Authors: Yin Hong Sheng
Published: June 2010

A simple proof of the classification of rational rotation

C
∗

{C^*}

-algebras is given.
On the Egoroff property of pointwise convergent sequences of functions
Authors: Blass Andreas , Jech Thomas
Published: June 2010

The space

L

(
x
)

\mathcal {L}(x)

of real-valued functions on

X
X

has the Egoroff property if for any

{

f

n
k

}

\{ {f_{nk}}\}

such that

0
⩽

f

n
k

↑
k

f

0 \leqslant {f_{nk}}{ \uparrow _k}f

(for every

n
n

), there exists

g
m

↑
f

{g_m} \uparrow f

such that, for each

m
m

and

n
n

,

g
m

⩽

n
k

{g_m}{ \leqslant _{nk}}

for some

k
k

. We show that

L

(
X
)

\mathcal {L}(X)

has the Egoroff property if and only if the cardinality of

X
X

is smaller than the minimum cardinality of an unbounded family of functions from the set of natural numbers to itself. Therefore, the statement that there is an uncountable set

X
X

such that

L

(
X
)

\mathcal {L}(X)

has the Egoroff property is independent of the axioms of set theory.
Local contractibility, cell-like maps, and dimension
Authors: van Mill Jan
Published: June 2010

We consider the existence of cell-like maps

f
:

f:

:

I
n

→
X

{I^n} \to X

such that no nonempty open subset of

X
X

is contractible in

X
X

. From the Taylor Example, it is easy to construct such a map for

n
=
∞

n = \infty

. We show that there exists such a map for some finite

n
n

if (and only if) there exists a dimension raising cell-like map of a compactum.
Some remarks on affine rings
Authors: Montgomery S. , Small L. W.
Published: June 2010

Various topics on affine rings are considered, such as the relationship between Gelfand-Kirillov dimension and Krull dimension, and when a "locally affine" algebra is affine. The dimension result is applied to study prime ideals in fixed rings of finite groups, and in identity components of group-graded rings.
On two theorems of Thompson
Authors: Zhang Guang Xiang
Published: June 2010

Let

G
G

be a finite group. Theorem. Let

P
∈

Syl
p

(
G
)

P \in {\operatorname {Syl} _p}(G)

with

Ω
1

(
P
)
≤
Z
(
P
)

{\Omega _1}(P) \leq Z(P)

. If

N
G

(
Z
(
P
)
)

{N_G}(Z(P))

has a normal

p
p

-complement, then so does

G
G

. Corollary. Let

M
M

be a nilpotent maximal subgroup of

G
G

and

P
∈

Syl
2

(
M
)

P \in {\operatorname {Syl} _2}(M)

with

Ω
2

(
P
)
≤
Z
(
P
)

{\Omega _2}(P) \leq Z(P)

. Then

G
G

is solvable. This extends Thompson’s solvability theorem [9]. We also give two other results generalizing Thompson’s theorem.
Three theorems on form rings
Authors: Ratliff Louis J.
Published: June 2010

Three theorems concerning the form ring (= associated graded ring)

F

(

R
,
I

)

{\mathbf {F}}\left ( {R,I} \right )

of an ideal

I
I

in a Noetherian ring

R
R

are proved. The first characterizes, for a

P
P

-primary ideal in a locally quasi-unmixed ring, when

F

(

R
,
I

)

red

{\mathbf {F}}{\left ( {R,I} \right )_{\operatorname {red}}}

is an integral domain in terms of when

F

(

R
P

,
I

R
P

)

red

{\mathbf {F}}{\left ( {{R_P},I{R_P}} \right )_{\operatorname {red}}}

is an integral domain. For an aribtrary Noetherian ring

R
R

the second gives a somewhat similar characterization for

F

(

R
,
J

)

{\mathbf {F}}\left ( {R,J} \right )

to have only one prime divisor of zero for some ideal

J
J

that is projectively equivalent to

I
I

. And the third characterizes unmixed semilocal rings in terms of the existence of an open ideal

I
I

such that the zero ideal in

F

(

R
,
I

)

{\mathbf {F}}\left ( {R,I} \right )

is isobathy.
Marx-Strohhäcker differential subordination systems
Authors: Miller Sanford S. , Mocanu Petru T.
Published: June 2010

Let

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

be analytic in the unit disc

U
U

and let

k
(
z
)
=
z

/

(
1
−
z
)

k(z) = z/(1 - z)

. The classic Marx-Strohhäcker result, that a convex (univalent) function

f
f

is starlike of order

1
2

\frac {1}{2}

, can be written in terms of differential subordinations as
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z f left-parenthesis z right-parenthesis slash f prime left-parenthesis z right-parenthesis precedes z k left-parenthesis z right-parenthesis slash k prime left-parenthesis z right-parenthesis right double arrow z f prime left-parenthesis z right-parenthesis slash f left-parenthesis z right-parenthesis precedes z k prime left-parenthesis z right-parenthesis slash k left-parenthesis z right-parenthesis period">

z
f
(
z
)

/

f
′

(
z
)
≺
z
k
(
z
)

/

k
′

(
z
)
⇒
z

f
′

(
z
)

/

f
(
z
)
≺
z

k
′

(
z
)

/

k
(
z
)
.

zf(z)/f’(z) \prec zk(z)/k’(z) \Rightarrow zf’(z)/f(z) \prec zk’(z)/k(z).

\]
The authors determine general conditions on

k
k

for which this relation holds. They also determine a different set of general conditions on

k
k

for which
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z f prime left-parenthesis z right-parenthesis slash f left-parenthesis z right-parenthesis precedes z k prime left-parenthesis z right-parenthesis slash k left-parenthesis z right-parenthesis right double arrow f left-parenthesis z right-parenthesis slash z precedes k left-parenthesis z right-parenthesis slash z period">

z

f
′

(
z
)

/

f
(
z
)
≺
z

k
′

(
z
)

/

k
(
z
)
⇒
f
(
z
)

/

z
≺
k
(
z
)

/

z
.

zf’(z)/f(z) \prec zk’(z)/k(z) \Rightarrow f(z)/z \prec k(z)/z.

\]
Finally, differential subordinations with starlike superordinate functions are considered.
Compact and weakly compact composition operators on spaces of vector valued continuous functions
Authors: Singh R. K. , Summers W. H.
Published: June 2010

We characterize the compact and weakly compact weighted composition operators on the Banach space

C
p

(
T
,
X
)

{C_p}(T,X)

of those continuous functions from a completely regular Hausdorff space

T
T

into a Banach space

X
X

which have relatively compact range.
On some spectral properties of elliptic pseudodifferential operators
Authors: Wong M. W.
Published: June 2010

We prove that the minimal and maximal operators associated with an elliptic pseudodifferential operator coincide in

L
p

(

R

n

)
,
1
>
p
>
∞

{L^p}({{\mathbf {R}}^n}),1 > p > \infty

. We obtain a set of necessary and sufficient conditions for a measurable function

q
q

on

R

n

{{\mathbf {R}}^n}

to be compact relative to some integral power of a constant coefficient elliptic pseudodifferential operator.
Note on compatible vector topologies
Authors: Kąkol Jerzy
Published: June 2010

Let

⟨

X
,
Y

⟩

\left \langle {X,Y} \right \rangle

be a dual pair. Then

X
X

admits the finest locally convex topology

μ
\mu

which is compatible with

⟨

X
,
Y

⟩

\left \langle {X,Y} \right \rangle

. In contrast, it is proved that there is no finest vector topology on

X
X

which is compatible with

⟨

X
,
Y

⟩

\left \langle {X,Y} \right \rangle

provided

X
X

contains a

μ
\mu

-dense subspace of infinite codimension.
Characterizing 𝑆^{𝑚} by the spectrum of the Laplacian on 2-forms
Authors: Goldberg S. I. , Gauchman H.
Published: June 2010

The Euclidean sphere

S

2
n
+
1

{S^{2n + 1}}

is characterized by the spectrum of the Laplacian on

2
2

-forms in all dimensions.
On the solvability of the equation ∑ⁿᵢ₌₁𝑥ᵢ/𝑑ᵢ≡0(𝑚𝑜𝑑1) and its application
Authors: Sun Qi , Wan Da Qing
Published: June 2010

In this paper, we obtain a necessary and sufficient condition under which the equation of the title is unsolvable. More precisely, for the equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction x 1 Over d 1 EndFraction plus StartFraction x 2 Over d 2 EndFraction plus midline-horizontal-ellipsis plus StartFraction x Subscript n Baseline Over d Subscript n Baseline EndFraction identical-to 0 left-parenthesis mod 1 right-parenthesis comma x Subscript i Baseline integral comma 1 less-than-or-equal-to x Subscript i Baseline greater-than d Subscript i Baseline left-parenthesis 1 less-than-or-equal-to i less-than-or-equal-to n right-parenthesis comma">

x
1

d
1

+

x
2

d
2

+
⋯
+

x
n

d
n

≡
0

(
mod
1
)
,

x
i

 integral

,

1

≤

x
i

>

d
i

(
1
≤
i
≤
n
)
,

\frac {{{x_1}}}{{{d_1}}} + \frac {{{x_2}}}{{{d_2}}} + \cdots + \frac {{{x_n}}}{{{d_n}}} \equiv 0\quad (\bmod 1),\quad {x_i}{\text { integral}},{\text {1}} \leq {x_i} > {d_i}(1 \leq i \leq n),

\]
where

d
1

,
…
,

d
n

{d_1}, \ldots ,{d_n}

are fixed positive integers, we prove the following result: The above equation is unsolvable if and only if 1. For some

d
i

,
(

d
i

,

d
1

d
2

⋯

d
n

/

d
i

)
=
1

{d_i},({d_i},{d_1}{d_2} \cdots {d_n}/{d_i}) = 1

, or 2. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d Subscript i 1 Baseline comma ellipsis comma d Subscript i Sub Subscript k Subscript Baseline left-parenthesis 1 less-than-or-equal-to i greater-than midline-horizontal-ellipsis greater-than i Subscript k Baseline less-than-or-equal-to n right-parenthesis">

d

i
1

,
…
,

d

i
k

(
1
≤
i
>
⋯
>

i
k

≤
n
)

{d_{{i_1}}}, \ldots ,{d_{{i_k}}}(1 \leq i > \cdots > {i_k} \leq n)

is the set of all even integers among

{

d
1

,
…
,

d
n

}

\left \{ {{d_1}, \ldots ,{d_n}} \right \}

, then

2
∤
k
,

d

i
1

/

2
,
…
,

d

i
k

/

2

2\nmid k,{d_{{i_1}}}/2, \ldots ,{d_{{i_k}}}/2

are pairwise prime, and

d

i
j

{d_{{i_j}}}

is prime to any odd number in

{

d
1

,
…
,

d
n

}
(
j
=
1
,
…
,
k
)

\{ {d_1}, \ldots ,{d_n}\} (j = 1, \ldots ,k)

.
Positive 𝑝-summing operators on 𝐿_{𝑝}-spaces
Authors: Blasco Oscar
Published: June 2010

It is shown that for any Banach space

B
B

every positive

p
p

-summing operator from

L

p
′

(
μ
)

{L^{p’}}(\mu )

in

B
B

,

1

/

p
+
1

/

p
′

=
1

1/p + 1/p’ = 1

, is also cone absolutely summing. We also prove here that a necessary and sufficient condition that

B
B

has the Radon-NikodṔm property is that every positive

p
p

-summing operator

T
:

L

p
′

(
μ
)
→
B

T:{L^{p’}}(\mu ) \to B

is representable by a function

f
f

in

L
p

(
μ
,
B
)

{L^p}(\mu ,B)

.
The Thurston norm and 2-handle addition
Authors: Scharlemann Martin
Published: June 2010

Suppose a

2
2

-handle is attached to a compact orientable

3
3

-manifold

M
M

along an annulus

A
A

contained in a subsurface

N
N

of

∂
M

\partial M

. If

N
N

is compressible in

M
M

, but

N
−
A

N - A

is not, then the Thurston norm is unaffected. This generalizes a series of results due to Przytycki, Jaco, and Johannson.
Interlacing polynomials
Authors: Johnson Charles R.
Published: June 2010

Let

A
A

be an

n
n

-by-

n
n

Hermitian matrix. We note that the set of all monic, degree

n
−
1

n - 1

polynomials whose roots interlace the eigenvalues of

A
A

is exactly the classical field of values of

adj
⁡
(
λ
I
−
A
)

\operatorname {adj}(\lambda I - A)

.
Isomorphisms of graph groups
Authors: Droms Carl
Published: June 2010

Given a graph

X
X

, define the presentation

P
X

PX

to have generators the vertices of

X
X

, and a relation

x
y
=
y
x

xy = yx

for each pair

x
,
y

x,y

of adjacent vertices. Let

G
X

GX

be the group with presentation

P
X

PX

, and given a field

K
K

, let

K
X

KX

denote the

K
K

-algebra with presentation

P
X

PX

. Given graphs

X
X

and

Y
Y

and a field

K
K

, it is known that the algebras

K
X

KX

and

K
Y

KY

are isomorphic if and only if the graphs

X
X

and

Y
Y

are isomorphic. In this paper, we use this fact to prove that if the groups

G
X

GX

and

G
Y

GY

are isomorphic, then so are the graphs

X
X

and

Y
Y

.
Level sets for functions convex in one direction
Authors: Brown Johnny E.
Published: June 2010

Goodman and Saff conjectured that ifffis convex in the direction of the imaginary axis then so are the functions1rf(rz)\frac {1}{r}f(rz)for all0>r>2−10 > r > \sqrt 2 - 1, i.e., the level setsf(|z|>r)f(|z|\, > r)are convex in the direction of the imaginary axis for0>r>2−10 > r > \sqrt 2 - 1. A weak form of this conjecture is proved and a question of Brannan is answered negatively.
Geodesics and deformed spheres
Authors: Faridi Abbas M. , Schucking E. L.
Published: June 2010

A simple class of deformed

2
2

-spheres is given whose intrinsic geometry and geodesies are described by elementary functions. Simple, explicit conditions for closed geodesies are obtained.
Boundary limits of Green potentials of general order
Authors: Mizuta Yoshihiro
Published: June 2010

Our aim in this note is to discuss the boundary limits of Green potentials in a half-space of the

n
n

-dimensional euclidean space. As a corollary, our result below includes the recent result of Stoll [5] concerning the boundary limits of superharmonic functions along curves tending to the boundary.
The locally finite topology on 2^{𝑋}
Authors: Beer G. A. , Himmelberg C. J. , Prikry K. , Van Vleck F. S.
Published: June 2010

Let

X
X

be a metrizable space. A Vietoris-type topology, called the locally finite topology, is defined on the hyperspace

2
X

{2^X}

of all closed, nonempty subsets of

X
X

. We show that the locally finite topology coincides with the supremum of all Hausdorff metric topologies corresponding to equivalent metrics on

X
X

. We also investigate when the locally finite topology coincides with the more usual topologies on

2
X

{2^X}

and when the locally finite topology is metrizable.
On positive solutions of semilinear elliptic equations
Authors: Dancer E. N. , Schmitt Klaus
Published: June 2010

This paper is concerned with necessary conditions for the existence of positive solutions of the semilinear problem

Δ
u
+
f
(
u
)
=
0
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω

\Delta u + f(u) = 0,x \in \Omega ,u = 0,x \in \partial \Omega

, whose supremum norm bears a certain relationship to zeros of the nonlinearity

f
f

. We first discuss the smooth case (i.e.,

f
f

and

∂
Ω

\partial \Omega

smooth) and then show how to obtain similar results in the nonsmooth case.
Symmetry and uniform approximation by analytic functions
Authors: Khavinson Dmitry
Published: June 2010

In this paper we treat the problem of finding all the domains in

C

{\mathbf {C}}

for which the uniform distance from the function

z
¯

\bar z

to the space of analytic functions is equal precisely to (2 area/perimeter). We show that for simply connected domains it occurs if and only if the domain is a disk. We also discuss the relation of the above problem to certain types of symmetry in potential theory and to the theory of Schwarz functions.
Some problems on splittings of groups. II
Authors: Szabó Sándor
Published: June 2010

If

G
G

is an additive abelian group,

S
S

is a subset of

G
,
M

G,M

is a set of nonzero integers, and if each element of

G
∖

{
0
}

G\backslash \left \{ 0 \right \}

is uniquely expressible in the form

m
s

ms

, where

m
∈
M

m \in M

and

s
∈
S

s \in S

, then we say that

M
M

splits

G
G

. A splitting is nonsingular if every element of

M
M

is relatively prime to the order of

G
G

; otherwise it is singular. In this paper we discuss the singular splittings of cyclic groups of prime power orders and the direct sum of isomorphic copies of groups.
On the zeros of polynomials of minimal 𝐿_{𝑝}-norm
Authors: Kroó András , Peherstorfer Franz
Published: June 2010

It is shown in this note that the zeros of the minimal polynomials in the

L
p

{L_p}

-norm interlace with those of the Chebyshev polynomials of the first and second kinds.
Embeddings of differential operator rings and Goldie dimension
Authors: Quinn Declan
Published: June 2010

The differential operator ring

S
=
R
[
x
;
δ
]

S = R[x;\delta ]

can be embedded in

A
1

(
R
)

{A_1}(R)

, the first Weyl algebra over

R
R

, where

R
R

is a

Q

{\mathbf {Q}}

-algebra and

δ
\delta

is a locally nilpotent derivation on

R
R

. Furthermore

A
1

(
R
)

{A_1}(R)

is again a differential operator ring over the image of

S
S

. We consider similar embeddings of the smash product

R
#
U
(
L
)

R\# U(L)

, where

L
L

is a finite dimensional Lie algebra and

U
(
L
)

U(L)

is its universal enveloping algebra. We show that the Weyl algebra over

R
R

has the same Goldie dimension as

R
R

itself and use this to prove that

R
R

and

R
[
x
;
δ
]

R[x;\delta ]

have the same Goldie dimension where

R
R

is again a

Q

{\mathbf {Q}}

-algebra and

δ
\delta

is locally nilpotent.
On large Zsigmondy primes
Authors: Feit Walter
Published: June 2010

If

a
a

and

m
m

are integers greater than 1, then a large Zsigmondy prime is a prime

l
l

such that

l

|

a
m

−
1
,
l
∤

a
i

−
1

l\left | {{a^m} - 1,l\nmid {a^i} - 1} \right .

for

1
⩽
i
⩽
m
−
1

1 \leqslant i \leqslant m - 1

and either

l
2

|

a
m

−
1

{l^2}\left | {{a^m} - 1} \right .

or

l
>
m
+
1

l > m + 1

. The main result of this paper lists all the pairs

(

a
,
m

)

\left ( {a,m} \right )

for which no large Zsigmondy prime exists.
Ergodic properties that lift to compact group extensions
Authors: Robinson E. Arthur
Published: June 2010

Let

T
T

and

R
R

be measure preserving,

T
T

weakly mixing,

R
R

ergodic, and let

S
S

be conservative ergodic and nonsingular. Let

T
~

\tilde T

be a weakly mixing compact abelian group extension of

T
T

. If

T
×
S

T \times S

is ergodic then

T
~

×
S

\tilde T \times S

is ergodic. A corollary is a new proof that if

T
T

is mildly mixing then so is

T
~

\tilde T

. A similar statement holds for other ergodic multiplier properties. Now let

T
~

\tilde T

be a weakly mixing type

α
\alpha

compact affine

G
G

extension of

T
T

where

α
\alpha

is an automorphism of

G
G

. If

T
T

and

R
R

are disjoint and

α
\alpha

or

R
R

has entropy zero, then

T
~

\tilde T

and

R
R

are disjoint.

T
~

\tilde T

is uniquely ergodic if and only if

T
T

is uniquely ergodic and

α
\alpha

has entropy zero. If

T
T

is mildly mixing and

T
~

\tilde T

is weakly mixing then

T
~

\tilde T

is mildly mixing. We also provide a new proof that if

T
~

\tilde T

is weakly mixing then

T
~

\tilde T

has the

K
K

-property if

T
T

does.
Hereditary 𝐶*-subalgebras of 𝐶*-crossed products
Authors: Kusuda Masaharu
Published: June 2010

Let

(
A
,
G
,
α
)

(A,G,\alpha )

be a

C
∗

{C^ * }

-dynamical system. Assume that

B
B

is an

α
\alpha

-invariant

C
∗

{C^ * }

- subalgebra of

A
A

. Then we shall give a necessary and sufficient condition for

B

×
α

G

B{ \times _\alpha }G

to be a

C
∗

{C^ * }

-subalgebra of

A

×
α

G

A{ \times _\alpha }G

, where

B

×
α

G

B{ \times _\alpha }G

(resp.

A

×
α

G

A{ \times _\alpha }G

) denotes a

C
∗

{C^ * }

-crossed product of

B
B

(resp.

A
A

) by a locally compact group

G
G

. Moreover, we shall show that if

B
B

is an

α
\alpha

-invariant hereditary

C
∗

{C^ * }

-subalgebra of

A
A

, then

B

×
α

G

B{ \times _\alpha }G

is a hereditary

C
∗

{C^ * }

-subalgebra of

A

×
α

G

A{ \times _\alpha }G

.
A note concerning the 3-manifolds which span certain surfaces in the 4-ball
Authors: Trace Bruce
Published: June 2010

We consider surfaces of the form

F

∪
K

D

F{ \cup _K}D

where

F
F

is a Seifert surface and

D
D

is a slicing disk for the knot

K
K

. We show that, in general, there is no

3
3

-manifold

M
M

which spans

F

∪
K

D

F{ \cup _K}D

in the

4
4

-ball such that

F
F

can be compressed to a disk in

M
M

.
A generalization of the Vietoris-Begle theorem
Authors: Dydak Jerzy , Kozlowski George
Published: June 2010

A theorem is proved which generalizes both the Vietoris-Begle theorem and the cell-like theorem for spaces of finite defomation dimension. The proof is geometric and uses a double mapping cylinder trick.
Symmetric functions: a bijective identity
Authors: Metropolis N. , Rota Gian-Carlo
Published: June 2010

We give a bijective proof of a classical identity which we have named the cyclotomic identity.
Elliptically embedded subgroups of polycyclic groups
Authors: Rhemtulla A. H. , Wilson J. S.
Published: June 2010

A subgroup

H
H

of a group

G
G

is elliptically embedded in

G
G

if for each subgroup

K
K

of

G
G

there is an integer

n
=
n
(
K
)

n = n(K)

such that

⟨

H
,
K

⟩

=
H
K
⋯
H
K

\left \langle {H,K} \right \rangle = HK \cdots HK

, where the product has

2
n

2n

factors. It is shown that a subgroup

H
H

of a polycyclic by finite group

G
G

is elliptically embedded in

G
G

if and only if

H
H

is subnormal in some subgroup of finite index in

G
G

.
Smooth convex 𝑡-norms do not exist
Authors: Alsina C. , Tomás M. S.
Published: June 2010

We show that smooth convextt-norms do not exist.
On extreme points of families described by subordination
Authors: Younis Rahman
Published: June 2010

Let

s

(
F
)

s\left ( F \right )

denote the set of analytic functions in

D
=
{
z
:

|

z

|

>
1
}

D = \{ z:|z| > 1\}

subordinate to an analytic function

F
F

. It is shown that if

F
F

is a polynomial then the extreme points of the closed convex hull of

s
(
F
)
⊂
{
F
∘
ϕ
:
ϕ
∈

extreme

points

of

B
(

H
0
∞

)
}

s(F) \subset \{ F \circ \phi :\phi \in {\text {extreme}}\;{\text {points}}\;{\text {of}}\;B(H_0^\infty )\}

. Also if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis z right-parenthesis equals left-parenthesis left-parenthesis z minus alpha right-parenthesis slash left-parenthesis 1 minus alpha overbar z right-parenthesis right-parenthesis Superscript n Baseline comma StartAbsoluteValue alpha EndAbsoluteValue greater-than 1">

F
(
z
)
=

(
(
z
−
α
)

/

(
1
−

α
¯

z
)

)
n

,

|

α

|

>
1

F(z) = {((z - \alpha )/(1 - \bar \alpha z))^n},|\alpha | > 1

and

n
n

is a positive integer then the extreme points of the closed convex hull of

s
(
F
)
=
{
F
∘
ϕ
:
ϕ
∈

extreme

points

of

B
(

H
0
∞

)
}

s(F) = \{ F \circ \phi :\phi \in {\text {extreme}}\;{\text {points}}\;{\text {of}}\;B(H_0^\infty )\}

. An analogue of Ryff’s theorem, and other results related to subordination in Bergman spaces have been obtained.
Tautness and locally finitely-valued Alexander-Spanier cochains
Authors: Deo Satya , Roy Atul N.
Published: June 2010

In this paper we prove that the following two subspaces are tautly embedded with respect to Alexander-Spanier cohomology based on locally finitely-valued cochains: (i) any neighborhood retract of arbitrary topological space, and (ii) any compact subset of a Hausdorff space.
A note on quadrics through an algebraic curve
Authors: Serrano Fernando
Published: June 2010

In this note we describe the intersection of all quadric hypersurfaces containing a given linearly normal smooth projective curve of genus

n
n

and degree

2
n
+
1

2n + 1

.
Approximation theorems and fixed point theorems in cones
Authors: Lin Tzu-Chu
Published: June 2010

In this paper, we investigate the validity of an interesting theorem of Fan [3, Theorem 2] in cones. We prove that it is true for a continuous condensing map defined on a closed ball in cones. A more interesting case is that we prove that it is true on an annulus if suitable inner boundary conditions are posed. As applications of our theorems, some new fixed point theorems in the norm form are derived.
Characterizations of denting points
Authors: Lin Bor-Luh , Lin Pei-Kee , Troyanski S. L.
Published: June 2010

Let

x
x

be a PC (point of continuity) for a bounded closed convex set

K
K

of a Banach space. Then

x
x

is a denting point of

K
K

if and only if

x
x

is an extreme point (resp. strong extreme point; weak*-extreme point) of

K
K

. A new definition for denting point is also given.
On midconvex functions with midconcave bounds
Authors: Ng C. T.
Published: June 2010

A representation theorem for midconvex functions with midconcave upper bounds is obtained. It solves a problem posed by K. Nikodem.
𝐻^{𝑝} joint approximation
Authors: Pérez-González F.
Published: June 2010

We show that Mergelyan sets and Farrell sets for

H
p

(

D

)

(

1
≤
p
>
∞

)

{H^p}\left ( {\mathbf {D}} \right )\left ( {1 \leq p > \infty } \right )

are just the same if either the weak topology or the norm topology is considered. This answers a question posed by Rubel and Stray in [3].
On the cardinality of a topology
Authors: Yongbin Ruan
Published: June 2010

Let

o

(
X
)

o\left ( X \right )

denote the cardinality of topology of a space

X
X

. I. Juhasz proves that

o

(
X
)

ω

=
o

(
X
)

o{\left ( X \right )^\omega } = o\left ( X \right )

for regular hereditarily paracompact spaces. We prove it for more general classes of spaces.
Finite type cones shaped on spherical submanifolds
Authors: Garay Oscar J.
Published: June 2010

The only finite type cones shaped on spherical submanifolds are minimal ones.
Rees algebras and mixed multiplicities
Authors: Verma J. K.
Published: June 2010

Let

(
R
,
m
)

(R,m)

be a local ring of positive dimension

d
d

and

I
I

and

J
J

two

m
m

-primary ideals of

R
R

. Let

T
T

denote the Rees algebra

R
[
J
t
]

R[Jt]

localized at the maximal homogeneous ideal

(
m
,
J
t
)

(m,Jt)

. It is proved that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e left-parenthesis left-parenthesis upper I comma upper J t right-parenthesis upper T equals e 0 left-parenthesis upper I vertical-bar upper J right-parenthesis plus e 1 left-parenthesis upper I vertical-bar upper J right-parenthesis plus midline-horizontal-ellipsis plus e Subscript d minus 1 Baseline left-parenthesis upper I vertical-bar upper J right-parenthesis comma">

e
(
(
I
,
J
t
)
T
=

e
0

(
I

|

J
)
+

e
1

(
I

|

J
)
+
⋯
+

e

d
−
1

(
I

|

J
)
,

e((I,Jt)T = {e_0}(I|J) + {e_1}(I|J) + \cdots + {e_{d - 1}}(I|J),

\]
where

e
i

(
I

|

J
)
,
i
=
0
,
1
,
…
,
d
−
1

{e_i}(I|J),i = 0,1, \ldots ,d - 1

are the first

d
d

mixed multiplicities of

I
I

and

J
J

. A formula due to Huneke and Sally concerning the multiplicity of the Rees algebra (of a complete zero-dimensional ideal of a two dimensional regular local ring) at its maximal homogeneous ideal is recovered.
Semiconfluent maps and continua containing no 𝑛-ods
Authors: Vought Eldon J.
Published: June 2010

J. F. Davis proved that the semiconfluent image of an atriodic continuum is atriodic and asked if this result could be generalized to

n
n

-ods. In this paper the question is answered in the affirmative. It is proved that the semiconfluent image of a Hausdorff continuum containing no

n
n

-ods must contain no

n
n

-ods
An example of nonhomotopic solutions to the Dirichlet problem for harmonic maps in two dimensions
Authors: Oswald Luc , Paulik George
Published: June 2010

Dirichlet problems of the harmonic map system from the disk into the sphere are presented which have multiple nonhomotopic solutions. In particular, it is shown that for any natural number

k
k

there is a Dirichlet problem which has at least

k
+
1

k + 1

nonhomotopic solutions.
On complementation of vector-valued Hardy spaces
Authors: Hensgen Wolfgang
Published: June 2010

Let

X
X

be a complex Banach space and

1
>
p
>
∞

1 > p > \infty

.

H
p

(
X
)

{H^p}(X)

resp.

h
p

(
X
)

{h^p}(X)

denote the Hardy spaces of

X
X

-valued analytic resp. harmonic functions on the disc.

L
p

(
X
)

{L^p}(X)

is the Lebesgue-Bochner space of

X
X

-valued integrable functions on the circle and

H

p

(
X
)

{{\mathbf {H}}^p}(X)

its Hardy-type subspace

{
f
∈

L
p

(
X
)
:

f
^

(
n
)
=
0
∀
n
>
0
}

\{ f \in {L^p}(X):\hat f(n) = 0\forall n > 0\}

. It is proved that the following four conditions are equivalent:

H
p

(
X
)

{H^p}(X)

is complemented in

h
p

(
X
)

{h^p}(X)

; the canonical analytic (or Riesz) projection is a bounded operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript p Baseline left-parenthesis upper X right-parenthesis right-arrow upper H Superscript p Baseline left-parenthesis upper X right-parenthesis semicolon bold upper H Superscript p Baseline left-parenthesis upper X right-parenthesis">

h
p

(
X
)
→

H
p

(
X
)
;

H

p

(
X
)

{h^p}(X) \to {H^p}(X);{{\mathbf {H}}^p}(X)

is complemented in

L
p

(
X
)

{L^p}(X)

; analytic projection is a bounded operator

L
p

(
X
)
→

H

p

(
X
)

{L^p}(X) \to {{\mathbf {H}}^p}(X)

. It is well known that the last condition, in turn, is equivalent to the UMD property of

X
X

.
Metacyclic 𝑝-algebras
Authors: Kang Ming Chang
Published: June 2010

Let

K
K

be a field of char

K
=
p
>
0
,
m
,
r

K = p > 0,m,r

any positive integers with

(
p
,
m
)
=
1

(p,m) = 1

, and

L
L

a metacyclic extension of

K
K

with degree

p
r

m

{p^r}m

, i.e.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Gal left-parenthesis upper L slash upper K right-parenthesis equals mathematical left-angle sigma comma tau colon sigma Superscript p Super Superscript r Superscript Baseline equals tau Superscript m Baseline equals 1 comma tau sigma tau Superscript negative 1 Baseline equals sigma Superscript e Baseline mathematical right-angle">

Gal(

L

/

K
)
=

⟨

σ
,
τ
:

σ

p
r

=

τ
m

=
1
,
τ
σ

τ

−
1

=

σ
e

⟩

{\text {Gal(}}L/K) = \left \langle {\sigma ,\tau :{\sigma ^{{p^r}}} = {\tau ^m} = 1,\tau \sigma {\tau ^{ - 1}} = {\sigma ^e}} \right \rangle

for some integer

e
e

. If

A
A

is a central simple

K
K

-algebra of degree

p
r

{p^r}

and is split by

L
L

, then

A
A

is a cyclic algebra. For

r
=
1

r = 1

, the theorem has been proved by A. A. Albert [1].
Some properties of measures of noncompactness in paranormed spaces
Authors: Hadžić Olga
Published: June 2010

This paper presents new properties of important measures of noncompactness in paranormed spaces. Using these properties some fixed point theorems for multivalued mappings in general topological vector spaces are obtained in a straightforward way.
A fixed point theorem revisited
Authors: Bollenbacher Alberta , Hicks T. L.
Published: June 2010

A version of a theorem commonly referred to as Caristi’s Theorem is given. It has an elementary constructive proof and it includes many generalizations of Banach’s fixed point theorem. Several examples illustrate the diversity that can occur.
Convolution operators on groups and multiplier theorems for Hermite and Laguerre expansions
Authors: Długosz Jolanta
Published: June 2010

Using harmonic analysis on nilpotent Lie groups the following theorem is proved. Let a sequence

{

a

n

}

\{ {a_{\text {n}}}\}

be defined by a function

K
∈

C
N

(

R

+

)

K \in {C^N}({{\mathbf {R}}_ + })

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript lamda greater-than 0 Endscripts StartAbsoluteValue upper K Superscript left-parenthesis j right-parenthesis Baseline left-parenthesis lamda right-parenthesis lamda Superscript j Baseline EndAbsoluteValue greater-than normal infinity comma j equals 0 comma 1 comma ellipsis comma upper N">

sup

λ
>
0

|

K

(
j
)

(
λ
)

λ
j

|

>
∞
,
j
=
0
,
1
,
…
,
N

{\sup _{\lambda > 0}}|{K^{(j)}}(\lambda ){\lambda ^j}| > \infty ,j = 0,1, \ldots ,N

, for

N
N

sufficiently large, putting

a

n

=
K
(

|

n

|

+
m

/

2
)

{a_{\text {n}}} = K(|{\text {n}}| + m/2)

. Let

φ

n

{\varphi _{\text {n}}}

be either Hermite or Laguerre functions. Then the operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n Endscripts left-parenthesis f comma phi Subscript n Baseline right-parenthesis phi Subscript n Baseline right-arrow sigma-summation Underscript n Endscripts a Subscript n Baseline left-parenthesis f comma phi Subscript n Baseline right-parenthesis phi Subscript n">

∑

n

(
f
,

φ

n

)

φ

n

→

∑

n

a

n

(
f
,

φ

n

)

φ

n

\sum \limits _{\text {n}} {(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}} \to \sum \limits _{\text {n}} {{a_{\text {n}}}(f,{\varphi _{\text {n}}}){\varphi _{\text {n}}}} }

\]
is bounded on

L
p

(

R

m

)

{L^p}\left ( {{\mathbb {R}^m}} \right )

or

L
p

(

R

+
m

)

{L^p}(\mathbb {R}_ + ^m)

respectively,

1
>
p
>
∞

1 > p > \infty

.
Analytic families of operators on some quasi-Banach spaces
Authors: Cwikel Michael , Sagher Yoram
Published: June 2010

An interpolation theorem for analytic families of operators on some quasi-Banach spaces is proved. The result is applicable to spaces whose quasi-norm is defined by means of a maximal function, for example the various

H
p

{H_p}

spaces on locally compact groups.
Trigonometric approximation and uniform distribution modulo one
Authors: Cochrane Todd
Published: June 2010

We construct

n
n

-dimensional versions of the Beurling and Selberg majorizing and minorizing functions and use them to prove results on trigonometric approximation and to prove an

n
n

-dimensional version of the Erdös-Turán inequality. Finally, an application is given to counting solutions of polynomial congruences.
Conditional asymptotic formulae for a class of arithmetic functions
Authors: Nowak Werner Georg , Schmeier Michael
Published: June 2010

Under the assumption of Riemann’s Hypothesis, a general asymptotic formula for sums

∑

n
≤
x

a

(
n
)

\sum \nolimits _{n \leq x} {a\left ( n \right )}

is established which applies (e.g.) to arithmetic functions

a

(
n
)

a\left ( n \right )

defined by a kind of convolution of the Möbius function with some divisor function.
von Neumann regularity of 𝑉-rings with Artinian primitive factor rings
Authors: Baccella Giuseppe
Published: June 2010

LetRRbe a ring whose right primitive factor rings are artinian. It is a known result that ifRRis von Neumann regular, thenRRis a rightVV-ring, that is, all simple rightRR-modules are injective. In the present work we prove that the converse holds.
Subnormal composition operators
Authors: Lambert Alan
Published: June 2010

Let

C
C

be the composition operator on

L
2

(
X
,
Σ
,
m
)

{L^2}(X,\Sigma ,m)

given by

C
f
=
f
∘
T

Cf = f \circ T

, where

T
T

is a

Σ
\Sigma

-measurable transformation from

X
X

onto

X
X

and

T

−
1

/

d
m

{T^{ - 1}}/dm

is strictly positive and bounded. It is shown that

C
C

is a subnormal operator if and only if the sequence

d
m
∘

T

−
n

/

d
m

dm \circ {T^{ - n}}/dm

is a moment sequence for almost every point in

X
X

. Several examples of subnormal composition operators are included.
Strong convergence of resolvents of monotone operators in Banach spaces
Authors: Kido Kazuo
Published: June 2010

Let

E
∗

{E^*}

be a real strictly convex dual Banach space with a Fréchet differentiable norm, and

A
A

a maximal monotone operator from

E
E

into

E
∗

{E^*}

such that

A

−
1

0
≠
∅

{A^{ - 1}}0 \ne \emptyset

. Fix

x
∈
E

x \in E

. Then

J
λ

x

{J_\lambda }x

converges strongly to

P
x

Px

as

λ
→
∞

\lambda \to \infty

, where

J
λ

{J_\lambda }

is the resolvent of

A
A

, and

P
P

is the nearest point mapping from

E
E

onto

A

−
1

0

{A^{ - 1}}0

.
Hyponormality of Toeplitz operators
Authors: Cowen Carl C.
Published: June 2010

For

φ
\varphi

in

L
∞

(
∂

D

)

{L^\infty }(\partial {\mathbf {D}})

, let

φ
=
f
+

g
¯

\varphi = f + \bar g

where

f
f

and

g
g

are in

H
2

{H^2}

. In this note, it is shown that the Toeplitz operator

T
φ

T\varphi

is hyponormal if and only if

g
=
c
+

T

h
¯

f

g = c + {T_{\bar h}}f

for some constant

c
c

and some function

h
h

in

H
∞

(

D

)

{H^\infty }({\mathbf {D}})

with

‖
h
‖

∞

≤
1

{\left \| h \right \|_\infty } \leq 1

.
Dual complementors on Banach algebras
Authors: Tomiuk B. J.
Published: June 2010

We study semisimple annihilator Banach algebras

A
A

with a right complementor

p
p

such that the mapping

q
q

on the closed left ideals

J
J

of

A
A

given by

J
q

=

l
A

(

[

r
A

(
J
)

]
p

)

{J^q} = {l_A}({[{r_A}(J)]^p})

is a left complementor on

A
A

. A right complementor

p
p

with this property is called dual, and the pair

(
p
,
q
)

(p,q)

is called a dual pair of complementors on

A
A

.
The space of cross sections of a bundle
Authors: Sakai Katsuro
Published: June 2010

Let

B
B

be a nondiscrete compactum,

Y
Y

a separable complete metrizable ANR with no isolated point and

p
:
X
→
B

p:X \to B

a locally trivial bundle with fiber

Y
Y

admitting a section. It is proved that the space

Γ
(
X
)

\Gamma (X)

of all cross sections of

p
:
X
→
B

p:X \to B

is an

l
2

{l_2}

-manifold.
Continuous images of arcs
Authors: Nikiel Jacek
Published: June 2010

A characterization of continuous images of (Hausdorff) arcs is used to describe their cyclic elements as inverse limits of inverse systems of ’nice’ spaces with ’nice’ bonding mappings.
Invariant Lagrangian subspaces
Authors: Andersson Lars
Published: June 2010

It is proved that on Hilbert spaces with strong symplectic form, every symplectic operatorI+CI + CwithCCcompact has an invariant Lagrangian subspace.
A commutator estimate for pseudo-differential operators
Authors: Marschall Jürgen
Published: June 2010

For the commutator

B
⋅
A
−

Op

(
b
a
)

B \cdot A - {\text {Op}}(ba)

of two pseudo-differential operators

A
A

and

B
B

an estimate on weighted Sobolev spaces is proved under minimal regularity assumptions on the symbols

a
a

and

b
b

.
Gaps between prime numbers
Authors: Hildebrand Adolf , Maier Helmut
Published: June 2010

Letdn=pn+1−pn{d_n} = {p_{n + 1}} - {p_n}denote thennth gap in the sequence of primes. We show that for every fixed integerkkand sufficiently largeTTthe set of limit points of the sequence{(dn/log⁡n,…,dn+k−1/log⁡n)}\{ ({d_n}/\log n, \ldots ,{d_{n + k - 1}}/\log n)\}in the cube[0,T]k{[0,T]^k}has Lebesgue measure≥c(k)Tk\geq c(k){T^k}, wherec(k)c(k)is a positive constant depending only onkk. This generalizes a result of Ricci and answers a question of Erdös, who had asked to prove that the sequence{dn/log⁡n}\{ {d_n}/\log n\}has a finite limit point greater than 1.
Elliptic curves with no rational points
Authors: Nakagawa Jin , Horie Kuniaki
Published: June 2010

The existence of infinitely many elliptic curves with no rational points except the origin

∞
\infty

is proved by refining a theorem of Davenport-Heilbronn. The existence of infinitely many quadratic fields with the Iwasawa invariant

λ
3

=
0

{\lambda _3} = 0

is proved at the same time.
The dominated ergodic estimate for mean bounded, invertible, positive operators
Authors: Martín-Reyes F. J. , De la Torre A.
Published: June 2010

We characterize those positive linear operators with positive inverse for which the dominated ergodic estimate holds. We also prove that for such operators one has mean and a.e. convergence.
Exponential dichotomies and Fredholm operators
Authors: Palmer Kenneth J.
Published: June 2010

It is shown that if the operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L x right-parenthesis left-parenthesis t right-parenthesis equals ModifyingAbove x With dot left-parenthesis t right-parenthesis minus upper A left-parenthesis t right-parenthesis x left-parenthesis t right-parenthesis">

(

L
x

)

(
t
)

=

x
˙

(
t
)

−
A

(
t
)

x

(
t
)

\left ( {Lx} \right )\left ( t \right ) = \dot x\left ( t \right ) - A\left ( t \right )x\left ( t \right )

is semi-Fredholm, then the differential equation

x
˙

=
A

(
t
)

x

\dot x = A\left ( t \right )x

has an exponential dichotomy on both

[
0
,
∞
)

[0,\infty )

and

(
−
∞
,
0
]

( - \infty ,0]

. This gives a converse to an earlier result.