bluuebunny/crossref_metadata_2025 huggingface dataset:
Norms of Hankel operators on a bidisc
Authors: Nakazi Takahiko
Published: June 2010

In the Hardy space on the bidisc

T
2

{T^2}

, if

ϕ
\phi

is a bounded function in the Lebesgue space and if its Fourier series vanishes on half of

Z

2

{{\mathbf {Z}}^2}

, then the norm of the Hankel operator

H
ϕ

{H_\phi }

is equal to the quotient norm of

ϕ
\phi

by the Hardy space

H
∞

(

T
2

)

{H^\infty }({T^2})

.
Properties on the unit circle of polynomials with unimodular coefficients
Authors: Newman Donald J. , Giroux André
Published: June 2010

A concrete explicit construction of a unimodular polynomial with prescribed zeros on the unit circle is given. More precisely a polynomial

P
(
z
)
=

a
0

+

a
1

z
+
⋯

a
N

z
N

P(z) = {a_0} + {a_1}z + \cdots {a_N}{z^N}

is produced for which

|

a
i

|

=
1

|{a_i}| = 1

for all

i
=
0
,
1
,
…
,
N

i = 0,1, \ldots ,N

and for which

P
(

α
j

)
=
0

P({\alpha _j}) = 0

for a given set of

α
j

,
j
=
1
,
2
,
…
,
n
,

|

α
j

|

=
1

{\alpha _j},j = 1,2, \ldots ,n,|{\alpha _j}| = 1

, and

P
(
z
)
≠
0

P(z) \ne 0

elsewhere on

|

z

|

=
1

|z| = 1

. It is further shown how to extend this construction so as to maintain these properties and force the maximum of

|

P
(
z
)

|

|P(z)|

to occur at any given number

β
≠

α
j

,
j
=
1
,
2
,
…
,
n

\beta \ne {\alpha _j},j = 1,2, \ldots ,n

and

|

β

|

=
1

|\beta | = 1

. The dependence of

N
N

on

n
n

is exponential, but there is rėason to believe that this is actually necessary and not just a weakness of the method.
A linearization of the circular maximal operator
Authors: Oberlin Daniel M.
Published: June 2010

An interesting linearization of the circular maximal operator is of restricted weak type

(
2
,
2
)

(2,2)

.
Inductive limits of normed algebras and 𝑚-convex structures
Authors: Oudadess M.
Published: June 2010

In this note, we give a negative answer to a question raised by A. Arosio in 1974. It concerns the multiplicativity of the inductive limit topology associated to a family of normed algebras.
The Hausdorff dimension of the graphs of continuous self-affine functions
Authors: Urbański M.
Published: June 2010

The exact formula for the Hausdorff dimension of the graph of a continuous self-affine function is obtained. The Hausdorff dimension of some class of Borel probability measures is computed. The Hausdorff measures corresponding to the functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript c Baseline left-parenthesis t right-parenthesis equals t Superscript upper H upper D left-parenthesis graph left-parenthesis f right-parenthesis right-parenthesis Baseline exp left-parenthesis c StartRoot log 1 slash t log log log 1 slash t EndRoot">

φ
c

(
t
)
=

t

H
D
(

graph

(
f
)
)

exp
⁡
(
c

log
⁡
1

/

t
log
⁡
log
⁡
log
⁡
1

/

t

{\varphi _c}(t) = {t^{HD({\text {graph}}(f))}}\exp (c\sqrt {\log 1/t\log \log \log 1/t}

are studied.
An oscillation theorem for second order sublinear differential equations
Authors: Wong James S. W.
Published: June 2010

An oscillation criterion is given for the second order sublinear differential equation

x
+
a
(
t
)

|
x
|

γ

sgn
⁡
x
=
0
,
0
>
γ
>
1

x + a(t){\left | x \right |^\gamma }\operatorname {sgn} x = 0,0 > \gamma > 1

, where the coefficient

a
(
t
)

a(t)

is not assumed to be nonnegative for all large values of

t
t

. The result extends a condition recently discovered by Butler, Erbe, and Mingarelli for the linear equation.
The Matheron representation theorem for gray-scale morphological filters
Authors: Crombez G.
Published: June 2010

We present a new proof of the Matheron representation theorem for gray-scale morphological filters, without using either the representation theorem for subsets of the plane or the umbra transform.
A short proof of Rohlin’s theorem for complex surfaces
Authors: Arapura Donu
Published: June 2010

We prove that the signature of a two dimensional compact complex spin manifold is divisible by 16.
The 𝐶*-algebras of Reeb foliations are not AF-embeddable
Authors: Fack Thierry , Wang Xiaolu
Published: June 2010

It is shown that the

C
∗

{C^ * }

-algebras of Reeb foliations cannot be embedded into AF-algebras.
A note on the number of primes in short intervals
Authors: Goldston D. A. , Gonek S. M.
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis beta comma upper T right-parenthesis equals integral Subscript 1 Superscript upper T Superscript beta Baseline Baseline left-parenthesis sigma-summation Underscript x greater-than p Superscript k Baseline less-than-or-equal-to x plus x slash upper T Endscripts log p minus x slash upper T right-parenthesis squared d x slash x squared">

J

(

β
,
T

)

=

∫
1

T
β

(

∑

x
>

p
k

≤
x
+
x

/

T

log
⁡
p
−
x

/

T

)

2

d
x

/

x
2

J\left ( {\beta ,T} \right ) = \int _1^{{T^\beta }} {{{\left ( {\sum \nolimits _{x > {p^k} \leq x + x/T} {\log p - x/T} } \right )}^2}dx/{x^2}}

, where the sum is over prime powers. H. L. Montgomery has shown that on the Riemann hypothesis, there is a positive constant

C
0

{C_0}

such that for each

β
≥
1
,
J

(

β
,
T

)

≤

C
0

β

log
2

T

/

T

\beta \geq 1,J\left ( {\beta ,T} \right ) \leq {C_0}\beta {\log ^2}T/T

, provided that

T
T

is sufficiently large. Here we prove a slightly stronger result from which we deduce a lower bound of the same order.
On the smoothed Toeplitz extensions and 𝐾-theory
Authors: Ji Ronghui
Published: June 2010

We construct a smoothed version of the Toeplitz extension for a

C
∗

{C^ * }

-dynamical system

(
A
,
α
,

R

)

(A,\alpha ,{\mathbf {R}})

. It defines a Thom class

[
τ
]

[\tau ]

in

K

K
1

(
A
,
A

×
α

R

)

K{K^1}(A,A{ \times _\alpha }{\mathbf {R}})

. The Wiener-Hopf-Rieffel extension for

(
A
,
α
,

R

)

(A,\alpha ,{\mathbf {R}})

is shown to be a smoothed Toeplitz extension which is the

K
K

KK

-inverse of

[
τ
]

[\tau ]

.
Seifert fibered surgery manifolds of composite knots
Authors: Kalliongis John , Tsau Chichen M.
Published: June 2010

A classification is given for the composite knots and the Dehn surgery on these knots which yield Seifert fibered surgery manifolds. We prove that if a knot

K
K

is the composition of two torus knots, then some (unique) integral surgery on

K
K

yields a Seifert fibered manifold, and conversely if the surgery manifold of a composite knot

K
K

is Seifert fibered, then

K
K

is the composition of two torus knots and the surgery must be integral surgery, which is uniquely determined.
Michael selection theorem under weak lower semicontinuity assumption
Authors: Przesławski Krzysztof , Rybiński Longin E.
Published: June 2010

We give a continuous selection theorem for convex-valued multifunction satisfying slightly weaker lower semicontinuity assumptions than those which are adopted in the famous Michael Theorem [4] and in [1].
Daugavet’s equation and orthomorphisms
Authors: Schmidt Klaus D.
Published: June 2010

The main result of this paper asserts that every Dunford-Pettis operator on an AL-space having no discrete elements satisfies Daugavet’s equation

‖

I
+
T

‖

=
1
+

‖
T
‖

\left \| {I + T} \right \| = 1 + \left \| T \right \|

; this extends a recent result of Holub on weakly compact operators. The proof is based on properties of orthomorphisms on a Banach lattice which also yield a short proof of another result of Holub concerning Daugavet’s equation for bounded operators on an arbitrary AL- or AM-space.
The hereditary Dunford-Pettis property for 𝑙₁(𝐸)
Authors: Cembranos Pilar
Published: June 2010

A Banach space

E
E

is said to be hereditarily Dunford-Pettis if all of its closed subspaces have the Dunford-Pettis property. In this note we prove that the Banach space

l
1

(
E
)

{l_1}(E)

, of all absolutely summing sequences in

E
E

with the usual norm, is hereditarily Dunford-Pettis if and only if

E
E

is also.
Two generalizations of Titchmarsh’s convolution theorem
Authors: Doss Raouf
Published: June 2010

Titchmarsh’s convolution theorem states that if the functions

f
,
g

f,g

vanish on

(
−
∞
,
0
)

( - \infty ,0)

and if the convolution

f
∗
g
(
t
)
=
0

f * g(t) = 0

on an interval

(
0
,
T
)

(0,T)

, then there are two numbers

α
,
β
≥
0

\alpha ,\beta \geq 0

such that

α
+
β
=
T
,
f
=
0

\alpha + \beta = T,f = 0

a.e. on

(
0
,
α
)

(0,\alpha )

, and

g
=
0

g = 0

a.e. on

(
0
,
β
)

(0,\beta )

.

T
T

may be infinite. For the case

T
=
∞

T = \infty

we prove that if

f
∗
g
=
0

f * g = 0

on

R
R

and one of the two functions

f
,
g

f,g

is 0 on

(
−
∞
,
0
)

( - \infty ,0)

, then either

f
f

or

g
g

is 0 a.e. on

R
R

. Next we consider the integro-differential-difference equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f asterisk g left-parenthesis t right-parenthesis plus sigma-summation lamda Subscript p sigma Baseline f Superscript left-parenthesis p right-parenthesis Baseline left-parenthesis t minus a Subscript p sigma Baseline right-parenthesis equals 0">

f
∗
g
(
t
)
+
∑

λ

p
σ

f

(
p
)

(
t
−

a

p
σ

)
=
0

f * g(t) + \sum {{\lambda _{p\sigma }}{f^{(p)}}(t - {a_{p\sigma }}) = 0}

for

t
t

in

(
0
,
T
)

(0,T)

, where

a

ρ
σ

≥
0
,

λ

p
σ

{a_{\rho \sigma }} \geq 0,{\lambda _{p\sigma }}

are constants. Conclusions similar to Titchmarsh’s hold with the additional information that

α
≥
T
−

a

ρ
σ

\alpha \geq T - {a_{\rho \sigma }}

whenever

λ

ρ
σ

≠
0

{\lambda _{\rho \sigma }} \ne 0

.
Global 𝐶^{𝑟} structural stability of vector fields on open surfaces with finite genus
Authors: Kotus Janina
Published: June 2010

A vector field

X
X

on the open manifold

M
M

is globally

C
r

{C^r}

structurally stable if

X
X

has a neighborhood

∪
\cup

in the Whitney

C
r

{C^r}

topology such that the trajectories of every vector field

Y
∈
∪

Y \in \cup

can be mapped onto trajectories of

X
X

by a homeomorphism

h
:
M
→
M

h:M \to M

which is in a preassigned compact-open neighborhood of the identity. In [2] it was proved the theorem formulating the sufficient conditions for global

C
r

(
r
≥
1
)

{C^r}(r \geq 1)

structural stability of vector fields on open surfaces

(
dim
⁡
M
=
2
)

(\dim M = 2)

. These conditions are also necessary for global

C
r

{C^r}

structural stability on the plane if

r
≥
1

r \geq 1

(see [2]) and for

r
=
1

r = 1

on any open surface of finite genus [1]. Here we will generalize it for

C
r

(
r
≥
1
)

{C^r}(r \geq 1)

vector fields defined on open orientable surface with finite genus and countable space of ends

E
E

.
On ordered Λ-bounded variation
Authors: Prus-Wiśniowski Franciszek
Published: June 2010

An example is given of a continuous real function that is of ordered

Λ
\Lambda

-bounded variation but not of

Λ
\Lambda

-bounded variation. No special assumptions on

Λ
\Lambda

are required.
A finite-energy bound on the approach of a diffusion to the zeros of its density
Authors: Wallstrom Timothy C.
Published: June 2010

Since the drift coefficients of a finite-energy diffusion are undefined on the nodes, or zeros of the density, an important problem has been to show that the sample paths stay away from the nodes. In this paper it is shown that if the drift is locally a gradient and smooth on the complement of the nodes, and if the density is smooth, then the closest approach to the nodes can be bounded solely in terms of the time-integrated energy.
A note on equivariant eta invariants
Authors: Zhang Wei Ping
Published: June 2010

We prove the regulairty of equivariant eta functions near the origin. We also propose an equivariant version of the Cheeger-Chou index theorem on spaces with conelike singularities.
The semimartingale structure of reflecting Brownian motion
Authors: Bass Richard F. , Hsu Pei
Published: June 2010

We prove that reflecting Brownian motion on a bounded Lipschitz domain is a semimartingale. We also extend the well-known Skorokhod equation to this case.
Semiderivations of prime rings
Authors: Brešar Matej
Published: June 2010

A semiderivation of a ring

R
R

is an additive mapping

f
:
R
→
R

f:R \to R

together with a function

g
:
R
→
R

g:R \to R

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x y right-parenthesis equals f left-parenthesis x right-parenthesis g left-parenthesis y right-parenthesis plus x f left-parenthesis y right-parenthesis equals f left-parenthesis x right-parenthesis y plus g left-parenthesis x right-parenthesis f left-parenthesis y right-parenthesis">

f
(
x
y
)
=
f
(
x
)
g
(
y
)
+
x
f
(
y
)
=
f
(
x
)
y
+
g
(
x
)
f
(
y
)

f(xy) = f(x)g(y) + xf(y) = f(x)y + g(x)f(y)

and

f
(
g
(
x
)
)
=
g
(
f
(
x
)
)

f(g(x)) = g(f(x))

for all

x
,
y
∈
R

x,y \in R

. We prove that the only semiderivations of prime rings are derivations and mappings of the form

f
(
x
)
=
λ
(
x
−
g
(
x
)
)

f(x) = \lambda (x - g(x))

, where

g
g

is an endomorphism and

λ
\lambda

is an element in the extended centroid.
On Rhoades’ open questions
Authors: Chang Shih-sen , Zhong Qi Chun
Published: June 2010

In this paper the open questions raised by Rhoades [1] are answered or partially answered.
Boundedness of the Calderón-Zygmund singular integral operators on Ba spaces
Authors: Chang Wen-Dong
Published: June 2010

A condition for the boundedness of a Calderón-Zygmund operator on a new class of Banach function spaces is studied.
The Schur product theorem in the block case
Authors: Choudhury Dipa
Published: June 2010

Let

H
H

be a positive semi-definite

m
n

mn

-by-

m
n

mn

Hermitian matrix, partitioned into

m
2

{m^2}

n
n

-square blocks

H

i
j

,
i
,
j
=
1
,
…
,
m

{H_{ij}},i,j = 1, \ldots ,m

. We denote this by

H
=
[

H

i
j

]

H = [{H_{ij}}]

. Consider the function

f
:

M
n

→

M
r

f:{M_n} \to {M_r}

given by

f
(
X
)
=

X
k

f(X) = {X^k}

(ordinary matrix product) and denote

H
f

=
[
f
(

H

i
j

)
]

{H_f} = [f({H_{ij}})]

. We shall show that if

H
H

is positive semi-definite then under some restrictions on

H

i
j

,

H
f

{H_{ij}},{H_f}

is also positive semi-definite. This generalizes familar results for Hadamard and ordinary products.
The operator inequality 𝑃≤𝐴*𝑃𝐴
Authors: Duggal B. P.
Published: June 2010

A short proof of the result that if

P
P

is a positive compact operator and

A
A

is a contraction such that

p
≤

A
∗

P
A

p \leq {A^*}PA

, then

P
=

A
∗

P
A
,

ran
¯

P

P = {A^*}PA,\overline {\operatorname {ran} } P

reduces

A
A

and

A

|

ran
¯

P

A|\overline {\operatorname {ran} } P

is unitary is given.
Instantaneous shrinking in nonlinear diffusion-convection
Authors: Gilding B. H. , Kersner R.
Published: June 2010

The Cauchy problem for a nonlinear diffusion-convection equation is studied. The equation may be classified as being of degenerate parabolic type with one spatial derivative and a time derivative. It is shown that under certain conditions solutions of the initial-value problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded above, although the support of the initial data function is not. This is a phenomenon which is normally only associated with nonlinear diffusion with strong absorption. In conjunction, a previously unreported phenomenon is revealed. It is shown that for a certain class of initial data functions there is a critical positive time such that the support of the solution is unbounded above at any earlier time, whilst the opposite is the case at any later time.
A non-completely regular quiet quasi-uniformity
Authors: Fletcher P. , Hejcman J. , Hunsaker W.
Published: June 2010

A complete quiet quasi-uniformity is constructed on A. Mysior’s regular but not completely regular space. This answers a question raised by D. Doitchinov at the Prague conference on categorial topology in 1988.
On rational reciprocity
Authors: Helou Charles
Published: June 2010

A method for deriving "rational"

n
n

th power reciprocity laws from general ones is described. It is applied in the cases

n
=
3
,
4
,
8

n = 3,4,8

, yielding results of von Lienen, Burde, Williams.
Three cardinal functions similar to net weight
Authors: Johnson Roy A. , Wajch Eliza , Wilczyński Władysław
Published: June 2010

The purpose of this paper is to introduce and investigate cardinal functions called pseudonet weight, weak net weight, and weak pseudonet weight. These are similar to but generally smaller than net weight. We look at how these cardinal functions relate to hereditary Lindelöf degree, hereditary density, and spread, and we study their behavior under products.
On the number of self-dual bases of 𝐺𝐹(𝑞^{𝑚}) over 𝐺𝐹(𝑞)
Authors: Jungnickel Dieter , Menezes Alfred J. , Vanstone Scott A.
Published: June 2010

Let

E
=
G
F
(

q
m

)

E = GF({q^m})

be the

m
m

-dimensional extension of

F
=
G
F
(
q
)

F = GF(q)

. We are concerned with the numbers

s
d
(
m
,
q
)

sd(m,q)

and

s
d
n
(
m
,
q
)

sdn(m,q)

of self-dual bases and self-dual normal bases of

E
E

over

F
F

, respectively. We completely determine

s
d
(
m
,
q
)

sd(m,q)

, en route giving a very simple proof for the Sempel-Seroussi theorem which states that

s
d
(
m
,
q
)
=
0

sd(m,q) = 0

iff

q
q

is odd and

m
m

is even. Using results of Lempel and Weinberger and MacWilliams, we can also determine

s
d
n
(
m
,
p
)

sdn(m,p)

for primes

p
p

.
A note on exponential integrability and pointwise estimates of Littlewood-Paley functions
Authors: Leckband Mark
Published: June 2010

Let

T
f

Tf

denote any one of the usual classical or generalized Littlewood-Paley functions. This paper derives a BLO norm estimate for

(
T
f

)
2

{(Tf)^2}

and a pointwise estimate for

T
f

Tf

.
On the equation 𝑑𝑖𝑣(|∇𝑢|^{𝑝-2}∇𝑢)+𝜆|𝑢|^{𝑝-2}𝑢=0
Authors: Lindqvist Peter
Published: June 2010

The first eigenvalue

λ
=

λ
1

\lambda = {\lambda _1}

for the equation

div
⁡
(

|

∇
u

|

p
−
2

∇
u

) + 

λ

|

u

|

p
−
2

u
=
0

\operatorname {div} ({\text {|}}\nabla u{{\text {|}}^{p - 2}}\nabla u{\text {) + }}\lambda {\text {|}}u{{\text {|}}^{p - 2}}u = 0

is simple in any bounded domain. (Through the nonlinear counterpart to the Rayleigh quotient

λ
1

{\lambda _1}

is related to the Poincaré inequality.)
Algebraic elements in group rings
Authors: Passi I. B. S. , Passman D. S.
Published: June 2010

In this brief note, we study algebraic elements in the complex group algebra

C

[
G
]

{\mathbf {C}}[G]

. Specifically, suppose

ξ
∈

C

[
G
]

\xi \in {\mathbf {C}}[G]

satisfies

f
(
ξ
)
=
0

f(\xi ) = 0

for some nonzero polynomial

f
(
x
)
∈

C

[
x
]

f(x) \in {\mathbf {C}}[x]

. Then we show that a certain fairly natural function of the coefficients of

ξ
\xi

is bounded in terms of the complex roots of

f
(
x
)

f(x)

. For

G
G

finite, this is a recent observation of [HLP]. Thus the main thrust here concerns infinite groups, where the inequality generalizes results of [K] and [W] on traces of idempotents.
A partition theorem for [0,1]
Authors: Prömel H. J. , Voigt B.
Published: June 2010

We prove a Hindman-type partition theorem for Baire partitions of

[
0
,
1
]

[0,1]

.
Inverse-closed Carleman algebras of infinitely differentiable functions
Authors: Siddiqi Jamil A.
Published: June 2010

We characterize the classes

C

M

(
I
)

{\mathcal {C}_M}(I)

and

C

M
∗

(
I
)

\mathcal {C}_M^ * (I)

of infinitely differentiable functions which are inverse-closed thereby giving a complete solution to a problem first posed by W. Rudin [11] and solved by him and J. Boman and L. Hörmander [2] for classes

C

M

(

R

)

{\mathcal {C}_M}({\mathbf {R}})

alone.
Generating functions for the numbers of abelian extensions of a local field
Authors: Travesa Artur
Published: June 2010

The aim of this paper is to give an explicit formula for the numbers of abelian extensions of a

p

{\mathbf {p}}

-adic number field and to study the generating function of these numbers. More precisely, we give the number of abelian extensions with given degree and ramification index, and the number of abelian extensions with given degree of any local field of characteristic zero. Moreover, we give a concrete expression of a generating function for these last numbers.
A note on some elementary measures of algebraic independence
Authors: Tubbs Robert
Published: June 2010

We investigate the algebraic independence of some numbers associated with elliptic functions when one of the numbers is a "Liouville-type" number. Suppose

℘
(
z
)

\wp (z)

is a Weierstrass elliptic function with algebraic invariants and

β
\beta

is an algebraic number, not belonging to the field of multiplications for

℘
(
z
)

\wp (z)

. We establish the algebraic independence of

℘
(
u
)

\wp (u)

and

℘
(
β
u
)

\wp (\beta u)

(respectively, of

u
u

and

℘
(
β
u
)

\wp (\beta u)

) when

℘
(
u
)

\wp (u)

(respectively,

u
u

) is a "Liouville-type" number. We also give quantitative versions of these results.
On the characterization of 𝐻^{𝑝}(𝑅ⁿ) in terms of Fourier multipliers
Authors: Uchiyama Akihito
Published: June 2010

Let

p
∈
(
0
,
1
]

p \in (0,1]

, let

f
∈

L
2

(

R
n

)
∩

H
p

(

R
n

)

f \in {L^2}({R^n}) \cap {H^p}({R^n})

and let

θ
∈
C
(

R
n

∖

{
0
}

)

\theta \in C({R^n}\backslash \left \{ 0 \right \})

be homogeneous of degree zero. We will give one sufficient condition in order for

f
f

and

θ
\theta

to satisfy
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript upper H Sub Superscript p Subscript Baseline less-than-or-equal-to upper C left-brace StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript Baseline plus StartAbsoluteValue EndAbsoluteValue script upper F Superscript negative 1 Baseline left-brace theta left-parenthesis xi right-parenthesis script upper F f left-parenthesis xi right-parenthesis right-brace StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript Baseline right-brace period">

|

|

f

|

|

H
p

≤
C

{

|

|

f

|

|

L
p

+

|

|

F

−
1

{

θ
(
ξ
)

F

f
(
ξ
)

}

|

|

L
p

}

.

||f|{|_{{H^p}}} \leq C\left \{ {||f|{|_{{L^p}}} + ||{\mathcal {F}^{ - 1}}\left \{ {\theta (\xi )\mathcal {F}f(\xi )} \right \}|{|_{{L^p}}}} \right \}.

\]
Explicit transitive automorphisms of the closed unit square
Authors: Xu Xiao Quan
Published: June 2010

We construct transitive automorphisms of the closed unit square.
Commutativity in operator algebras
Authors: Blecher David P.
Published: June 2010

By an ’operator algebra’ we shall mean a subalgebra

A

\mathcal {A}

of the algebra

B

(

H

)

\mathcal {B}(\mathcal {H})

of bounded operators on a Hilbert space

H

\mathcal {H}

, together with the matrix normed structure

A

\mathcal {A}

inherits from

B

(

H

)

\mathcal {B}(\mathcal {H})

. A unital operator algebra is an operator algebra with an identity of norm 1. Note that we do not require the algebra to be self-adjoint or uniformly closed. Such algebras were characterized abstractly by Ruan, Sinclair, and the author up to complete isometric isomorphism. In this paper we study commutativity for operator algebras, and we give a characterization of commutative unital operator algebras and a characterization of unital uniform algebras.
The ring of integer-valued polynomials of a Dedekind domain
Authors: Gilmer Robert , Heinzer William , Lantz David , Smith William
Published: June 2010

Let

D
D

be a Dedekind domain and

R
=
I
n
t
(
D
)

R = Int(D)

be the ring of integer-valued polynomials of

D
D

. We relate the ideal class groups of

D
D

and

R
R

. In particular we prove that, if

D
=

Z

D = \mathbb {Z}

is the ring of rational integers, then the ideal class group of

R
R

is a free abelian group on a countably infinite basis.
A nonzero complex sequence with vanishing power-sums
Authors: Lenard Andrew
Published: June 2010

In response to a question of W. M. Priestley, we construct a nonvanishing infinite sequence of complex numbers

(

a
j

)

({a_j})

such that all the power-sums

∑
j

a
j
r

(
r
=
1
,
2
,
3
,
…

)

\sum \nolimits _j {a_j^r(r = 1,2,3, \ldots } )

vanish.
Invariant subspaces of finite codimension for measures with thin support
Authors: Trent T. T.
Published: June 2010

A simple proof that

M
z

{M_z}

on

P
2

(
μ
)

{P^2}(\mu )

has a nontrivial invariant subspace is given. If

P
2

(
μ
)
≠

L
2

(
μ
)

{P^2}(\mu ) \ne {L^2}(\mu )

and if

μ
\mu

has "thin" support, then

P
2

(
μ
)

{P^2}(\mu )

has bounded point evaluations.
On the sectional curvature of compact hypersurfaces
Authors: Coghlan Leslie , Itokawa Yoe
Published: June 2010

We establish a sufficient condition for compact hypersurfaces of a complete riemannian manifold to be spherical. It is well known, from the works of Jacobowitz, Jorge and Koutroufiotis, and others, that the maximum sectional curvature of such hypersurfaces can be estimated from the curvature of the ambient space and the outer radius. Our result sharpens these estimates. It also implies a new nonimmersibility theorem of the Chern-Kuiper type.
On eigenvalue problems of 𝑝-Laplacian with Neumann boundary conditions
Authors: Huang Yin Xi
Published: June 2010

We study the nonlinear eigenvalue problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus normal upper Delta Subscript p Baseline u equals lamda m left-parenthesis x right-parenthesis StartAbsoluteValue u EndAbsoluteValue Superscript p minus 2 Baseline u in normal upper Omega comma StartFraction partial-differential u Over partial-differential n EndFraction equals 0 on partial-differential normal upper Omega comma where p greater-than 1 comma lamda element-of bold upper R period">

−

Δ
p

u
=
λ
m
(
x
)

|

u

|

p
−
2

u

in

Ω
,

∂
u

∂
n

=
0

on

∂
Ω
,

where

p
>
1
,
λ
∈

R

.

- {\Delta _p}u = \lambda m(x)|u{|^{p - 2}}u\,\quad {\text {in}}\,\Omega ,\quad \,\frac {{\partial u}} {{\partial n}} = 0\,\quad {\text {on}}\,\partial \Omega ,\,\;{\text {where}}\,p > 1,\lambda \in {\mathbf {R}}.

\]
For

∫
Ω

m
(
x
)
>
0

\int _\Omega {m(x) > 0}

, we prove that the first positive eigenvalue

λ
1

{\lambda _1}

exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction. In the case

∫
Ω

m
(
x
)
=
0

\int _\Omega {m(x) = 0}

, we prove that

λ
0

=
0

{\lambda _0} = 0

is the only eigenvalue with a positive eigenfunction.
On double cosine, sine, and Walsh series with monotone coefficients
Authors: Móricz Ferenc
Published: June 2010

We extend from one-dimensional to two-dimensional series the results by Hardy and Littlewood [6] on the

L
r

{L^r}

-integrability of the sum

f
f

and the results by Stechkin [10] on the

L
1

{L^1}

-integrability of the maximum partial sum

M
∗

{M^*}

in the case of cosine and sine series with monotone coefficients. Among others, we prove that the

L
r

{L^r}

-integrability of

f
f

and

M
∗

{M^*}

is essentially equivalent for

r
>
1

r > 1

in the two-dimensional setting, too. Simultaneously, we extend our earlier results in [7] from one-dimensional to two-dimensional Walsh series.
Hardy inequalities
Authors: Wannebo Andreas
Published: June 2010

Here the following Hardy inequalities are studied
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript k equals 0 Overscript m minus 1 Endscripts integral StartFraction StartAbsoluteValue nabla Superscript k Baseline u EndAbsoluteValue Superscript p Baseline d Subscript partial-differential normal upper Omega Baseline left-parenthesis x right-parenthesis Superscript t Baseline Over d Subscript partial-differential normal upper Omega Baseline left-parenthesis x right-parenthesis Superscript left-parenthesis m minus k right-parenthesis p Baseline EndFraction d x less-than-or-equal-to upper A Subscript normal upper Omega Baseline integral StartAbsoluteValue nabla Superscript m Baseline u EndAbsoluteValue Superscript p Baseline d Subscript partial-differential normal upper Omega Baseline left-parenthesis x right-parenthesis Superscript t Baseline d x">

∑

k
=
0

m
−
1

∫

|

∇
k

u

|

p

d

∂
Ω

(
x
)

t

d

∂
Ω

(
x
)

(
m
−
k
)
p

d
x
≤

A
Ω

∫

|

∇
m

u

|

p

d

∂
Ω

(
x
)

t

d
x

\sum \limits _{k = 0}^{m - 1} {\int {\frac {{|{\nabla ^k}u{|^p}{d_{\partial \Omega }}{{(x)}^t}}}{{{d_{\partial \Omega }}{{(x)}^{(m - k)p}}}}dx \leq {A_\Omega }\int {|{\nabla ^m}u{|^p}{d_{\partial \Omega }}{{(x)}^t}dx} } }

\]
for

u
∈

C
0
∞

(
Ω
)

u \in C_0^\infty (\Omega )

, an open proper subset of

R

N

{{\mathbf {R}}^N}

and

t
>

t
0

t > {t_0}

, some small positive

t
0

{t_0}

. This inequality has previously been shown to hold for bounded Lipschitz domains. The question discussed is, How general can

Ω
\Omega

be and allow the same inequality? A sufficient condition is given in the form of a local Maz’ja capacity condition. In

R

2

{{\mathbf {R}}^2}

, or generally if

p
>
N
−
1

p > N - 1

, this is satisfied for any

Ω
\Omega

which is deformable to a point. Furthermore, if

p
>
N

p > N

the condition is satisfied for all

Ω
\Omega

.
Notes on the Baumslag-Solitar non-residually finite examples
Authors: Campbell Robert I.
Published: June 2010

We examine the abelianization of G. Baumslag and Solitar’s example of a one-generator group that is not residually finite. In particular, the nonfinitely-generated commutator subgroup is shown to be not residually finite. We also review a specific example of a cyclic extension of a residually finite group that is not residually finite.
When does 𝑐𝑎(Σ,𝑋) contain a copy of 𝑙_{∞} or 𝑐₀?
Authors: Drewnowski Lech
Published: June 2010

For a

σ
\sigma

-algebra

Σ
\Sigma

and a Banach space

X
,
c
a

(

Σ
,
X

)

X,ca\left ( {\Sigma ,X} \right )

is the Banach space of all vector measures from

Σ
\Sigma

to

X
X

. If

Σ
\Sigma

admits a nonzero atomless finite positive measure, then

c
a

(

Σ
,
X

)

⊃

l
∞

ca\left ( {\Sigma ,X} \right ) \supset {l_\infty }

(or

c
0

{c_0}

) if and only if there is a noncompact bounded linear operator from

l
2

{l_2}

to

X
X

(Theorem 1). Otherwise,

c
a

(

Σ
,
X

)

⊃

l
∞

ca\left ( {\Sigma ,X} \right ) \supset {l_\infty }

(or

c
0

{c_0}

) if and only if

X
⊃

l
∞

X \supset {l_\infty }

(or

c
0

{c_0}

) (Theorem 2).
A new family of Enneper type minimal surfaces
Authors: Fang Yi
Published: June 2010

An Enneper type surface is a complete immersed minimal surface in

R

3

{{\mathbf {R}}^3}

with only one end and finite total curvature. In this paper we construct a family of Enneper type surfaces of genus 1, total curvature

−
8
(
2
n
+
1
)
π
,
n
=
0
,
1
,
2
,
⋯

- 8(2n + 1)\pi ,n = 0,1,2, \cdots

. We use the Weierstrass

℘
\wp

elliptic function as a tool and also prove some results about

℘
\wp

on a square torus.
Copies of 𝑙_{∞} in 𝐿^{𝑝}(𝜇;𝑋)
Authors: Mendoza José
Published: June 2010

Let

X
X

be a Banach space and let

(
Ω
,
∑
,
μ
)

(\Omega ,\sum ,\mu )

be a measure space. For

1
≤
p
>
+
∞

1 \leq p > + \infty

we denote by

L
p

(
μ
;
X
)

{L^p}(\mu ;X)

the Banach space of all

X
X

-valued Bochner

p
p

-integrable functions on

Ω
\Omega

. In this note we show that

L
p

(
μ
;
X
)

{L^p}(\mu ;X)

contains an isomorphic copy of

l
∞

{l_\infty }

if and only if

X
X

does.
Convolution in the harmonic Hardy class ℎ^{𝑝} with 0<𝑝<1
Authors: Pavlović Miroslav
Published: June 2010

It is proved that if

u
∈

h
p

,
0
>
p
>
1

u \in {h^p},0 > p > 1

, and

v
∈

h
q

,
q
≥
p

v \in {h^q},q \geq p

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript q Baseline left-parenthesis u asterisk v comma r right-parenthesis equals 0 left-parenthesis left-parenthesis 1 minus r right-parenthesis Superscript 1 minus 1 slash p Baseline right-parenthesis comma r right-arrow 1 minus comma">

M
q

(
u
∗
v
,
r
)
=
0

(

(
1
−
r
)

1
−
1

/

p

)

,
r
→
1
−
,

{M_q}(u * v,r) = 0\left ( {{{(1 - r)}^{1 - 1/p}}} \right ),r \to 1 - ,

\]
where

u
∗
v

u*v

stands for the convolution of

u
u

and

v
v

.
Locally efficient monotone operators
Authors: Verona Andrei , Verona Maria Elena
Published: June 2010

We study monotone operators on quasi open or convex subsets of a real Banach space

X
X

(quasi open means that the contingent cone at each point equals

X
X

). Among others we characterize the maximality of such an operator in terms of its

w
∗

{w^*}

-upper semicontinuity properties and, in the case of a convex domain, also in terms of its behavior at the support points. We next give sufficient conditions for such an operator to be generically single valued, extending Kenderov’s theorems. As an application we reobtain generic Gâteaux and Fréchet differentiability results for convex functions defined on not necessarily open convex sets.
A characterization of continua that contain no 𝑛-ods and no 𝑊-sets
Authors: Vought Eldon J.
Published: June 2010

A proper, nondegenerate subcontinuum

K
K

of a continuum

Y
Y

is said to be a

W
W

-set if, for every continuum

X
X

and map

f
f

of

X
X

onto

Y
Y

, some subcontinuum of

X
X

is mapped by

f
f

onto

K
K

. Jim Davis asked whether a simple closed curve is the only atriodic continuum that contains no

W
W

-set. An affirmative answer is given to this question. The result follows as a corollary to the more general theorem that a continuum contains no

n
n

-od and has no

W
W

-set if and only if it is a graph in which every point is contained in a simple closed curve. Properties of this class of graphs are also described.
Higher-dimensional shift equivalence and strong shift equivalence are the same over the integers
Authors: Wagoner J. B.
Published: June 2010

Let

R
S
(
Λ
)

RS(\Lambda )

and

S
(
Λ
)

S(\Lambda )

denote, respectively, the spaces of strong shift equivalences and shift equivalences over a subset

Λ
\Lambda

of a ring which is closed under addition and multiplication. For example, let

Λ
\Lambda

be the integers

Z
Z

or the nonnegative integers

Z
+

{Z^ + }

. For any principal ideal domain

Λ
\Lambda

, we prove that the continuous map

R
S

(
Λ
)

→
S

(
Λ
)

RS\left ( \Lambda \right ) \to S\left ( \Lambda \right )

is a homotopy equivalence. The methods also show that any inert automorphism, i.e., an element in the kernel of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi 1 left-parenthesis upper R upper S left-parenthesis upper Z Superscript plus Baseline right-parenthesis comma upper A right-parenthesis right-arrow pi 1 left-parenthesis upper S left-parenthesis upper Z Superscript plus Baseline right-parenthesis comma upper A right-parenthesis">

π
1

(

R
S

(

Z
+

)

,
A

)

→

π
1

(

S

(

Z
+

)

,
A

)

{\pi _1}\left ( {RS\left ( {{Z^ + }} \right ),A} \right ) \to {\pi _1}\left ( {S\left ( {{Z^ + }} \right ),A} \right )

can be represented by a closed loop in

R
S

(

Z
+

)

RS\left ( {{Z^ + }} \right )

which in

S
R

(
Z
)

SR\left ( Z \right )

is spanned by a triangulated

2
2

-disc supporting a positive

1
1

-cocycle. These cocycles are used in work of Kim-Roush that leads to a counterexample to Williams’ lifting problem for automorphisms of finite subsystems of subshifts of finite type.
Centralizers of immersions of the circle
Authors: Arteaga Carlos
Published: June 2010

We prove here that for every element

f
f

of an open and dense subset of immersions of the circle

S
1

{S^1}

, either the centralizer

Z

(
f
)

Z\left ( f \right )

of

f
f

is trivial (i.e.

f
f

only commutes with its own powers) or

f
f

is topologically conjugate to a map

f
n

:

S
1

→

S
1

{f_n}:{S^1} \to {S^1}

given by

f
n

(
z
)

=

z
n

{f_n}\left ( z \right ) = {z^n}

and, in this case, if

h
h

is the conjugacy between

f
f

and

f
n

{f_n}

then

Z

(
f
)

Z\left ( f \right )

is a subgroup of

{

h

−
1

∘
ω

f
m

∘
h
;
m
∈

N

 and 

ω

n
−
1

=
1

}

\left \{ {{h^{ - 1}} \circ \omega {f_m} \circ h;m \in {\mathbf {N}}{\text { and }}{\omega ^{n - 1}} = 1} \right \}

.
Restrictions of open mappings of continua
Authors: Bula Witold D. , Tymchatyn E. D.
Published: June 2010

We give an example of an open map

f
:
X
→
Y

f:X \to Y

, where

X
X

is a locally connected one-dimensional continuum, such that there is a subcontinuum

K
K

of

Y
Y

for which the restriction of

f
f

to a certain component of

f

−
1

(
K
)

{f^{ - 1}}(K)

is not open.
Divisibility of generalized exponential and logarithmic coefficients
Authors: Davis Donald M.
Published: June 2010

We prove that for isomorphic formal group laws, the maximal

p
p

-power in the denominators of certain collections of coefficients of powers of their exponential series are equal. These numbers are significant in algebraic topology.
First countability and local simple connectivity of one point unions
Authors: Eda Katsuya
Published: June 2010

There exists a path-connected, simply connected, Tychonoff space

X
X

with

x
x

such that

X
X

is locally simply connected at

x
x

, but

(
X
,
x
)
∨
(
X
,
x
)

(X,x) \vee (X,x)

is not simply connected.
The distribution of Rademacher sums
Authors: Montgomery-Smith S. J.
Published: June 2010

We find upper and lower bounds for

Pr
(

Σ
±

x
n

≥
t
)

{\Pr (}\Sigma \pm {x_n} \geq t)

, where

x
1

,

x
2

,
…

{x_1},{x_2}, \ldots

are real numbers. We express the answer in terms of the

K
K

-interpolation norm from the theory of interpolation of Banach spaces.
Weak convergence and weak compactness for multifunctions with values in a separable Banach space
Authors: Papageorgiou Nikolaos S.
Published: June 2010

In this paper we extend the notion of weak convergence to Banach space valued integrable multifunctions. We prove some properties of this mode of convergence and we show that under certain hypotheses the space of uniformly bounded,

w
w

-compact, convex valued integrable multifunctions is sequentially weakly complete. Then we prove two Dunford-Pettis type weak compactness theorems and we also show that the set valued conditional expectation is weakly continuous. Finally we present an application from control theory.
Hilbert-Schmidt Hankel operators on the Bergman space
Authors: Zhu Ke He
Published: June 2010

We show that there are no nonzero Hilbert-Schmidt Hankel operators on the Bergman space of the open unit ball in

C

n

{{\mathbf {C}}^n}

with antiholomorphic symbols when

n
≥
2

n \geq 2

.
A Thomson’s principle for infinite, nonlinear resistive networks
Authors: De Michele L. , Soardi P. M.
Published: June 2010

Suppose

Γ
\Gamma

is an infinite resistive electrical network with resistors of the form (2). We prove an existence and uniqueness theorem for the current generated by external current sources by establishing the analogue of Thomson’s principle in suitable modular sequence spaces.
On the Mal′cev correspondence
Authors: Videla Carlos R.
Published: June 2010

We generalize Mal’cev’s correspondence [5] to other nilpotent groups, namely to certain maximal unipotent subgroups of Chevalley groups. If

L

\mathcal {L}

is a root system and

K
K

is an infinite field of characteristic different to 2 or 3, then we show that a group

G
G

elementarily equivalent to

U

L

(
K
)

{U_\mathcal {L}}(K)

is isomorphic to

U

L

(
F
)

{U_\mathcal {L}}(F)

, where

F
F

is a field elementarily equivalent to

K
K

.
Subdominant ultrametrics
Authors: Bayod José M. , Martínez-Maurica J.
Published: June 2010

Given a metric space

(

X
,
d

)

\left ( {X,d} \right )

we study the existence of an ultrametric

δ
\delta

which is dominated by

d
d

. We give a number of examples and prove, among other results, that such an ultrametric exists if

(

X
,
d

)

\left ( {X,d} \right )

is locally compact and totally disconnected.
Periodic solutions for nonlinear evolution equations in a Banach space
Authors: Vrabie Ioan I.
Published: June 2010

We prove an existence result for

T
T

-periodic mild solutions to nonlinear evolution equations of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u prime left-parenthesis t right-parenthesis plus upper A u left-parenthesis t right-parenthesis small-contains-as-member upper F left-parenthesis t comma u left-parenthesis t right-parenthesis right-parenthesis comma t element-of upper R Subscript plus Baseline period">

u
′

(
t
)
+
A
u
(
t
)
∍
F
(
t
,
u
(
t
)
)
,

t
∈

R
+

.

u’(t) + Au(t) \backepsilon F(t,u(t)),\quad t \in {R_ + }.

\]
Here

(
X
,

|

|

⋅

|

|

)

(X,|| \cdot ||)

is a real Banach space,

A
:
D
(
A
)
⊂
X
→

2
X

A:D(A) \subset X \to {2^X}

is an operator with

A
−
a
I

A - aI

m
m

-accretive for some

a
>
0

a > 0

and such that

−
A

- A

. generates a compact semigroup, while

F
:

R
+

×

D
(
A
)

¯

→
X

F:{R_ + } \times \overline {D(A)} \to X

is a Carathéodory mapping which is

T
T

-periodic with respect to its first argument and satisfies
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript r right-arrow plus normal infinity Endscripts StartFraction 1 Over r EndFraction sup left-brace StartAbsoluteValue EndAbsoluteValue upper F left-parenthesis t comma v right-parenthesis StartAbsoluteValue EndAbsoluteValue semicolon t element-of upper R Subscript plus Baseline comma v element-of ModifyingAbove upper D left-parenthesis upper A right-parenthesis With bar comma StartAbsoluteValue EndAbsoluteValue v StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to r right-brace greater-than a period">

lim

r
→
+
∞

1
r

sup

{

|

|

F
(
t
,
v
)

|

|

;
t
∈

R
+

,
v
∈

D
(
A
)

¯

,

|

|

v

|

|

≤
r

}

>
a
.

\lim \limits _{r \to + \infty } \tfrac {1}{r}\sup \left \{ {||F(t,v)||;t \in {R_ + },v \in \overline {D(A)} ,||v|| \leq r} \right \} > a.

\]
. As a consequence, we obtain an existence theorem for

T
T

-periodic solutions to the porous medium equation.
Minimal shell of a typical convex body
Authors: Zucco Andreana
Published: June 2010

We prove that "most" convex bodies in

E
d

{E^d}

touch the boundary of their minimal shell in precisely

d
+
2

d + 2

points.
Lifting homeomorphisms of a complex surface to its normalization
Authors: Adkins William A.
Published: June 2010

It is proved that any homeomorphism of a complex analytic surface lifts to a homeomorphism of the normalization.
Łojasiewicz’s support and doubly stochastic measures
Authors: Mikusiński P. , Sherwood H. , Taylor M. D.
Published: June 2010

We consider an alternative to the usual notion of the support of a measure which seems more suitable for the study of doubly stochastic measures.
Maps between orbifolds
Authors: Yamasaki Masayuki
Published: June 2010

Elementary homotopy theory on maps between orbifolds is discussed. For example, it is shown that, given a homomorphism

φ
\varphi

between orbifold fundamental groups of certain orbifolds, there exists a map (unique up to homotopy) between the orbifolds which induces

φ
\varphi

. We also study the properties of orbifolds preserved by homotopy-equivalences.
The Diophantine equation 𝑓(𝑥)=𝑔(𝑦)
Authors: Cochrane Todd
Published: June 2010

Let

f
(
x
)
,
g
(
y
)

f(x),g(y)

be polynomials over

Z

\mathbb {Z}

of degrees

n
n

and

m
m

respectively and with leading coefficients

a
n

,

b
m

{a_n},{b_m}

. Suppose that

m

|

n

m|n

and that

a
n

/

b
m

{a_n}/{b_m}

is the

m
m

th power of a rational number. We give two elementary proofs that the equation

f
(
x
)
=
g
(
y
)

f(x) = g(y)

has at most finitely many integral solutions unless

f
(
x
)
=
g
(
h
(
x
)
)

f(x) = g(h(x))

for some polynomial

h
(
x
)

h(x)

with rational coefficients taking integral values at infinitely many integers.
Two families of stable bundles with the same spectrum
Authors: Rao A. P.
Published: June 2010

We study stable rank two algebraic vector bundles on

P

3

{{\mathbf {P}}^3}

and show that the family of bundles with fixed Chern classes and spectrum may have more than one irreducible component. We also produce a component where the generic bundle has a monad with ghost terms which cannot be deformed away.
Lifting theorems in nonstandard measure theory
Authors: Ross David
Published: June 2010

1. A nonstandard capacity construction, analogous to Loeb’s measure construction, is developed. Using this construction and Choquet’s Capacitability theorem, it is proved that a Loeb measurable function into a general (not necessarily second countable) space has a lifting precisely when its graph is ’almost’ analytic. This characterization is used to generalize and simplify some known lifting existence theorems. 2. The standard notion of ’Lusin measurability’ is related to the nonstandard notion of admitting a ’two-legged’ lifting. An immediate consequence is a new and simple proof of the general Lusin theorem. Another consequence is the existence of a Loeb measurable function, not admitting a lifting, into a relatively small topological space.
Existence results for some initial and boundary value problems
Authors: O’Farrell Anthony G. , O’Regan Donal
Published: June 2010

Existence results for a variety of initial- and boundary-value problems are presented. For the problems considered, we show that the existence question depends on properties of the zero set of the nonlinearity. The analysis throughout is based upon a nonlinear alternative of A. Granas and the use of a priori bounds.
Rotation sets of toral flows
Authors: Franks John , Misiurewicz Michał
Published: June 2010

We consider the rotation set

ρ
(
Φ
)

\rho (\Phi )

for a lift

Φ
=

{

Φ
t

}

t
∈

R

\Phi = {\{ {\Phi _t}\} _{t \in \mathbb {R}}}

of a flow

φ
=

{

φ
t

:

T

2

→

T

2

}

t
∈

R

\varphi = {\{ {\varphi _t}:{\mathbb {T}^2} \to {\mathbb {T}^2}\} _{t \in \mathbb {R}}}

. Our main result is that

ρ
(
Φ
)

\rho (\Phi )

consists of either a single point, a segment of a line through 0 with rational slope, or a line segment with irrational slope and one endpoint equal to 0. Any set of one of these types is the rotation set for some flow.
Nonclosed pure subgroups of locally compact abelian groups
Authors: Takahashi Yuji
Published: June 2010

We study the existence of many nonclosed pure subgroups of nondiscrete locally compact abelian groups. It is shown that every nondiscrete locally compact abelian group has uncountably many nonclosed pure subgroups. This in particular solves a question of Armacost. It is also shown that, if

G
G

is a nondiscrete locally compact abelian group and if either

G
G

is a compact group or the torsion part

t

(
G
)

t\left ( G \right )

of

G
G

is nonopen, then

G
G

has

2
c

{2^c}

proper dense pure subgroups, where

c
c

denotes the power of the continuum. This in particular gives a partial answer to another question of Armacost.
A note on subdiagonality for triangular AF algebras
Authors: Ventura Belisario A.
Published: June 2010

We consider the analogue, for triangular AF algebras, of the notion of subdiagonality for subalgebras of von Neumann algebras. We show that a subalgebra

T

\mathcal {T}

of the AF algebra

A

\mathcal {A}

is subdiagonal if and only if it is strongly maximal.
A unicellular universal quasinilpotent weighted shift
Authors: Herrero Domingo A.
Published: June 2010

For a suitably chosen sequence of weights

{

α
n

}

\{ {\alpha _n}\}

, the unilateral weighted shift

Q
Q

on

l
p

(
1
≤
p
>
∞
)

{l^p}(1 \leq p > \infty )

, defined by

Q

e
n

=

α
n

e

n
+
1

(
n
≥
1
)

Q{e_n} = {\alpha _n}{e_{n + 1}}(n \geq 1)

, is a unicellular quasinilpotent operator such that

Q
k

{Q^k}

is not compact for any power

k
≥
1

k \geq 1

. As a corollary, several applications to approximation of Hilbert space operators are given.
Uniformity and uniformly continuous functions for locally compact groups
Authors: Milnes Paul
Published: June 2010

We show that a locally compact group

G
G

has equivalent right and left uniform structures if (and only if) the sets of bounded, complex-valued, right and left uniformly continuous functions on

G
G

coincide. Along the way it is seen that

G
G

has equivalent right and left uniform structures if (and only if) each

σ
\sigma

-compact subgroup of

G
G

has equivalent right and left uniform structures. We also note that a bounded function

f
:
G
→

C

f:G \to \mathbb {C}

is right uniformly continuous if (and only if)

f

|

H

f{|_H}

is right uniformly continuous for each

σ
\sigma

-compact subgroup

H
H

of

G
G

.

σ
\sigma

-compactness cannot be weakened to compact generation for these last results; a

σ
\sigma

-compact group is exhibited which has inequivalent right and left uniform structures, and for which each compactly generated subgroup has equivalent right and left uniform structures.
The normal holonomy group
Authors: Olmos Carlos
Published: June 2010

We prove that the restricted normal holonomy group of a submanifold of a space of constant curvature is compact and that the nontrivial part of its representation on the normal space is the isotropy representation of a semisimple Riemannian symmetric space.