bluuebunny/crossref_metadata_2025 huggingface dataset:
Power series and nonnormal functions
Authors: Hwang J. S. , Lappan Peter
Published: June 2010

We construct a sequence

{

a
n

}

\left \{ {{a_n}} \right \}

of positive numbers such that

a
n

→
0
,
∑

|

a
n

−

a

n
−
1

|

>
∞

{a_n} \to 0,\sum {\left | {{a_n} - {a_{n - 1}}} \right | > \infty }

, and the function

f
(
z
)
=
∑

a
n

z
n

f(z) = \sum {{a_n}{z^n}}

is not a normal function. This answers a question raised by the second author (Proc. Amer. Math. Soc. 85 (1982), 335-341).
A Poincaré-type inequality for solutions of elliptic differential equations
Authors: Ziemer William P.
Published: June 2010

A sharpened version of the Poincaré inequality is shown to hold for solutions of a large class of second order elliptic equations.
On the existence of complete parallel vector fields
Authors: Welsh David J.
Published: June 2010

In 1966, Chern asked which compact orientable manifolds carry a Riemannian metric and a vector field which is parallel under this metric. An earlier paper answered this without assuming orientability. The present paper applies techniques which work without compactness to identify manifolds which can carry complete parallel vector fields
Lindenbaum algebras and partial conservativity
Authors: Bennet Christian
Published: June 2010

A partial Lindenbaum algebra

Γ
A

{\Gamma ^A}

, where

A
A

is a theory extending Peano arithmetic and

Γ
∈
{

Π
n

,

Σ
n

}

\Gamma \in \{ {\Pi _n},{\Sigma _n}\}

, is the full Lindenbaum algebra for

A
A

restricted to sentences in

A
A

provably equivalent to

Γ
n

{\Gamma _n}

-sentences. Using a new result on pairs of partially conservative sentences, we show that

Π
n
A

\Pi _n^A

and

Σ
n
A

\Sigma _n^A

are not isomorphic.
A metric form of microtransitivity
Authors: Hohti Aarno
Published: June 2010

We prove that every homogeneous compact metrizable space

X
X

has a compatible metric for which

X
X

is Lipschitz homogeneous and for which the group

L
(
X
)

L(X)

of Lipschitz homeomorphisms of

X
X

acts Lipschitz microtransitively on

X
X

.
On supercomplete spaces. III
Authors: Hohti Aarno
Published: June 2010

We show that for each positive integer

n
n

there is a fine uniform space

X
X

, topologically a subspace of the real line, such that

X
n

{X^n}

is supercomplete, but

X

n
+
1

{X^{n + 1}}

is not supercomplete. The space

X
X

can also be chosen so that that

X
n

{X^n}

is supercomplete for all

n
∈
N

n \in N

, but the countably infinite power

X
N

{X^N}

is not supercomplete.
On Rhoades’ open questions and some fixed point theorems for a class of mappings
Authors: Chang Shih-Sen
Published: June 2010

In this paper the concept of

C
C

-mapping and some fixed point theorems for such mappings are introduced and presented. These results are then used to answer two questions of Rhoades.
A counterexample in dynamical systems of the interval
Authors: Chu Hsin , Xiong Jin Cheng
Published: June 2010

In [1] it was proved that if the recurrent points of a continuous map of the unit interval form a closed set, then this map has no periodic point with period not equal to a power of 2, i.e. this map is of type

2
∞

{2^\infty }

. In this paper we will construct a continuous map of the interval which is of type

2
∞

{2^\infty }

and for which the set of recurrent points is not closed. By such a counterexample it may be shown that some of the results announced in [2] are not correct.
A property of weakly compact operators on 𝐶[0,1]
Authors: Holub James R.
Published: June 2010

It is shown that if

T
T

is a bounded linear operator on

C
[
0
,
1
]

C[0,1]

then either

|

|

I
+
T

|

|

||I + T||

or

|

|

I
−
T

|

|

||I - T||

equals

1
+

|

|

T

|

|

1 + ||T||

. If

T
T

is a weakly compact operator then

|

|

I
+
T

|

|

=
1
+

|

|

T

|

|

||I + T|| = 1 + ||T||

, an extension of a result of Daugavet concerning compact operators on

C
[
0
,
1
]

C[0,1]

.
Coisometric extensions of contraction semigroups
Authors: Szymański Wacław
Published: June 2010

One more approach to dilation theorems for contraction semigroups is presented. The result is obtained by applying the

∗
*

-dilation theorem for

∗
*

-semigroups to a

∗
*

-semigroup with the algebraic operation defined in a special way.
A note on MT operators
Authors: Trautman D. A.
Published: June 2010

We give two characterizations of an MT operator

A
A

from a Banach space

X
X

into a quasi-Banach space

Y
Y

. Both characterizations are generalizations of results of Burkholder concerning MT spaces, find one characterization answers a question of Bourgain and Davis.
The Mackey continuity of the monotone rearrangement
Authors: Horsley Anthony , Wrobel Andrzej J.
Published: June 2010

Let

(
A
,

A

,
μ
)

(A, \mathcal {A},\mu )

be a probability space, and let mes denote the Lebesgue measure on the Borel

σ
\sigma

-algebra

B

\mathcal {B}

in

[
0
,
1
]

[0,1]

. The nondecreasing-rearrangement operator from the space

L
∞

(
μ
)
=

L
∞

(
A
,

A

,
μ
)

{L^\infty }(\mu ) = {L^\infty }(A, \mathcal {A}, \mu )

of real-valued essentially bounded functions into

L
∞

=

L
∞

(
[
0
,
1
]

{L^\infty } = {L^\infty }([0,1]

,

B

\mathcal {B}

, mes) is shown to be uniformly continuous in the Mackey topologies

τ
(

L
∞

(
μ
)

\tau ({L^\infty }(\mu )

,

L
1

(
μ
)
)

{L^1}(\mu ))

and

τ
(

L
∞

,

L
1

)

\tau ({L^\infty },{L^1})

on

L
∞

(
μ
)

{L^\infty }(\mu )

and

L
∞

{L^\infty }

, respectively.
Continuity of the Fenchel transform of convex functions
Authors: Back Kerry
Published: June 2010

Given a separated dual system

(
E
,

E
′

)

(E,E’)

, the Fenchel transform determines a pairing of the convex functions on

E
E

with the convex functions on

E
′

E’

. This operation is shown to have a continuity property. The result implies that the minimum set of a convex function varies in an upper-semicontinuous way with the function’s conjugate. Several convergence concepts for convex functions are discussed. It is shown for each of the two most useful that the Fenchel transform is not a homeomorphism.
Spectra of some domains in compact Lie groups and their applications
Authors: Urakawa Hajime
Published: June 2010

In this paper, we determine explicitly the spectra of the Dirichlet problems of some domains in simply connected compact simple Lie groups. As their applications, combining results of Hoffman [6] and Mori [10], we can state some stability conditions of these domains for the standard minimal isometric immersions into unit spheres.
Measurability properties of spectra
Authors: Levi S. , Slodkowski Z.
Published: June 2010

We study Borel measurability of the spectrum in topological algebras. We give some equivalences of the various properties, show that the spectrum in a Banach algebra is continuous on a dense

G
δ

{G_\delta }

, and prove that in a Polish algebra the set of invertible elements is an

F

σ
δ

{F_{\sigma \delta }}

and the inverse mapping is a Borel function of the second class.
Four counterexamples to Bloch’s principle
Authors: Rubel Lee A.
Published: June 2010

In this note, four counterexamples are given to Bloch’s heuristic principle in complex function theory. The first involves univalent functions, the second certain autonomous differential equations, and the remaining two involve certain autonomous differential expressions omitting certain values.
Probabilities on models of universal sentences
Authors: Hoover Douglas N.
Published: June 2010

We show that the asymptotics of conditional probabilities of first order universal sentences on finite models are the same as those of general first order sentences. This answers a question of R. Fagin.
Some classes of orthogonal polynomials associated with martingales
Authors: Feinsilver Philip
Published: June 2010

The classes of orthogonal polynomials which arise as iterated stochastic integrals of a process with stationary independent increments are discussed. They are classes of Meixner polynomials.
The cohomology of an isospectral flow
Authors: Fried David
Published: June 2010

Building on work of Tomei, we compute the cohomology of the manifold of real symmetric tridiagonal matrices with distinct fixed eigenvalues. The proof uses the global dynamical properties of the Toda flow on this isospectral manifold.
Monotone and open Whitney maps
Authors: Illanes Alejandro
Published: June 2010

In this paper we prove that if

X
X

is a locally connected continuum, then there exists a monotone and open Whitney map for

2
X

{2^X}

.
Densities for ranks of certain parts of 𝑝-class groups
Authors: Gerth Frank
Published: June 2010

Let

K
K

be a Galois extension of the field of rational numbers of prime degree

p
p

, and let

C
K

{C_K}

be the

p
p

-class group of

K
K

. In this paper densities for the ranks of certain parts of such

C
K

{C_K}

are calculated, and these densities suggest a way to extend conjectures of Cohen and Lenstra.
An upper bound for the permanent of a 3-dimensional (0,1)-matrix
Authors: Dow Stephen J. , Gibson Peter M.
Published: June 2010

Let

A
=
(

a

i
j
k

)

A = ({a_{ijk}})

be a

3
3

-dimensional matrix of order

n
n

. The permanent of

A
A

is defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p e r upper A equals sigma-summation Underscript sigma comma tau element-of upper S Subscript n Baseline Endscripts product Underscript i equals 1 Overscript n Endscripts a Subscript i sigma left-parenthesis i right-parenthesis tau left-parenthesis i right-parenthesis Baseline comma">

per
⁡
A
=

∑

σ
,
τ
∈

S
n

∏

i
=
1

n

a

i
σ
(
i
)
τ
(
i
)

,

\operatorname {per} A = \sum \limits _{\sigma ,\tau \in {S_n}} {\prod \limits _{i = 1}^n {{a_{i\sigma (i)\tau (i)}},} }

\]
where

S
n

{S_n}

is the symmetric group on

{
1
,
2
,
…
,
n
}

\{ 1,2, \ldots ,n\}

. Suppose that

A
A

is a (0,1)-matrix and that

r
i

=

∑

j
,
k
=
1

n

a

i
j
k

 for 

i
=
1
,
2
,
…
,
n

{r_i} = \sum \nolimits _{j,k = 1}^n {{a_{ijk}}} {\text { for }}i = 1,2, \ldots ,n

. In this paper it is shown that per

A
≤

∏

i
=
1

n

r
i

!

1

/

r
i

.

A \leq \prod \nolimits _{i = 1}^n {{r_i}{!^{1/{r_i}}}.}

A similar bound is then obtained for a second function, the

2
2

-permanent of a

3
3

-dimensional matrix, that is another analogue of the permanent of an ordinary (

2
2

-dimensional) matrix.
The Jantzen filtration of a certain class of Verma modules
Authors: Ku Jong Min
Published: June 2010

Let

G
=

N
+

⊕
H
⊕

N
−

G = {N_ + } \oplus H \oplus {N_ - }

be a Kac-Moody Lie algebra. For each

M
M

in the category

O

\mathcal {O}

of

G
G

-modules, there is a filtration

(

M
i

)

i
≥
0

{({M_i})_{i \geq 0}}

by

G
G

-submodules of

M
M

naturally associated with the set

{

υ
∈
M

|

N
+

υ

=
0

}

\left \{ {\upsilon \in M\left | {{N_ + }\upsilon } \right . = 0} \right \}

. If

G
G

is symmetrizable and

M
M

is a Verma module,

M
i

=

M
i

{M_i} = {M^i}

for all

i
i

if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper M colon upper L left-parenthesis mu right-parenthesis right-bracket equals dimension upper H o m Subscript upper G Baseline left-parenthesis upper M left-parenthesis mu right-parenthesis comma upper M right-parenthesis">

[

M
:
L
(
μ
)

]

=
dim
⁡

Hom
G

⁡
(
M
(
μ
)
,
M
)

\left [ {M:L(\mu )} \right ] = \dim \operatorname {Hom}_G(M(\mu ),M)

for all

μ
∈

H
∗

\mu \in {H^ * }

where

(

M
i

)

i
≥
0

{({M^i})_{i \geq 0}}

is the Jantzen filtration of

M
M

. The main tools used are the nondegenerate form on each

M
i

/

M

i
+
1

{M^i}/{M^{i + 1}}

together with the

Γ
\Gamma

-operator of

G
G

.
On analytic slit mappings in the class Σ
Authors: Abu-Muhanna Yusuf , Leung Y. J.
Published: June 2010

We show that any analytic slit mapping is a support point in

Σ
\Sigma

. As an application, we use Schiffer’s theory to show that a rational slit mapping must be of the form

z

 + 

a
0

+

a
1

/

(
z
−

z
1

)

z{\text { + }}{a_0} + {a_1}/(z - {z_1})

.
The fixed points of an analytic self-mapping
Authors: Fisher S. D. , Franks John
Published: June 2010

Let

R
R

be a hyperbolic Riemann surface embedded in a compact Riemann surface of genus

g
g

and let

f
f

be an analytic function mapping

R
R

into

R
,
f

R, f

not the identity function. Then

f
f

has as most

2
g
+
2

2g + 2

distinct fixed points in

R
R

; equality may hold. If

f
f

has 2 or more distinct fixed points, then

f
f

is a periodic conformal automorphism of

R
R

onto itself. This paper contains a proof of this theorem and several related results.
Asymptotic behavior of the solutions of the second order difference equation
Authors: Drozdowicz Andrzej , Popenda Jerzy
Published: June 2010

The second order difference equation (E)
\[

Δ
2

x
n

+

p
n

f
(

x
n

)
=
0

{\Delta ^2}{x_n} + {p_n}f({x_n}) = 0

\]
is considered. The results give a necessary and sufficient condition for some solution of (E) to have asymptotic behavior

x
n

∼
C
=
const

{x_n} \sim C = \text {const}

. as

n
n

approaches infinity.
A substitute of l’Hospital’s rule for matrices
Authors: Kratz W.
Published: June 2010

In this paper the following limit theorem is obtained: If

A
A

and

B
B

are

(
n
,
n
)

(n,n)

-matrices with

rank

(

A
T

,

B
T

)
=
n
,

A
T

B
=

B
T

A

{\text {rank}}({A^T},{B^T}) = n,\;{A^T}B = {B^T}A

, then

A

(
A
+
S
B

)

−
1

S
→
0

A{(A + SB)^{ - 1}}S \to 0

as

S
→
0
+
,

i

.e

.

S

→

0

S \to 0 + ,\;{\text {i}}{\text {.e}}{\text {.}}\;{\text {S}} \to {\text {0}}

where

S
S

is symmetric and positive definite. Some applications of this result are given to linear algebra (the behavior of

(
A
+
λ
B

)

−
1

{(A + \lambda B)^{ - 1}}

as

λ
→
0
)

\lambda \to 0)

and to differential equations (the asymptotic behavior of Hamiltonian systems and of selfadjoint differential equations of even order).
Relative trace formula and simple algebras
Authors: Flicker Yuval Z.
Published: June 2010

A Deligne-Kazhdan variant of the relative trace formula of Jacquet-Lai is given, thus generalizing the study of distinguished representations from the context of the quaternion algebra to that of any simple algebra.
Some characterizations of trivial parts for 𝐻^{∞}(𝐷)
Authors: Abram Thomas J. , Weiss Max L.
Published: June 2010

The unit disc in the complex plane is made into a locally compact topological group. This group acts as a transformation group on the maximal ideal space of the Banach algebra of bounded analytic functions on the disc. Among other characterizations the trivial parts are shown to be the minimal closed invariant sets of this transformation group. A point in the maximal ideal space is a trivial part if and only if it is the limit of a maximal invariant filter. An example shows that the correspondence between such points and filters is not one-to-one.
On the accumulation of limit cycles
Authors: Perko L. M.
Published: June 2010

Limit cycles of a planar analytic system can only accumulate at a critical point or on a graphic; the critical points are isolated; and the inner and outer boundaries of any continuous band of cycles consist of either a critical point or graphic.
A new proof of the Grauert direct image theorem
Authors: Levy R.
Published: June 2010

A new proof of the Grauert direct image theorem based on methods of functional analysis is given.
A note on a selection theorem
Authors: Michael E.
Published: June 2010

A recent selection theorem of O. N. Kolesnikov is generalized and its proof is simplified.
On the existence of exactly (2,1) maps
Authors: Smithson R. E.
Published: June 2010

The following two theorems concerning the existence of exactly 2 to 1 maps are proved. If

Y
Y

is a continuum such that each nondegenerate subcontinuum contains a cutpoint, then there does not exist a continuum

X
X

and an exactly 2 to 1 map on

X
X

onto

Y
Y

. Further, if

X
X

is an arcwise connected continuum and

Y
Y

is a nested continuum, then there does not exist an exactly 2 to 1 map on

X
X

onto

Y
Y

.
Smoothing curves in 𝑃³ with 𝑝ₐ=1
Authors: Sánchez Carmen A.
Published: June 2010

In [3] Tannenbaum proved that every connected, reduced curve in

P
3

{P^3}

of arithmetic genus 0 may be smoothed. Here we prove, using results of Hartshorne and Hirschowitz [1], that every connected, reduced curve in

P
3

{P^3}

of arithmetic genus 1 is also smoothable.
A structure theorem for completely regular semigroups
Authors: Petrich Mario
Published: June 2010

Structure theorems for arbitrary completely regular semigroups have been given by Yamada, Warne, Petrich, and Clifford. A new structure theorem for these semigroups is proved here. It is reminiscent of both the structure theorem of Clifford-Petrich and of the Rees construction for completely simple semigroups. It simplicity ought to prove useful in the study of various aspects of completely regular semigroups.
The row rank of a subring of a matrix ring
Authors: Li M. S. , Zelmanowitz J. M.
Published: June 2010

For

R
R

a subring of an

n
×
n

n \times n

matrix ring

M
n

(
Δ
)

{M_n}(\Delta )

over a division ring

Δ
\Delta

, we examine an invariant called the row rank of

R
R

. Roughly speaking, the row rank of

R
R

is the largest integer

k
k

such that

R
R

contains all

k
k

-rowed matrices over a left order in

Δ
\Delta

. The row rank of

R
R

is then an integer between 0 and

n
n

; and we will see that row rank

R
≥
1

R \geq 1

means that

M
n

(
Δ
)

{M_n}(\Delta )

is the maximal left quotient ring of

R
R

, while row rank

R
=
n

R = n

signifies that

M
n

(
Δ
)

{M_n}(\Delta )

is the classical left quotient ring of

R
R

. Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings

R
R

with row rank

R
≥
k

R \geq k

is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank

R
=
n

R = n

and row rank

R
≥
1

R \geq 1

.
Oscillatory and periodic solutions of delay differential equations with piecewise constant argument
Authors: Aftabizadeh A. R. , Wiener Joseph , Xu Jian-Ming
Published: June 2010

Oscillatory and periodic solutions of retarded functional differential equations are investigated. The study concerns equations with piecewise constant arguments which found applications in certain biomedical problems.
A probability distribution associated with the Hurwitz zeta function
Authors: Duncan R. L.
Published: June 2010

A new discrete probability distribution is defined in terms of the Hurwitz zeta function and its relationship to the Poisson distribution is demonstrated.
A note on cocycles in von Neumann algebras
Authors: Saito Kichi-Suke
Published: June 2010

In this note, we study the comparison theory for cocycles in von Neumann algebras. In particular, we investigate quasi-equivalent cocylces.
A reduction of algebraic representations of matroids
Authors: Lindström Bernt
Published: June 2010

We prove the following result conjectured by M. J. Piff in his thesis (1972). Theorem. Let

M
M

be a matroid with an algebraic representation over a field

F
(
t
)

F(t)

, where

t
t

is transcendental over

F
F

. Then

M
M

has an algebraic representation over

F
F

. The proof depends on Noether’s normalization theorem and the place extension theorem. We obtain the following corollary. Corollary. If a matroid is algebraic over a field

F
F

, then any minor of

M
M

is algebraic over

F
F

.
A translation principle for Kac-Moody algebras
Authors: Neidhardt Wayne
Published: June 2010

Let

g

\mathfrak {g}

be a Kac-Moody algebra defined by a symmetrizable generalized Cartan matrix. We show that the multiplicity of the irreducible module

L
(

w
1

⋅
λ
)

L({w_1} \cdot \lambda )

in the Verma module

M
(

w
2

⋅
λ
)

M({w_2}\cdot \lambda )

depends only on the elements

w
1

{w_1}

and

w
2

{w_2}

of the Weyl group, and not on the dominant integral weight

λ
\lambda

, generalizing the translation principle of Jantzen for finite-dimensional algebras.
Polynomials of an inner function which are exposed points in 𝐻¹
Authors: Inoue Jyunji , Nakazi Takahiko
Published: June 2010

It is known that if

p

(
z
)

p\left ( z \right )

is an analytic polynomial which has no zeros in the open unit disc and distinct zeros in the unit circle, then

p

(
z
)

/

‖

p

(
z
)

‖

1

p\left ( z \right )/{\left \| {p\left ( z \right )} \right \|_1}

is an exposed point of the unit ball of the Hardy space

H
1

{H^1}

. In this paper, it is proved that for a bounded analytic function

f
f

with

‖
f
‖

∞

⩽
1

{\left \| f \right \|_\infty } \leqslant 1

,

p

(
f
)

/

‖

p

(
f
)

‖

1

p\left ( f \right )/{\left \| {p\left ( f \right )} \right \|_1}

is also an exposed point.
On the converse of the nonexpansive map fixed point theorem for Hilbert space
Authors: Sine Robert
Published: June 2010

A simple proof is given of Ray’s theorem that a nonempty, closed, convex, and unbounded set in Hilbert space admits a fixed point free nonexpansive map.
Operators on Banach lattices and the Radon-Nikodým theorem
Authors: Feldman William
Published: June 2010

For positive operators between Banach lattices a concept of absolute continuity is considered. On an AM-space with order unit norm,

S
S

is shown to be absolutely continuous with respect to

T
T

if and only if there is an approximation of

S
S

by finite sums of operators of the type

q
∘
T
∘
h

q \circ T \circ h

where

h
h

and

q
q

are multiplication operators or orthomorphisms. Given

T
T

compact, compactness of

S
S

is characterized.
Infinitely many radially symmetric solutions to a superlinear Dirichlet problem in a ball
Authors: Castro Alfonso , Kurepa Alexandra
Published: June 2010

In this paper we show that a radially symmetric superlinear Dirichlet problem in a ball has infinitely many solutions. This result is obtained even in cases of rapidly growing nonlinearities, that is, when the growth of the nonlinearity surpasses the critical exponent of the Sobolev embedding theorem. Our methods rely on the energy analysis and the phase-plane angle analysis of the solutions for the associated singular ordinary differential equation.
A theorem on weighted 𝐿¹-approximation
Authors: Schmidt Darrell
Published: June 2010

It is proven that the

A
A

-property is necessary for a finitedimensional subspace to be Chebyshev in

C
(
K
)

C(K)

with respect to the weighted

L
1

{L^1}

-norm

|

|

f

|

|

w

=

∫
K

w

|

f

|

d
μ

||f|{|_w} = \int _K {w|f|d\mu }

for all weight functions

w
w

in certain classes of functions.
𝐴≥𝐵≥0 assures (𝐵^{𝑟}𝐴^{𝑝}𝐵^{𝑟})^{1/𝑞}≥𝐵^{(𝑝+2𝑟)/𝑞} for 𝑟≥0, 𝑝≥0, 𝑞≥1 with (1+2𝑟)𝑞≥𝑝+2𝑟
Authors: Furuta Takayuki
Published: June 2010

An operator means a bounded linear operator on a Hilbert space. This paper proves the assertion made in its title. Theorem 1 yields the famous result that

A
≥
B
≥
0

A \geq B \geq 0

assures

A
α

≥

B
α

{A^\alpha } \geq {B^\alpha }

for each

α
∈
[
0
,
1
]

\alpha \in [0,1]

when we put

r
=
0

r = 0

in Theorem 1. Also Corollary 1 implies that

A
≥
B
≥
0

A \geq B \geq 0

assures

(
B

A
p

B

)

1

/

p

≥

B

(
p
+
2
)

/

p

{(B{A^p}B)^{1/p}} \geq {B^{(p + 2)/p}}

for each

p
≥
1

p \geq 1

and this inequality for

p
=
2

p = 2

is just an affirmative answer to a conjecture posed by Chan and Kwong. We cite three counterexamples related to Theorem 1 and Corollary 1.
Weighted norm inequalities for the Fourier transform on certain totally disconnected groups
Authors: Quek T. S.
Published: June 2010

Let

G
G

be a locally compact totally disconnected Abelian group with dual group

Γ
\Gamma

. Let

U
U

and

V
V

be nonnegative measurable functions on

Γ
\Gamma

and

G
G

, respectively. In this paper we give, in terms of

U
U

and

V
V

, a necessary condition and some sufficient conditions for the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue ModifyingAbove f With caret upper U StartAbsoluteValue EndAbsoluteValue Subscript q Baseline less-than-or-equal-to upper C StartAbsoluteValue EndAbsoluteValue f upper V StartAbsoluteValue EndAbsoluteValue Subscript p">

|

|

f
^

U

|

|

q

≤
C

|

|

f
V

|

|

p

||\hat fU|{|_q} \leq C||fV|{|_p}

to hold for all

f
f

in

L
1

(
G
)

{L_1}\left ( G \right )

, where

f
^

\hat f

denotes the Fourier transform of

f
f

and

1
>
p
≤
q
>
∞

1 > p \leq q > \infty

. If

U
U

and

V
V

are both radial, we give a necessary and sufficient condition for the above norm inequality to hold for all

f
f

in

L
1

(
G
)

{L_1}\left ( G \right )

.
On the intersection of varieties with a totally real submanifold
Authors: Jacobowitz Howard
Published: June 2010

In their work on uniqueness in the Cauchy problem for CR functions, Baouendi and Treves have utilized a condition on a totally real submanifold

M
M

in a neighborhood of one of its points

p
p

: There should exist a variety

X
X

such that the component containing

p
p

of

M
−

(

M
∩
X

)

M - \left ( {M \cap X} \right )

has compact closure in

M
M

. All real analytic submanifolds satisfy this condition. In this paper, a

C
∞

{C^\infty }

submanifold is constructed which does not. Uniqueness in the corresponding Cauchy problem remains unresolved.
Orthogonal complex structures on 𝑆⁶
Authors: LeBrun Claude
Published: June 2010

A complex structure on the six-sphere is called orthogonal if the standard metric is Hermitian with respect to it. While such structures locally exist in profusion, there is no such complex structure on the entire sphere.
A simple construction of Robinson equivalence relations
Authors: Mundici Daniele
Published: June 2010

We present the simplest constructions of nontrivial Robinson equivalence relations.
An adjoint representation for polynomial algebras
Authors: Mitchell Stephen A. , Stong Robert E.
Published: June 2010

This paper shows that a graded polynomial algebra over

F
2

{F_2}

with Steenrod algebra action possesses an analog of the adjoint representation for the cohomology of the classifying space of a compact connected Lie group.
Multicoherence of spaces of the form 𝑋/𝑀
Authors: Illanes M. Alejandro
Published: June 2010

Let

X
X

be a connected, locally connected, normal

T
1

{T_1}

-space and let

M
M

be a closed connected, locally connected subspace of

X
X

. Suppose that

X

/

M

X/M

denotes the space obtained by identifying

M
M

in a single point, and that, for a connected space

Y
Y

,

ı
(
Y
)

\imath (Y)

denotes the multicoherence degree of

Y
Y

. In this paper, we prove that if

M
M

is unicoherent, then

ı
(
X
)
=
ı
(
X

/

M
)

\imath (X) = \imath (X/M)

. As an application of this result we prove that if

X
=
A
∪
B

X = A \cup B

, where

A
,
B

A,B

are closed subsets of

X
X

and

A
∩
B

A \cap B

is connected, locally connected and unicoherent, then

ı
(
X
)
=
ı
(
A
)
+
ı
(
B
)

\imath (X) = \imath (A) + \imath (B)

. Also, we prove that if

X

/

M

X/M

is unicoherent, then

ı
(
X
)
⩽
ı
(
M
)

\imath (X) \leqslant \imath (M)

.
On the quadratic subfield of a 𝑍₂-extension of an imaginary quadratic number field
Authors: Endô Akira
Published: June 2010

We determine explicitly the quadratic subfield of a noncyclotomic

Z
2

{Z_2}

-extension of an imaginary quadratic number field and get a congruence property of the integer solution of a certain indeterminate equation.
Maps which preserve graphs
Authors: Nall Van C.
Published: June 2010

In 1976 Eberhart, Fugate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent. In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.
Ultrametrically injective spaces
Authors: Bayod José M. , Martínez-Maurica J.
Published: June 2010

We study the injective objects in the category of ultrametric spaces with contractive mappings. As a result, the property of spherical completeness is characterized in several new ways related to extensions of uniformly continuous mappings and to extensions of isometries. A construction of the injective envelope is also given.
Integer parts of powers of quadratic units
Authors: Cass Daniel
Published: June 2010

Let

α
>
1

\alpha > 1

be a unit in a quadratic field. The integer part of

α
n

{\alpha ^n}

, denoted

[

α
n

]

[{\alpha ^n}]

, is shown to be composite infinitely often. Provided

α
≠
(
1
+

5

)

/

2

\alpha \ne (1 + \sqrt 5 )/2

, it is shown that the number of primes among

[
α
]
,
[

α
2

]
,
…
,
[

α
n

]

[\alpha ],[{\alpha ^2}], \ldots ,[{\alpha ^n}]

is bounded by a function asymptotic to

c
⋅

log
2

n

c \cdot {\log ^2}n

, with

c
=
1

/

(
2
log
⁡
2
⋅
log
⁡
3
)

c = 1/(2\log 2 \cdot \log 3)

.
Exact sequences for generalized Toeplitz operators
Authors: Sundberg Carl
Published: June 2010

Let

T

\mathcal {T}

be the

C
∗

{C^*}

-algebra generated by the Toeplitz operators on

H
2

{H^2}

of the unit circle, and let

C
C

be the

T

\mathcal {T}

-ideal generated by

{

T
φ

T
ψ

−

T

φ
ψ

:
φ
,
ψ
∈

L
∞

}

\{ {T_\varphi }{T_\psi } - {T_{\varphi \psi }}:\varphi ,\psi \in {L^\infty }\}

. It is well known that

T

/

C

\mathcal {T} / C

is naturally

∗
*

-isomorphic to

L
∞

{L^\infty }

. Several authors have obtained a similar result for other classes of Toeplitz operators. In the present paper a general theorem is proved which establishes the relevant isomorphism for a wide class of generalized Toeplitz operators.
An indecomposable Daniell integral
Authors: Jefferies B. R. F. , Fremlin D. H.
Published: June 2010

An example of an indecomposable Daniell integral is given.
A sharp estimate for dyadic martingales with multiple indices
Authors: Morrow Gregory J.
Published: June 2010

A variant of Doob’s maximal inequality is obtained for dyadic martingales with multiple indices. The inequality furnishes a precise estimate of the

L
p

{L^p}

norm of the maximal function in terms of the

L
p

{L^p}

norms of the jumps,

p
≥
2

p \geq 2

.
Strong limit theorems for blockwise 𝑚-dependent and blockwise quasi-orthogonal sequences of random variables
Authors: Móricz F.
Published: June 2010

Let

{

X
k

:
k
=
1
,
2
,
…
}

\{ {X_k}:k = 1,2, \ldots \}

be a sequence of random variables with zero mean and finite variance

σ
k
2

\sigma _k^2

. We say that

{

X
k

}

\{ {X_k}\}

is blockwise

m
m

-dependent if for each

p
p

large enough the following is true: if we remove

m
m

or more consecutive

X
X

’s from the dyadic block

{

X

2

p
−
1

+
1

,
…
,

X

2
p

}

\{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\}

, then the two remaining portions are independent. We say that

{

X
k

}

\{ {X_k}\}

is blockwise quasiorthogonal if for each

p
p

, the expectations

E
(

X
k

X
l

)

E({X_k}{X_l})

are small in a certain sense again within the dyadic block

{

X

2

p
−
1

+
1

,
…
,

X

2
p

}

\{ {X_{{2^{p - 1}} + 1}}, \ldots ,{X_{{2^p}}}\}

. Blockwise independence and blockwise orthogonality are particular cases of the above notions, respectively. We study the a.s. behavior of the series

∑

k
=
1

∞

X
k

\sum \nolimits _{k = 1}^\infty {{X_k}}

and that of the first arithmetic means

(
1

/

n
)

∑

k
=
1

n

X
k

(1/n)\sum \nolimits _{k = 1}^n {{X_k}}

. It turns out that the classical strong limit theorems, with one exception, remain valid in this more general setting, too.
Partitioning topological spaces into countably many pieces
Authors: Komjáth P. , Weiss W.
Published: June 2010

We assume

X
→
(
top
⁡
ω
+
1

)
ω
1

X \to (\operatorname {top}\omega + 1)_\omega ^1

and determine which larger

α
\alpha

can replace

ω
+
1

\omega + 1

. If

X
X

is first countable, any countable

α
\alpha

can replace

ω
+
1

\omega + 1

. If the character of

X
X

is

ω
1

{\omega _1}

, it is consistent and independent whether

ω
2

+
1

{\omega ^2} + 1

can always replace

ω
+
1

\omega + 1

. Consistently

ω
1

{\omega _1}

cannot replace

ω
+
1

\omega + 1

for any

X
X

of size

ω
1

{\omega _1}

.
On a family of elliptic surfaces with Mordell-Weil rank 4
Authors: Schwartz Charles F.
Published: June 2010

In this paper, we find bases for the Mordell-Weil groups of a family of elliptic surfaces. In particular, let

E

(
a
,
b
)

→
B

{E_{(a,b)}} \to B

be the elliptic surface given by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y squared equals 4 left-bracket x cubed minus sigma-summation Underscript i equals 0 Overscript 2 Endscripts a Subscript i Baseline u Superscript i Baseline x plus sigma-summation Underscript j equals 0 Overscript 3 Endscripts b Subscript j Baseline u Superscript j Baseline right-bracket period">

y
2

=
4

[

x
3

−

∑

i
=
0

2

a
i

u
i

x
+

∑

j
=
0

3

b
j

u
j

]

.

{y^2} = 4\left [ {{x^3} - \sum \limits _{i = 0}^2 {{a_i}{u^i}x + } \sum \limits _{j = 0}^3 {{b_j}{u^j}} } \right ].

\]
If the elliptic surface has Mordell-Weil rank 4 over

C

{\mathbf {C}}

, then we find a basis

{

σ
i

=
(

x
i

,

y
i

)

|

1
≤
i
≤
4
}

\{ {\sigma _i} = ({x_i},{y_i})|1 \leq i \leq 4\}

with

x
i

{x_i}

and

y
i

{y_i}

, linear in

u
u

. We do this by finding a parametrization of this family of elliptic surfaces; furthermore, if the parameters are rational numbers, then the Mordell-Weil group is rational over

Q

{\bf {Q}}
Vanishing theorems for 𝑉-manifolds
Authors: Arapura Donu
Published: June 2010

Generalizations to Kähler

V
V

-manifolds, are given for the Kodaira-Nakano-Sommese and Le Potier vanishing theorems. The proof uses Hodge theory for open varieties.
Counterexamples involving growth series and Euler characteristics
Authors: Parry Walter
Published: June 2010

This note presents examples for which the value of a finitely generated group’s growth series at 1 is not the reciprocal of the group’s Euler characteristic.
On normality of Blaschke quotients
Authors: Colonna Flavia
Published: June 2010

In this paper we give conditions on the zeros and poles of a Blaschke quotient in order to obtain a normal meromorphic function, extending the condition for normality given by Cima and Colwell to the case of zeros and poles forming interpolating sequences and of bounded multiplicities. It is also shown that a similar characterization does not apply in the case of unbounded multiplicities.
Multiplicities of the eigenvalues of the discrete Schrödinger equation in any dimension
Authors: Burghelea Dan , Kappeler Thomas
Published: June 2010

The following von Neumann-Wigner type result is proved: The set of potentials

a
:

Γ
→

R

(
Γ
⊆

Z

N

)

a:\;\Gamma \to {\mathbf {R}}(\Gamma \subseteq {{\mathbf {Z}}^N})

, with the property that the corresponding discrete Schrödinger equation

Δ
d

+
a

{\Delta _d} + a

has multiple eigenvalues when considered with certain boundary conditions, is an algebraic set of

codimension

≥

2

{\text {codimension}} \geq {\text {2}}

within

R

Γ

{{\mathbf {R}}^\Gamma }

.
Extensions of certain compact operators on vector-valued continuous functions
Authors: Khurana Surjit Singh
Published: June 2010

For any compact Hausdorff spaces

X
,
Y

X,Y

with

φ
:

X
→
Y

\varphi :\;X \to Y

a continuous onto mapping,

E
,
F

E,F

, Hausdorff locally convex spaces with

F
F

complete,

C
(
X
,
E
)

(
C
(
Y
,
E
)
)

C(X,E)\;(C(Y,E))

all

E
E

-valued continuous functions on

X
(
Y
)

X(Y)

, and

L
:
C
(
Y
,
E
)
→
F

L:C(Y,E) \to F

a

T

\mathcal {T}

-compact continuous operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sigma left-parenthesis upper F comma upper F Superscript prime Baseline right-parenthesis less-than-or-equal-to script upper T less-than-or-equal-to tau left-parenthesis upper F comma upper F prime right-parenthesis right-parenthesis">

(
σ
(
F
,

F
′

)
≤

T

≤
τ
(
F
,

F
′

)
)

(\sigma (F,F’) \leq \mathcal {T} \leq \tau (F,F’))

, it is proved there exists a

T

\mathcal {T}

-compact continuous operator

L
0

:
C
(
X
,
E
)
→
F

{L_0}:C(X,E) \to F

such that

L
0

(
f
∘
φ
)
=
L
(
f
)

{L_0}(f \circ \varphi ) = L(f)

for every

f
∈
C
(
Y
,
E
)

f \in C(Y,E)

.
An estimate for certain meromorphic univalent functions
Authors: Xie Ming Qin
Published: June 2010

In this paper the coefficient problem for the family of univalent functions
\[

g

(
z
)

=
z
+

∑

n
=
1

∞

b
n

z

−
n

g\left ( z \right ) = z + \sum \limits _{n = 1}^\infty {{b_n}{z^{ - n}}}

\]
in

{

|
z
|

>
1
}

\{ \left | z \right | > 1\}

has been studied. The author obtained the sharp estimate

b
7

≤
1

/

4
+
3

/

280

{b_7} \leq 1/4 + 3/280

when

g
(
z
)

g(z)

is an odd function and all its coefficients are real.
Weighted norm inequalities for multipliers
Authors: Gutiérrez Cristian E.
Published: June 2010

We consider the two-weight function problem for a class of multiplier operators that include the Riesz and Bessel potentials.
Decidable sentences over polynomial rings
Authors: Tung Shih Ping
Published: June 2010

Let

R
R

be an algebraic number field or an algebraic integer ring. We prove that there is an algorithm to determine whether the sentence

∀
x
∃
y
ϕ

(

x
,
y

)

\forall x\exists y\phi \left ( {x,y} \right )

, with

ϕ

(

x
,
y

)

\phi \left ( {x,y} \right )

a quantifier free formula over

R

[
T
]

R\left [ T \right ]

, is true in the polynomial ring

R

[
T
]

R\left [ T \right ]

or not.
The separable representations of 𝑈(𝐻)
Authors: Pickrell Doug
Published: June 2010

In this paper we show that the separable representation theory of

U(

H
)

{\text {U(}}H)

is completely analogous to that for

U(

C

n

)

{\text {U(}}{{\mathbf {C}}^n})

, in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of

H
H

. This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup

U(

H

)
∞

{\text {U(}}H{)_\infty }

, consisting of operators which are compact perturbations of the identity, by Kirillov and Ol’shanskii. In particular we show that all nontrivial representations of the unitary Calkin group are nonseparable. The proof exploits the analogue of the following fact about the Calkin algebra: if

π
\pi

is a nontrivial representation of the Calkin algebra and

T
T

is a normal operator on

H
H

, then every point in the spectrum of

π
(
T
)

\pi (T)

is an eigenvalue.
Unstable compositions related to the image of 𝐽
Authors: Mahowald Mark , Thompson Robert
Published: June 2010

We study elements in the homotopy groups of spheres which are desuspensions of generators of the image of the

J
J

-homomorphism for

p
=
2

p = 2

.

J
∗

(

R

P
∞

)

{J_*}({\mathbf {R}}{P^\infty })

is used to detect the nontriviality of compositions and Toda brackets involving these elements. Some formulas are derived which show how various compositions of these elements are related to one another.
On forbidden minors for 𝐺𝐹(3)
Authors: Kahn Jeff , Seymour Paul
Published: June 2010

A new, surprisingly simple proof is given of the finiteness of the set of matroids minor-minimally not representable over

G
F
(
3
)

GF(3)

. It is, in fact, proved that every such matroid has rank or corank at most 3.
On the Frobenius endomorphisms of Fermat and Artin-Schreier curves
Authors: Coleman Robert F.
Published: June 2010

The following article offers an explanation of the relationship of Jacobi and Gauss sums to Fermat and Artin-Schreier curves which is an analogue of the proof of Stickleberger’s theorem.
A note on the minimum property
Authors: Shafrir Itai
Published: June 2010

It is shown that a Banach space

E
E

has a strictly convex dual if and only if for every accretive operator

A
⊂
E
×
E

A \subset E \times E

that satisfies the range condition at infinity,

cl

(
R
(
A
)
)

{\text {cl}}(R(A))

has the minimum property.
𝑄-sets do not necessarily have strong measure zero
Authors: Ihoda Jaime , Shelah Saharon
Published: June 2010

The purpose of this paper is to give a negative answer to the following question (see Miller [4]): Do all

Q
Q

-sets have strong measure zero?
A quadratic system with a nonmonotonic period function
Authors: Chicone Carmen , Dumortier Freddy
Published: June 2010

For a certain

c
∗

>
1.4

{c_ * } > 1.4

and

c
∈

(

1.4
,

c
∗

)

c \in \left ( {1.4,{c_ * }} \right )

the quadratic system

x
˙

=
−
y
+
x
y
,

y
˙

=
x
+
2

y
2

−
c

x
2

\dot x = - y + xy,\dot y = x + 2{y^2} - c{x^2}

has a center at the origin surrounded by a one-parameter family of periodic trajectories. We show the period is not a monotone function of the parameter.
Sums of incompressible surfaces
Authors: Oertel Ulrich
Published: June 2010

In an orientable Haken

3
3

-manifold, given two orientable incompressible surfaces

F
F

and

S
S

transverse to each other with intersections suitably simplified, certain cut-and-paste operations along curves of intersection yield embedded incompressible surfaces. We show in this paper that, no matter how

F
F

and

S
S

are isotoped, as long as intersections are suitably simplified, exactly the same finite (possibly empty) set of isotopy classes of incompressible surfaces result from cut-and-paste operations.
The set of balanced points with respect to 𝑆¹ and 𝑆³ actions of maps into Banach space
Authors: Mramor-Kosta Neža
Published: June 2010

Let

G
G

be the group of units in the field

F
F

, which is either

R
R

,

C
C

or

H
H

, let

X
X

be a free

G
G

-space, and let

f
f

be a map from

X
X

to a Banach space

E
E

over

F
F

. In this paper we give an estimate for the size of the subset of

X
X

consisting of points at which the average of

f
f

is equal to zero. The result represents an extension of the Borsuk-Ulam-Yang theorem.
Number versus size
Authors: Watson Stephen
Published: June 2010

We show, for example, that normal collectionwise Hausdorff spaces which are collectionwise normal with respect to families of cardinality

ℵ
1

{\aleph _1}

are collectionwise normal with respect to any number of copies of

[
0
,
1
]

[0,1]

. We also show that normal collectionwise Hausdorff spaces are collectionwise normal with respect to scattered paracompact sets.
Greedily partitioning the natural numbers into sets free of arithmetic progressions
Authors: Gerver Joseph , Propp James , Simpson Jamie
Published: June 2010

We describe a "greedy" algorithm for partitioning the natural numbers into sets free of arithmetic progressions of length 3. A recursive formula governing the resulting partition is proved, and some features of its asymptotic behavior are discussed.
Relation between growth and regularity of solutions of hypoelliptic equations
Authors: Shafii-Mousavi M. , Zielezny Z.
Published: June 2010

For a class of linear partial differential equations with variable coefficients, it is shown that the Gevrey regularity of solutions depends on their growth at infinity.
Joint continuity of measurable biadditive mappings
Authors: Christensen Jens Peter Reus , Fischer Pal
Published: June 2010

The main result of this paper is the following theorem. If

G
1

,

G
2

{G_1},{G_2}

and

G
3

{G_3}

are abelian Polish groups and

C
:

G
1

×

G
2

→

G
3

C:{G_1} \times {G_2} \to {G_3}

is a Christensen measurable biadditive mapping, then

C
C

is jointly continuous.
A remark on angular complex dilatations of quasiconformal mappings
Authors: Fehlmann Richard
Published: June 2010

By a theorem of Ortel an angular complex dilatation

κ
\kappa

is extremal iff it is Teichmüllèr (quadratic differential with finite norm) or if it satisfies an integral condition involving the angular limits

λ
x

(
ϑ
)

{\lambda _x}(\vartheta )

. We show that this second case occurs iff

λ
x

(
ϑ
)

{\lambda _x}(\vartheta )

can be given explicitly at a certain point

x
x

, namely by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript x Baseline left-parenthesis theta right-parenthesis equals double-vertical-bar kappa double-vertical-bar Subscript normal infinity Baseline e Superscript 2 i Baseline left-parenthesis theta minus theta 0 right-parenthesis">

λ
x

(
ϑ
)
=

‖
κ
‖

∞

e

2
i

(
ϑ
−

ϑ
0

)

{\lambda _x}(\vartheta ) = {\left \| \kappa \right \|_\infty }{e^{2i}}(\vartheta - {\vartheta _0})

. Moreover, we investigate this statement under the weaker condition of angularity when the uniformity part in its definition is dropped.
On almost periodic solutions of the competing species problems
Authors: Ahmad Shair
Published: June 2010

This paper considers the two-dimensional Volterra-Lotka competition equations which are almost periodic in time. Conditions for the existence of an asymptotically stable almost periodic solution with positive components are given.
Complex convexity and finitely additive vector measures
Authors: Lešnjak Gorazd
Published: June 2010

We prove that complex uniform convexity of a complex normed space

X
X

is equivalent to complex uniform convexity of

bva
⁡

(

F

,X)

\operatorname {bva} {\text {(}}\mathcal {F}{\text {,X)}}

. Similar results hold also for complex uniform convexity at a point. Some consequences for duals of certain operator spaces are discussed.
Schrödinger equations: pointwise convergence to the initial data
Authors: Vega Luis
Published: June 2010

Let

u
(
x
,
t
)

u(x,t)

be the solution of the Schrödinger equation with initial data

f
f

in the Sobolev space

H
s

(

R

n

)

{H^s}({{\mathbf {R}}^n})

with

s
>

1
2

s > \frac {1}{2}

. The a.e. convergence of

u
(
x
,
t
)

u(x,t)

to

f
(
x
)

f(x)

follows from a weighted estimate of the maximal function

u
∗
(
x
,
t
)
=

su

p

t
>
0

|

u
(
x
,
t
)

|

u * (x,t) = {\text {su}}{{\text {p}}_{t > 0}}|u(x,t)|

.
Operator theoretic characterizations of [𝐼𝑁]-groups and inner amenability
Authors: Lau Anthony To Ming , Paterson Alan L. T.
Published: June 2010

Let

G
G

be a locally compact group and

p
∈
[
1
,
∞
]

p \in [1,\infty ]

. Let

π
p

{\pi _p}

be the isometric representation of

G
G

on

L
p

(
G
)

{L_p}(G)

given by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript p Baseline left-parenthesis x right-parenthesis f left-parenthesis t right-parenthesis equals f left-parenthesis x Superscript negative 1 Baseline t x right-parenthesis normal upper Delta left-parenthesis x right-parenthesis Superscript 1 slash p">

π
p

(
x
)
f
(
t
)
=
f
(

x

−
1

t
x
)
Δ

(
x

)

1

/

p

{\pi _p}(x)f(t) = f({x^{ - 1}}tx)\Delta {(x)^{1/p}}

. Let

A

p
′

{\mathcal {A}’_p}

be the commutant of

A

p

{\mathcal {A}_p}

in

B
(

L
p

(
G
)
)

B({L_p}(G))

. In this paper we determine those

G
G

for which: (*)

A

p
′

{\mathcal {A}’_p}

contains a nonzero compact operator. We prove, among other things, that if

p
∈
[
1
,
∞
)

p \in [1,\infty )

, then (*) holds if and only if

G
∈
[
I
N
]

G \in [IN]

, and that if

p
=
∞

p = \infty

, then (*) holds if and only if

G
G

is inner amenable.
Some stationary subsets of 𝒫(𝜆)
Authors: Donder Hans-Dieter , Koepke Peter , Levinski Jean-Pierre
Published: June 2010

Let

κ
\kappa

and

λ
\lambda

be uncountable cardinals such that

κ
≤
λ

\kappa \leq \lambda

, and set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S left-parenthesis kappa comma lamda right-parenthesis equals left-brace upper X element-of script upper P Subscript kappa Baseline left-parenthesis lamda right-parenthesis StartAbsoluteValue EndAbsoluteValue upper X intersection kappa StartAbsoluteValue greater-than EndAbsoluteValue upper X vertical-bar right-brace">

S
(
κ
,
λ
)
=

{

X
∈

P

κ

(
λ
)

|

|

X
∩
κ

|

>

|

X

|

}

S(\kappa ,\lambda ) = \left \{ {X \in {\mathcal {P}_\kappa }(\lambda )|\;|X \cap \kappa | > |X|} \right \}

. We determine the consistency strength of the statement "

(

∃
λ
≥
κ

)

\left ( {\exists \lambda \geq \kappa } \right )

(

(
S
(
κ
,
λ
)

(S(\kappa ,\lambda )

is stationary in

P

κ

(
λ
)

{\mathcal {P}_\kappa }(\lambda )

)" using a new type of partition cardinals. In addition, we show that the property "

S
(
κ
,

κ
+

)

S(\kappa ,{\kappa ^ + })

is stationary in

P

κ

(

κ
+

)

{\mathcal {P}_\kappa }({\kappa ^ + })

" is much stronger.