bluuebunny/crossref_metadata_2025 huggingface dataset:
Quasisimilar operators in the commutant of a cyclic subnormal operator
Authors: Raphael Marc
Published: June 2010

Compactly supported positive regular Borel measures on the complex plane that share "certain" properties with normalized arclength measure on the boundary of the unit disk are called

m
m

-measures. Let

μ
\mu

be an

m
m

-measure and let

S
μ

{S_\mu }

be the cyclic subnormal operator of multiplication by

z
z

on the closure of the polynomials in

L
2

(
μ
)

{L^2}(\mu )

. Necessary and sufficient conditions for an operator in the commutant of

S
μ

{S_\mu }

to be quasisimilar to

S
μ

{S_\mu }

are investigated. In particular it is shown that if the Bergman shift and an operator in its commutant are quasisimilar, then they are unitarily equivalent.
Symmetric cut loci in Riemannian manifolds
Authors: Vannini W. , Rubinstein J. H.
Published: June 2010

Let

M
M

be a compact Riemannian manifold with

H
1

(
M
,
Z
)
=
0

{H_1}(M,Z) = 0

. We show that, for a point

p
∈
M

p \in M

, the cut locus and conjugate locus of

p
p

must intersect if

M
M

admits a group of isometries which fixes

p
p

and has principal orbits of codimension at most 2. This is a classical theorem of Myers [5] in the case when

M
M

has dimension 2.
A problem of Sallee on equidecomposable convex bodies
Authors: Gardner R. J.
Published: June 2010

We show that equidecomposable planar convex bodies need not be convex equidecomposable. This answers a question of Sallee. We also show that convex and scissors equidecomposability are equivalent notions for convex bodies in the plane, and include a discussion of these concepts in higher dimensions.
Remarks on the homotopy type of groups of symplectic diffeomorphisms
Authors: McDuff Dusa
Published: June 2010

Let

(
X
,
ω
)

(X,\omega )

be a symplectic manifold without boundary,

G
(
X
)

G(X)

the identity component of its group of compactly supported diffeomorphisms, and

H
ω

(
X
)

{H_\omega }(X)

the subgroup of

G
(
X
)

G(X)

consisting of all symplectic diffeomorphisms. In this note, we give examples in which

H
ω

(
X
)

{H_\omega }(X)

is not homotopy equivalent to

G
(
X
)

G(X)

.
On |\overline𝑁,𝑝_{𝑛}|_{𝑘} summability factors
Authors: Bor Hüseyin
Published: June 2010

In this paper a theorem on

|

N
¯

,

p
n

|

k

|\overline N ,{p_n}{|_k}

summability factors, which generalizes the theorem of Singh [5], has been proved.
On supportless convex sets
Authors: Borwein J. M. , Tingley D. W.
Published: June 2010

We give some general constructions of supportless convex subsets of normed spaces and pose a number of open questions.
A regular counterexample to the 𝛾-space conjecture
Authors: Fox Ralph , Kofner Jacob
Published: June 2010

This paper presents a completely regular counterexample to the conjecture that every

γ
\gamma

-space is quasi-metrizable. Junnila has shown that developable

γ
\gamma

-spaces are quasi-metrizable; this example shows that "developable" cannot be replaced by "quasi-developable". In the process we provide a method for constructing non-

n
n

-pretransitive spaces.
Periodic points of positively expansive maps
Authors: Sakai Kazuhiro
Published: June 2010

Stability of fixed points of a uniformly convergent sequence of open

ε
\varepsilon

-locally expansive maps of a compact connected locally connected metric space is studied by Hu and Rosen. Under the same assumption we prove a stability result stronger than theirs.
The catenarian property of power series rings over a Prüfer domain
Authors: Arnold J. T.
Published: June 2010

Let

D
D

be a Prüfer domain that has the SFT-property. It is shown that the power series ring

D
[
[
x
]
]

D[[x]]

is catenarian. If

n
>
1

n > 1

and dim

D
>
1

D > 1

then the ring

D
[
[

x
1

,
…
,

x
n

]
]

D[[{x_1}, \ldots ,{x_n}]]

is not catenarian.
Fine embeddings of finite-dimensional subspaces of 𝐿_{𝑝},1≤𝑝<2, into 𝑙^{𝑚}₁
Authors: Schechtman Gideon
Published: June 2010

Every

m
m

-dimensional subspace of

L
p

{L_p}

,

1
>
p
>
2

1 > p > 2

,

(
1
+
ε
)

(1 + \varepsilon )

-embeds into

l
1
n

l_1^n

as long as

n
⩾
η

m

1
+
(
1

/

p
)

(
log
⁡
m

)

−
1

n \geqslant \eta {m^{1 + (1/p)}}{(\log m)^{ - 1}}

, where

η
=
η
(
p
,
ε
)
>
∞

\eta = \eta (p,\varepsilon ) > \infty

. For subspaces of

L
1

{L_1}

we get a somewhat weaker result.
High order coefficient estimates in the class Σ
Authors: Leung Y. J. , Schober G.
Published: June 2010

Estimates are given for the coefficients

b
n

{b_n}

of functions in the class

∑
\sum

in terms of

Re
⁡
{
1
−

b
1

}

\operatorname {Re} \{ 1 - {b_1}\}

. As a consequence, there is an explicit finite number

λ
\lambda

such that

Re
⁡
{
λ

b
1

−

b
n

}
⩽
λ

\operatorname {Re} \{ \lambda {b_1} - {b_n}\} \leqslant \lambda

.
The asymptotic-norming and the Radon-Nikodým properties are equivalent in separable Banach spaces
Authors: Ghoussoub N. , Maurey B.
Published: June 2010

We show that the asymptotic-norming and the Radon-Nikodym properties are equivalent, settling a problem of James and Ho [9]. In the process, we give a positive solution to two questions of Edgar and Wheeler [6] concerning Cech-complete Banach spaces. We also show that a separable Banach space with the Radon-Nikodym property semi-embeds in a separable dual whenever it has a norming space not containing an isomorphic copy of

l
1

{l_1}

. This gives a partial answer to a problem of Bourgain and Rosenthal [3].
On the Riesz-transforms along surfaces in 𝑅³
Authors: El Kohen A.
Published: June 2010

For

x
=
(

x
1

,

x
2

,

x
3

)

x = ({x_1},{x_2},{x_3})

in

R
3

{R^3}

,

t
=
(

t
1

,

t
2

)

t = ({t_1},{t_2})

in

R
2

{R^2}

,

|

t

|

=

(

t
1
2

+

t
2
2

)

1

/

2

|t| = {(t_1^2 + t_2^2)^{1/2}}

and

a
>
0

a > 0

, we define
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript a Baseline f left-parenthesis x right-parenthesis equals integral Underscript upper R squared Endscripts f left-parenthesis x 1 minus t 1 comma x 2 minus t 2 comma x 3 minus StartAbsoluteValue t EndAbsoluteValue Superscript a Baseline right-parenthesis StartFraction t 1 Over StartAbsoluteValue t EndAbsoluteValue cubed EndFraction d t period">

R
a

f
(
x
)
=

∫

R
2

f
(

x
1

−

t
1

,

x
2

−

t
2

,

x
3

−

|

t

|

a

)

t
1

|

t

|

3

d
t
.

{R_a}f(x) = \int _{{R^2}} {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})\frac {{{t_1}}}{{|t{|^3}}}dt.}

\]
The transformation

R
a

{R_a}

can be thought of as a Riesz-transform along the surface

(

t
1

,

t
2

,

|

t

|

a

)

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

in

R
3

{R^3}

. Our purpose here is to show that, for all

a
>
0

a > 0

, the operator

R
a

{R_a}

is bounded on

L
p

{L^p}

for

6

/

5
>
p
>
6

6/5 > p > 6

. Singular integrals along lower dimensional nonaffine varieties have been studied by several authors. See for example [3, 5, 6]. In the proof of our result, we use techniques similar to the ones used in [1, 2].
Paramétrisations boréliennes
Authors: Debs Gabriel
Published: June 2010

Given a Borel subset

B
B

of the product

X
×
Y

X \times Y

of two Polish spaces

X
X

and

Y
Y

such that

{
x
∈
X
:
B
(
x
)
∩
V
≠
∅
}

\{ x \in X:B(x) \cap V \ne \emptyset \}

is Borel for any open subset

V
V

of

Y
Y

, then: (1) If for any

x
∈
X

x \in X

the section

B
(
x
)

B(x)

is a dense in itself

G
δ

{G_\delta }

-subset of

Y
Y

, we prove that there exists a Borel isomorphism

f
:
X
×

N

N

→
B

f:X \times {{\mathbf {N}}^{\mathbf {N}}} \to B

such that

f
(
x
,
⋅
)

f(x, \cdot )

is one-to-one and continuous from

N

N

{{\mathbf {N}}^{\mathbf {N}}}

onto

B
(
x
)

B(x)

. (2) If for any

x
∈
X

x \in X

,

B
(
x
)

¯

\overline {B(x)}

is a

0
0

-dimensional compact space and

B
(
x
)

B(x)

is a dense in itself

G
δ

{G_\delta }

, we prove that

f
f

may be chosen such that

f
(
x
,
⋅
)

f(x, \cdot )

is a homeomorphism from

N

N

{{\mathbf {N}}^{\mathbf {N}}}

onto

G
(
x
)

G(x)

for any

x
∈
X

x \in X

.
Recognizing 𝜎-manifolds
Authors: Henderson James P.
Published: June 2010

Denote by

σ
\sigma

the subspace of the Hilbert cube consisting of

{
(

x
i

)
:

x
i

=
0

\{ (x_i): x_i = 0

for all but finitely many

i
i

}. Then following characterization of manifolds modeled on

σ
\sigma

is proven and applied to cell-like, upper semicontinuous decompositions of

σ
\sigma

-manifolds. An ANR

X
X

is a

σ
\sigma

-manifold if and only if (a)

X
X

is the countable union of finite-dimensional compacta, (b) each compact subset of

X
X

is a strong

Z
Z

-set, and (c) for each positive integer

k
k

, every mapping

f
:

R
k

→
X

f:{R^k} \to X

can be arbitrarily closely approximated by an injection.
Relative Lubin-Tate groups
Authors: de Shalit Ehud
Published: June 2010

We construct a class of formal groups that generalizes Lubin-Tate groups. We formulate the major properties of these groups and indicate their relation to local class field theory.
Some algebraic sets of high local cohomological dimension in projective space
Authors: Lyubeznik Gennady
Published: June 2010

Let

V
0

,
…
,

V

[
n

/

t
]

{V_0}, \ldots ,{V_{[n/t]}}

be algebraic sets of pure codimension

t
t

in

P
n

{P^n}

, and suppose

∩

V
i

\cap {V_i}

is empty. Then

P
n

−
∪

V
i

{P^n} - \cup {V_i}

has cohomological dimension

n
−
[
n

/

t
]

n - [n/t]

.
The sum of two Radon-Nikodým-sets need not be a Radon-Nikodým-set
Authors: Schachermayer Walter
Published: June 2010

It was shown by C. Stegall that, if

C
C

is a Radon-Nikodym-set and

K
K

weakly compact, then

K
+
C

K + C

is a Radon-Nikodym-set. We show that there are closed, bounded, convex Radon-Nikodym-sets

C
1

{C_1}

and

C
2

{C_2}

such that

C
1

+

C
2

{C_1} + {C_2}

is closed but contains an isometric copy of the unit-ball of

c
0

{c_0}

. In fact, we give two examples, one following the lines of one due to McCartney and O’Brian, the other due to Bourgain and Delbaen. We also give an easy example of a non-Radon-Nikodym-set

C
C

such that, for every

ε
>
0

\varepsilon > 0

, there is a Radon-Nikodym-set

C
ε

{C_\varepsilon }

such that

C
C

is contained in the sum of

C
ε

{C_\varepsilon }

and the ball of radius

ε
\varepsilon

.
Minimal superalgebras of weak-* Dirichlet algebras
Authors: Nakazi Takahiko
Published: June 2010

Let

A
A

be a weak-

∗
*

Dirichlet algebra in

L
∞

(
m
)

{L^\infty }(m)

and let

H
∞

(
m
)

{H^\infty }(m)

be the weak-

∗
*

closure of

A
A

in

L
∞

(
m
)

{L^\infty }(m)

. It may happen that there are minimal weak-

∗
*

closed subalgebras of

L
∞

(
m
)

{L^\infty }(m)

that contain

H
∞

(
m
)

{H^\infty }(m)

properly. In this paper it is shown that if there is a minimal, proper, weak-

∗
*

closed superalgebra of

H
∞

(
m
)

{H^\infty }(m)

, then, in fact, that algebra is the unique least element in the lattice of all proper weak-

∗
*

closed superalgebras of

H
∞

(
m
)

{H^\infty }(m)

.
Complete distributivity and ordered group lattices
Authors: Laurie Cecelia
Published: June 2010

Arveson has recently generalized an important result of Andersen’s about continuous nests to a larger class of lattices. Andersen’s result is a base for much of the recent interesting work on compact perturbations and similarity of nest algebras. This paper investigates further the structure of such lattices. It is shown that the ordered group lattices with

Σ
\Sigma

-continuous measures introduced by Arveson are completely distributive. This immediately implies various nice properties of

Alg

L

{\text {Alg}}\mathcal {L}

, the associated algebra of operators leaving such a lattice

L

\mathcal {L}

invariant. (Among these are the fact that the rank one operators in

Alg

L

{\text {Alg}}\mathcal {L}

are dense in

Alg

L

{\text {Alg}}\mathcal {L}

and that

Alg

L

+

K

{\text {Alg}}\mathcal {L} + \mathcal {K}

is norm closed where

K

\mathcal {K}

denotes the compact operators.)
Horrors of topology without 𝐴𝐶: a nonnormal orderable space
Authors: van Douwen Eric K.
Published: June 2010

In the absence of AC there can be a space which is not normal, yet which is orderable and is the topological sum of countably many compact countable metrizable spaces.
An 𝑛-dimensional subgroup of 𝑅ⁿ⁺¹
Authors: Keesling James
Published: June 2010

A construction given by R. D. Anderson and J. E. Keisler is modified to show that there exists an

n
n

-dimensional subgroup

G
G

in

R

n
+
1

{R^{n + 1}}

such that

dim
⁡

G
k

=
n

\dim {G^k} = n

for all

k
k

. The group

G
G

is connected, locally connected, and divisible.
On the classes Λ𝐵𝑉 and 𝑉[𝜈]
Authors: Avdispahić M.
Published: June 2010

We prove inclusion relations between Waterman’s and Chanturiya’s classes and point to some corollaries thereof. The situation which occurs in connection with Zygmund’s theorem for Waterman’s classes is clarified.
Pettis integrability and the equality of the norms of the weak* integral and the Dunford integral
Authors: Bator Elizabeth M.
Published: June 2010

If

(
Ω
,
∑
,
μ
)

(\Omega ,\sum ,\mu )

is a perfect finite measure space and

X
X

is a Banach space, then it is shown that

X
∗

{X^ * }

has the

μ
\mu

-Pettis Integral Property if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar left-parenthesis wea k Superscript asterisk Baseline right-parenthesis minus integral Underscript normal upper Omega Endscripts f d mu double-vertical-bar equals double-vertical-bar left-parenthesis Dunford right-parenthesis minus integral Underscript normal upper Omega Endscripts f d mu double-vertical-bar">

‖

(

wea

k

∗

)
−

∫
Ω

f
d
μ

‖

=

‖

(

Dunford

)
−

∫
Ω

f
d
μ

‖

\left \| {({\text {wea}}{{\text {k}}^ * }) - \int \limits _\Omega {fd\mu } } \right \| = \left \| {({\text {Dunford}}) - \int \limits _\Omega {fd\mu } } \right \|

\]
for every bounded weakly measurable function

f
:
Ω
→

X
∗

f:\Omega \to {X^ * }

.
The number of 𝑇₂-precompact group topologies on free groups
Authors: Remus Dieter
Published: June 2010

It is shown that every free group

F
F

admits exactly

2

2

|

F

|

T

2
−

{2^{{2^{|F|}}}}{T_{{2^ - }}}

precompact group topologies. Those topologies can even be chosen to be finer than the finite-index topology on

F
F

.
On a question of Archangelskij concerning Lindelöf spaces with countable pseudocharacter
Authors: Alster K.
Published: June 2010

We give a negative solution to Archangelskij’s problem by showing that there exists a Lindelöf space with countable pseudocharacter which does not admit a continuous one-to-one mapping onto a first countable Hausdorff space.
Weight spaces of Lie algebra modules
Authors: Smith Martha K.
Published: June 2010

Let

V
V

be a finite-dimensional module for the finite-dimensional Lie algebra

L
L

over a field of characteristic zero. If
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V Superscript lamda Baseline equals left-brace v element-of upper V vertical-bar all x element-of upper L comma left-bracket x minus lamda left-parenthesis x right-parenthesis right-bracket Superscript i Baseline v equals 0 for some right-brace">

V
λ

=
{
v
∈
V

|

all 

x
∈
L
,

[
x
−
λ
(
x
)

]
i

v
=
0

for some 

}

{V^\lambda } = \{ v \in V|\;{\text {all }}x \in L,{[x - \lambda (x)]^i}v = 0\;{\text {for some }}\}

is nonzero, then

λ
∈

L
∗

\lambda \in {L^*}

and is a character of

L
L

. Moreover, the corresponding eigenspace

{
v
∈
V

|

all 

x
∈
L
,
x
v
=
λ
(
x
)
v
}

\{ v \in V|{\text {all }}x \in L,xv = \lambda (x)v\}

is nonzero and

V
λ

{V^\lambda }

is an

L
L

submodule of

V
V

.
Generalization of two results of the theory of uniform distribution
Authors: Proĭnov Petko D.
Published: June 2010

For a sequence

x
1

,
…
,

x
N

{x_1}, \ldots ,{x_N}

of points in

[
0
,
1
]

[0,1]

and a sequence

p
1

,
…
,

p
N

(

p
1

+

p
2

+
⋯
+

p
N

=
1
)

{p_1}, \ldots ,{p_N}({p_1} + {p_2} + \cdots + {p_N} = 1)

of nonnegative numbers, define the distribution function
\[

g
(
x
)
=
x
−

∑

x
k

>
x

p
k

.

g(x) = x - \sum \limits _{{x_k} > x} {{p_k}.}

\]
Let

φ
\varphi

be an increasing function on

[
0
,
1
]

[0,1]

and

φ
(
0
)
=
0

\varphi (0) = 0

. The main result of the paper is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis upper D Subscript upper N Baseline right-parenthesis less-than-or-slanted-equals integral Subscript 0 Superscript 1 Baseline phi left-parenthesis StartAbsoluteValue g left-parenthesis x right-parenthesis EndAbsoluteValue right-parenthesis d x less-than-or-slanted-equals phi left-parenthesis upper D Subscript upper N Baseline right-parenthesis comma">

F
(

D
N

)
⩽

∫
0
1

φ

(

|

g
(
x
)

|

)

d
x
⩽
φ
(

D
N

)
,

F({D_N}) \leqslant \int _0^1 {\varphi \left ( {\left | {g(x)} \right |} \right )} dx \leqslant \varphi ({D_N}),

\]
where

D
N

{D_N}

is the supremum norm of

g
g

on

[
0
,
1
]

[0,1]

and

F
F

is the antiderivative of

φ
\varphi

with

F
(
0
)
=
0

F(0) = 0

. This result generalizes and improves an estimate of Niederreiter [1] for the

L
2

{L^2}

discrepancy of the sequence

x
1

,
…
,

x
N

{x_1}, \ldots ,{x_N}

. Applying the above inequality we also obtain a new criterion for uniform distribution modulo one.
Remark on the generalized Putnam-Fuglede theorem
Authors: Yoshino Takashi
Published: June 2010

If

A
∗

{A^ * }

is

M
M

-hyponormal,

B
B

is dominant, and if

C
A
=
B
C

CA = BC

, then

C

A
∗

=

B
∗

C

C{A^ * } = {B^ * }C

.
Characterizations of Baire*1 functions in general settings
Authors: Peek Darwin E.
Published: June 2010

Baire* 1 functions from

[

0
,
1

]

\left [ {0,1} \right ]

to

R
R

were defined by R. J. O’Malley. For a general topological space

X
X

, a function

f
:
X
→
R

f:X \to R

will be said to be Baire* 1 if and only if for every nonempty closed subset

H
H

of

X
X

, there is an open set

U
U

such that

U
∩
H
≠
∅

U \cap H \ne \emptyset

and

f

|
H

f\left | H \right .

is continuous on

U
U

. Several characterizations of Baire* 1 functions are found by altering the well-known Baire 1 characterization: If

H
H

is a nonempty closed subset of the domain of

f
f

, then

f

|
H

f\left | H \right .

has a point where

f

|
H

f\left | H \right .

is continuous. These conditions simply replace "closed subset of the preceding characterization with "subset", "countable subset" or "dense-in-itself subset". The relationships of these characterizations are examined with the domain of

f
f

being various spaces. The independence of these conditions from the discrete convergence condition described by Á. Császár and M. Laczkovich is discussed.
Projective modules with free multiples and powers
Authors: Bass Hyman , Guralnick Robert
Published: June 2010

Let

P
P

be a finitely generated projective module over a commutative ring. Some tensor power of

P
P

is free iff some sum of copies of

P
P

is free.
A note on integral means of the derivative in conformal mapping
Authors: Makarov N. G.
Published: June 2010

There exists a number

p
0

>
1

/

3

{p_0} > 1/3

such that among the derivatives of univalent functions, that of the Koebe function ceases to have the greatest order of growth of

L
p

{L^p}

-means for all

p
≤

p
0

p \leq {p_0}

.
Oscillatory behavior of orthogonal polynomials
Authors: Máté Attila , Nevai Paul , Totik Vilmos
Published: June 2010

Let

d
α

d\alpha

be a positive Borel measure in [-1,1] with

α
′

>
0

\alpha ’ > 0

a.e. It is shown that the polynomials

p
n

{p_n}

orthonormal with respect to this measure oscillate almost everywhere in [-1,1]. A function

F
F

is also described that is a pointwise bound for

p
n

{p_n}

, exceeded only on sets of small measure. It is shown that

F
F

is the best possible.
On the distortion of 𝑛-dimensional quasiconformal mappings
Authors: Vuorinen Matti
Published: June 2010

A new upper bound

c
(
n
,
K
)

c(n,K)

for the linear dilatation of a

K
K

-quasiconformal mapping of a domain in

R
n

{R^n}

is proved. This upper bound substantially improves the previously known

n
n

-dimensional bound and it is asymptotically sharp when

n
=
2

n = 2

.
A structure theorem for the commutant of a class of cyclic subnormal operators
Authors: Raphael Marc
Published: June 2010

An

m
m

-measure is defined to be a measure

μ
\mu

such that the analytic bounded point evaluations of

P
2

(
μ
)

{P^2}(\mu )

is the open unit disk

D

{\mathbf {D}}

in the complex plane, and the weak* closure of the analytic polynomials in

L
∞

(
μ
)

{L^\infty }(\mu )

is the set of bounded analytic functions on

D

{\mathbf {D}}

. A complete characterization of

P
2

(
μ
)
∩

L
∞

(
μ
)

{P^2}(\mu ) \cap {L^\infty }(\mu )

, the commutant of the cyclic subnormal operator of multiplication by

z
z

on

P
2

(
μ
)

{P^2}(\mu )

, is then obtained.
On metabelian two-knot groups
Authors: Hillman Jonathan A.
Published: June 2010

We show that if the commutator subgroup of a

2
2

-knot group is abelian but not finitely generated, then it is isomorphic to the additive group of dyadic rationals, thus eliminating the one possibility left open in recent work of Yoshikawa. It follows that the examples given by Cappell and Fox provide a complete list of metabelian

2
2

-knot groups.
Every curve on a nonsingular surface can be defined by two equations
Authors: Boratyński M.
Published: June 2010

Let

R
R

be a smooth two-dimensional affine algebra over an algebraically closed field, and let

I
I

be an unmixed ideal of height one in

R
R

. Then there exist

a
,
b

a,b

in

I
I

such that

rad

(
I
)
=

rad

(
a
,
b
)

{\text {rad}}(I) = {\text {rad}}(a,b)

.
Small commutators and invariant subspaces
Authors: Lin Hua Xin
Published: June 2010

Let

A
A

be an operator on a Banach space, and let

T
T

be a nonzero compact (polynomial compact) operator. We prove that if

T
A
−
A
T

TA - AT

is "small", then

A
A

has a nontrivial (hyper)invariant subspace.
Extension of a theorem of Baayen and Helmberg on monothetic groups
Authors: Armacost D. L.
Published: June 2010

Let

G
G

and

K
K

be compact monothetic groups and let

ϕ
\phi

be a continuous homomorphism from

G
G

onto

K
K

. If

k
k

is a generator of

K
K

, must there exist a generator

g
g

of

G
G

such that

ϕ

(
g
)

=
k
?

\phi \left ( g \right ) = k?

? A useful theorem of Baayen and Helmberg provides an affirmative answer if

K
K

is the circle

T
T

. We show that the answer remains affirmative as long as

K
K

is metrizable. We also provide an example to show that the answer may be negative for nonmetrizable

K
K

.
Algebraic numbers and topologically equivalent measures in the Cantor set
Authors: Huang K. J.
Published: June 2010

It is known that the transcendental and rational numbers in the unit interval are not binomial numbers. In this article we will show that the algebraic integers of degree 2 are not binomial numbers either. Therefore two shift invariant measures

u
(
s
)
,
u
(
r
)

u(s),u(r)

with

r
r

being an algebraic integer of degree 2 in the unit interval are topologically equivalent if and only if

s
=
r

s = r

or

s
=
1
−
r

s = 1 - r

. We also show that for each positive integer

n

 > 2

n{\text { > 2}}

, there are algebraic integers and fractionals of degree

n
n

in the unit interval that are binomial numbers.
Sur la dérivée de fonctions harmoniques par rapport à un champ variable
Authors: Detraz J.
Published: June 2010

In

R

2

{{\mathbf {R}}^2}

, the derivative of a harmonic function with respect to a variable field has planar mean property.
A Gaussian measure for certain continued fractions
Authors: Kalpazidou Sofia
Published: June 2010

We solve a variant of Gauss’ problem for grotesque continued fraction using the approach of dependence with complete connections.
𝐴_{𝑝}-weight properties of real analytic functions in 𝑅ⁿ
Authors: Garofalo Nicola , Garrett Paul B.
Published: June 2010

In this note we show that given any real analytic function

u
u

in

R

n

{{\mathbf {R}}^n}

, there exists some

p
>
1

p > 1

for which

|
u
|

\left | u \right |

is locally an

A
p

{A_p}

-weight of Muckenhoupt.
Measures of Carleson type on filtrated probability spaces and the corona theorem on complex Brownian spaces
Authors: Arai Hitoshi
Published: June 2010

N. Th. Varopoulos showed the existence of corona solutions for bounded holomorphic martingales (cf. [7]). In this paper we investigate measures of Carleson type on usual filtrated probability spaces and as its application we estimate the corona solutions by a probabilistic modification of Wolff’s approach.
Nonnegatively curved contact manifolds
Authors: Goldberg S. I.
Published: June 2010

A compact simply connected contact Riemannian manifold with positive sectional curvature is homeomorphic with a sphere if its contact structure is normal and regular. In dimension 3 the regularity condition is omitted, and a stronger theorem is proved.
Homology of closed geodesics in certain Riemannian manifolds
Authors: Katsuda Atsushi , Sunada Toshikazu
Published: June 2010

It is shown, by using the trace formula of Selberg type, that every primitive, one-dimensional homology class of a negatively curved compact locally symmetric space contains infinitely many prime closed geodesics.
Rowbottom-type properties and a cardinal arithmetic
Authors: Tryba Jan
Published: June 2010

Assuming Rowbottom-type properties, we estimate the size of certain families of closed disjoint functions. We show that whenever

k
k

is Rowbottom and

2
ω

>

ℵ

ω
1

(
k
)

{2^\omega } > {\aleph _{{\omega _1}}}(k)

, then

2

>
k

=

2
ω

{2^{ > k}} = {2^\omega }

or

k
k

is the strong limit cardinal. Next we notice that every strongly inaccessible Jónsson cardinal

k
k

is

v
v

-Rowbottom for some

v
>
k

v > k

. In turn, Shelah’s method allows us to construct a Jónsson model of cardinality

k
+

{k^ + }

provided

k

cf(

k

)

=

k
+

{k^{{\text {cf(}}k{\text {)}}}} = {k^ + }

. We include some additional remarks.
Co-Blumberg spaces
Authors: Brown J. B. , Piotrowski Z.
Published: June 2010

A pair

(
X
,
Y
)

(X,Y)

of topological spaces is said to be a Blumberg pair ("BP") if for every

f
:
X
→
Y

f:X \to Y

, there exists a dense subset

D
D

of

X
X

such that

f

|

D

f|D

is continuous.

X
X

is a Blumberg space if

(
X
,
R
)

(X,R)

is BP, where

R
R

denotes the reals.

Y
Y

is co-Blumberg if

(
R
,
Y
)

(R,Y)

is BP. We survey the literature concerning the relationships between Blumberg spaces and Baire spaces and then study the relationships between co-Blumberg spaces and separability properties.
The homotopy type of certain laminated manifolds
Authors: Daverman R. J. , Tinsley F. C.
Published: June 2010

Let

M
M

denote a connected

(
n
+
1
)

(n + 1)

-manifold

n
⩾
5

n \geqslant 5

. A lamination

G
G

of

M
M

is an use decomposition of

M
M

into closed connected

n
n

-manifolds. Daverman has shown that the decomposition space

M

/

G

M/G

is homeomorphic to a

1
1

-manifold possibly with boundary. If

M

/

G
=

R
1

M/G = {R^1}

, we prove that

M
M

has the homotopy type of an

n
n

-manifold if and only if

∏
1

(
M
)

{\prod _1}(M)

is finitely presented. In the case that

M

/

G
=

S
1

M/G = {S^1}

we use the above result to construct an approximate fibration

f
:
M
→

S
1

f:M \to {S^1}

. We then discuss the important interactions of this study with that of perfect subgroups of finitely presented groups.
Factorization of measures and perfection
Authors: Adamski Wolfgang
Published: June 2010

It is proved that a probability measure

P
P

defined on a countably generated measurable space

(

Y
,

C

)

\left ( {Y,\mathcal {C}} \right )

is perfect iff every probability measure on

R

×
Y

{\mathbf {R}} \times Y

having

P
P

as marginal can be factored. This result leads to a generalization of a theorem due to Blackwell and Maitra.
Uniform differentiability, compactness, and 𝑙¹
Authors: Bilyeu Russell G. , Lewis Paul W.
Published: June 2010

In an earlier paper the authors have shown that conditionally compact subsets of

l
1

{l^1}

are characterized by uniform Gâteaux differentiability. Results in this paper show that this equivalence characterizes spaces which contain

l
1

{l^1}

.
Integral representation without additivity
Authors: Schmeidler David
Published: June 2010

Let

I
I

be a norm-continuous functional on the space

B
B

of bounded

Σ
\Sigma

-measurable real valued functions on a set

S
S

, where

Σ
\Sigma

is an algebra of subsets of

S
S

. Define a set function

v
v

on

Σ
\Sigma

by:

v
(
E
)

v (E)

equals the value of

I
I

at the indicator function of

E
E

. For each

a
a

in

B
B

let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis a right-parenthesis equals integral Subscript negative normal infinity Superscript 0 Baseline left-parenthesis v left-parenthesis a greater-than-or-equal-to alpha right-parenthesis minus v left-parenthesis upper S right-parenthesis right-parenthesis d alpha plus integral Subscript 0 Superscript normal infinity Baseline v left-parenthesis a greater-than-or-equal-to alpha right-parenthesis d alpha period">

J
(
a
)
=

∫

−
∞

0

(
v
(
a
≥
α
)
−
v
(
S
)
)
d
α
+

∫
0
∞

v
(
a
≥
α
)
d
α
.

J(a) = \int _{ - \infty }^0 {(v (a \geq \alpha ) - v (S))d\alpha + \int _0^\infty {v (a \geq \alpha )d\alpha .} }

\]
Then

I
=
J

I = J

on

B
B

if and only if

I
(
b
+
c
)
=
I
(
b
)
+
I
(
c
)

I(b + c) = I(b) + I(c)

whenever
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis b left-parenthesis s right-parenthesis minus b left-parenthesis t right-parenthesis right-parenthesis left-parenthesis c left-parenthesis s right-parenthesis minus c left-parenthesis t right-parenthesis right-parenthesis greater-than-or-slanted-equals 0">

(
b
(
s
)
−
b
(
t
)
)
(
c
(
s
)
−
c
(
t
)
)
⩾
0

(b(s) - b(t))(c(s) - c(t)) \geqslant 0

for all

s
s

and

t
t

in

S
S

.
Best 𝐿₁-approximation of 𝐿₁-approximately continuous functions on (0,1)ⁿ by nondecreasing functions
Authors: Darst R. B. , Fu Shu Sheng
Published: June 2010

For

n
≥
1

n \geq 1

, let

Ω
\Omega

denote the open unit

n
n

-cube,

(
0
,
1

)
n

{(0,1)^n}

. Let

μ
\mu

denote Lebesgue measure, let

Σ
\Sigma

consist of the Lebesgue measurable subsets of

Ω
\Omega

, and let

L
1

=

L
1

(
Ω
,
Σ
,
μ
)

{L_1} = {L_1}(\Omega ,\Sigma ,\mu )

. Let

A
1

{A_1}

consist of the approximately continuous functions in

L
1

{L_1}

and let

M
M

consist of the equivalence classes in

L
1

{L_1}

which contain nondecreasing functions. Let

f
∈

A
1

f \in {A_1}

. It is shown that there is a unique best

L
1

{L_1}

-approximation

[
g
]

[g]

to

f
f

in

M
M

. If

n
=
1
,
g

n = 1,g

is continuous, but if

n
>
1
,
[
g
]

n > 1,[g]

may not consist of a continuous function.
On the oscillation of almost-periodic Sturm-Liouville operators with an arbitrary coupling constant
Authors: Halvorsen S. G. , Mingarelli A. B.
Published: June 2010

In this paper we characterize those (Bohr) almost periodic functions

V
V

on

R

{\mathbf {R}}

for which the Sturm-Liouville equations
\[

−
y
+
λ
V
(
x
)
y
=
0
,

x
∈

R

,

- y + \lambda V(x)y = 0,\quad x \in \mathbf {R},

\]
are oscillatory at

±
∞

\pm \infty

for every real

λ
≠
0

\lambda \ne 0

, or, equivalently, for which there exists a real

λ
≠
0

\lambda \ne 0

such that the equation has a positive solution on

R

{\mathbf {R}}

.
On four-dimensional ℎ-cobordism
Authors: Kwasik Sławomor
Published: June 2010

The influence of the Poincaré Conjecture on the following problem of M. Cohen is examined. Problem. Does there exist a four-dimensional

h
h

-cobordism with nontrivial Whitehead torsion? It is proved (assuming the Poincarè Conjecture) that none of the following groups may be the fundamental group of a

4
4

-dimensional

h
h

-corbordism with nontrivial Whitehead torsion: generalized quaternion groups, cyclic groups

Z
8

{Z_8}

,

Z

12

{Z_{12}}

, the binary octahedral group, the binary tetrahedral group.
A natural partial order for semigroups
Authors: Mitsch H.
Published: June 2010

A partial order on a semigroup

(
S
,
⋅
)

(S, \cdot )

is called natural if it is defined by means of the multiplication of

S
S

. It is shown that for any semigroup

(
S
,
⋅
)

(S, \cdot )

the relation

a
≤
b

a \leq b

iff

a
=
x
b
=
b
y

a = xb = by

,

x
a
=
a

xa = a

for some

x
x

,

y
∈

S
1

y \in {S^1}

, is a partial order. It coincides with the well-known natural partial order for regular semigroups defined by Hartwig [4] and Nambooripad [10]. Similar relations derived from the natural partial order on the regular semigroup

(

T
X

,
∘
)

({T_X}, \circ )

of all maps on the set

X
X

are investigated.
A geometrical characterization of singly generated Douglas algebras
Authors: Izuchi Keiji
Published: June 2010

If

B
B

is a Douglas algebra with

B
≠
⊃

H
∞

+
C

B_ \ne ^ \supset {H^\infty } + C

, then

B
B

is singly generated if and only if ball

(
B

/

H
∞

+
C
)

(B/{H^\infty } + C)

has an extreme point.
Dense 𝐺_{𝛿}’s contain orthonormal bases
Authors: Mercer Richard
Published: June 2010

We show that a dense

G
δ

{G_\delta }

in the unit sphere of a Hilbert space necessarily contains an orthonormal basis. This allows us to choose an orthonormal basis of cyclic and separating vectors for a standard von Neumann algebra.
The Euler characteristic and inequalities for Kleinian groups
Authors: Abikoff William
Published: June 2010

A class of torsion-free finitely generated Kleinian groups is defined. For groups

G
G

in this class, sharp versions—with error terms—of standard inequalities are given. Also, sharp estimates are given for the number of cusps and the sum of the ranks of maximal nonconjugate cyclic parabolic subgroups.
Finite rank perturbations of singular spectra
Authors: Howland James S.
Published: June 2010

Let

T
T

be selfadjoint, and

V
V

nonnegative of finite rank, with the range of

V
V

cyclic for

T
T

. Then the singular parts of

T
T

and

H
=
T
+
V

H = T + V

are supported on two sets

S
T

{S_T}

and

S
H

{S_H}

such that the multiplicity of

T
T

on

S
T

∩

S
H

{S_T} \cap {S_H}

is less than the rank of

V
V

.
Subspaces of small codimension of finite-dimensional Banach spaces
Authors: Pajor Alain , Tomczak-Jaegermann Nicole
Published: June 2010

Given a finite-dimensional Banach space

E
E

and a Euclidean norm on

E
E

, we study relations between the norm and the Euclidean norm on subspaces of

E
E

of small codimension. Then for an operator taking values in a Hilbert space, we deduce an inequality for entropy numbers of the operator and its dual.
The Borsuk-Ulam theorem for a 𝑍_{𝑞}-map from a 𝑍_{𝑞}-space to 𝑆²ⁿ⁺¹
Authors: Kobayashi Teiichi
Published: June 2010

J. W. Walker obtained in [2] a generalization of the Borsuk-Ulam theorem. The purpose of this note is to prove a

mod

q

\mod q

version of Walker’s theorem.
Existence of a volume preserving diffeomorphism without periodic points on three-dimensional manifolds
Authors: Watanabe Nobuya
Published: June 2010

In this paper we show that on a smooth oriented closed

3
3

-manifold there exists a volume preserving diffeomorphism without periodic points.
Engulfing and finitely generated groups
Authors: Skora Richard
Published: June 2010

Let

M
M

be a simply connected

3
3

-manifold and

K
K

a piecewiselinear, simple loop in the interior of

M
M

. It is shown that there is a piecewiselinear, homotopy

3
3

-ball

B

⊂

M
˚

\mathcal {B} \subset \mathring {M}

, such that

K
⊂

B

˚

K \subset \mathring {\mathcal {B}}

if and only if

π
1

(
M
∖
K
)

{\pi _1}(M\backslash K)

is finitely generated.
On the number of generators and the modular group-ring of a finite 𝑝-group
Authors: Manz Olaf , Staszewski Reiner
Published: June 2010

We consider the Loewy-series

J

(
K
P

)
i

J{(KP)^i}

of a finite

p
p

-group

P
P

over a field

K
K

of characteristic

p
p

. We point that the series is not ’monotonic’ in general, but we can show that the dimensions of the Loewy-factors

J

(
K
P

)
i

/

J

(
K
P

)

i
+
1

J{(KP)^i}/J{(KP)^{i + 1}}

(except the first and the last one) are greater than or equal to the minimal number of generators of

P
P

.
Note on symbolic powers and going down
Authors: McAdam S. , Ratliff L. J.
Published: June 2010

For primes

P
⊆
Q

P \subseteq Q

of a Noetherian ring

R
R

, we consider when, for all

k
≥
1

k \geq 1

, there is an

m
m

with

P

(
m
)

⊆

Q

(
k
)

{P^{(m)}} \subseteq {Q^{(k)}}

and reprove a relevant theorem of Schenzel. If

R
R

is a domain, we consider sufficient conditions for

P
⊆
Q

P \subseteq Q

to satisfy going down for all primes

Q
Q

containing

P
P

.
On Gorenstein ideals of codimension four
Authors: Vasconcelos W. V. , Villarreal R.
Published: June 2010

We describe two computations showing how the Koszul homology impacts on the structure of Cohen-Macaulay ideals of height 3 and Gorenstein ideals of height 4.
Smash products and 𝐺-Galois actions
Authors: Osterburg James
Published: June 2010

We show that duality for coactions follows because the smash product is a

G
G

-Galois extension. We study

X
X

-inner and

X
X

-outer actions of the smash product and prove that if

A
A

is a semiprime

G
G

-graded ring such that

G
G

is

X
X

-outer on the smash product then the center of

A
A

is contained in the homogeneous component of the identity element of

G
G

.
The 𝐾-functional for 𝐻^{𝑝} and BMO in the poly-disk
Authors: Jawerth Björn
Published: June 2010

Peetre’s

K
K

-functional for the Hardy space

H
p

{H^p}

,

0
>
p
>
+
∞

0 > p > + \infty

, and the space BMO of functions of bounded mean oscillation is explicitly characterized in the case of a product of upper half-spaces.
On the indecomposable elements of the bar construction
Authors: Hain Richard M.
Published: June 2010

An explicit formula for a canonical splitting

s
:
Q

B

(

E

⋅

)
→

B

(

E

⋅

)

s:Q\mathcal {B}({\mathcal {E}^ \cdot }) \to \mathcal {B}({\mathcal {E}^ \cdot })

of the projection

B

(

E

⋅

)
→
Q

B

(

E

⋅

)

\mathcal {B}({\mathcal {E}^ \cdot }) \to Q\mathcal {B}({\mathcal {E}^ \cdot })

of the bar construction on a commutative d.g. algebra

E

⋅

{\mathcal {E}^\cdot }

onto its indecomposables is given. We prove that

s
s

induces a d.g. algebra isomorphism
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda left-parenthesis upper Q script upper B left-parenthesis script upper E Superscript dot Baseline right-parenthesis right-parenthesis right-arrow script upper B left-parenthesis script upper E Superscript dot Baseline right-parenthesis">

Λ
(
Q

B

(

E

⋅

)
)
→

B

(

E

⋅

)

\Lambda (Q\mathcal {B}({\mathcal {E}^\cdot })) \to \mathcal {B}({\mathcal {E}^\cdot })

and that

H
(
Q

B

(

E

⋅

)
)

H(Q\mathcal {B}({\mathcal {E}^\cdot }))

is isomorphic with

Q
H
(

B

(

E

⋅

)
)

QH(\mathcal {B}({\mathcal {E}^\cdot }))

.
Generalized refinable maps
Authors: Grace E. E.
Published: June 2010

Relationships among refinability, two successively greater generalizations, and a related property are studied, for maps between compacta.
Better bounds for periodic solutions of differential equations in Banach spaces
Authors: Busenberg Stavros N. , Fisher David C. , Martelli Mario
Published: June 2010

Let

f
f

be Lipschitz with constant

L
L

in a Banach space and let

x
(
t
)

x(t)

be a

P
P

-periodic solution of

x
′

(
t
)
=
f
(
x
(
t
)
)

x’(t) = f(x(t))

. We show that

P
⩾
6

/

L

P \geqslant 6/L

. An example is given with

P
=
2
π

/

L

P = 2\pi /L

, so the bound is nearly strict. We also give a short proof that

P
⩾
2
π

/

L

P \geqslant 2\pi /L

in a Hilbert space.
Order relation in Jordan rings and a structure theorem
Authors: González Santos , Martínez Consuelo
Published: June 2010

It is shown that the relation

⩽
\leqslant

defined by

x
⩽
y

x \leqslant y

if and only if

x
y
=

x
2

xy = {x^2}

,

x
2

y
=
x

y
2

=

x
3

{x^2}y = x{y^2} = {x^3}

is an order relation for a class of Jordan rings and we prove that a Jordan ring

R
R

is isomorphic to a direct product of Jordan division rings if and only if

⩽
\leqslant

is a partial order on

R
R

such that

R
R

is hyperatomic and orthogonally complete.
A property of ideals in polynomial rings
Authors: Lyubeznik Gennady
Published: June 2010

Every ideal in the polynomial ring in

n
n

variables over an infinite field has a reduction generated by

n
n

elements.
On the primitivity of polynomial rings with nonprimitive coefficient rings
Authors: Hodges T. J.
Published: June 2010

For a hereditary noetherian prime ring

R
R

with classical quotient ring

Q
Q

, various necessary and sufficient conditions are given for the polynomial ring

R
[

X
1

,
…
,

X
n

]

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

to be primitive when

R
R

itself is not primitive. It is shown that if

R
R

is a local hereditary noetherian prime ring, then

R
[
X
]

R[X]

is primitive if and only if

Q
[
X
]

Q[X]

is primitive. Similarly, for a semilocal hereditary noetherian prime ring

R
R

whose Jacobson radical contains a nonzero central element, it is shown that

R
[

X
1

,
…
,

X
n

]

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

is primitive if and only if

Q
[

X
1

,
…
,

X
n

]

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

is primitive.
Morrey space
Authors: Zorko Cristina T.
Published: June 2010

For

1
≤
p
>
∞

1 \leq p > \infty

,

Ω
\Omega

an open and bounded subset of

R
n

{R^n}

, and a nonincreasing and nonnegative function

φ
\varphi

defined in

(
0
,

ρ
0

]

(0,{\rho _0}]

,

ρ
0

=
diam
⁡
Ω

{\rho _0} = \operatorname {diam} \Omega

, we introduce the space

M

φ
,
0

p

(
Ω
)

\mathcal {M}_{\varphi ,0}^p(\Omega )

of locally integrable functions satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf Underscript c element-of upper C Endscripts left-brace integral Underscript upper B left-parenthesis x 0 comma rho right-parenthesis intersection normal upper Omega Endscripts StartAbsoluteValue f left-parenthesis x right-parenthesis minus c EndAbsoluteValue Superscript p Baseline d x right-brace less-than-or-equal-to upper A StartAbsoluteValue upper B left-parenthesis x 0 comma rho right-parenthesis EndAbsoluteValue phi Superscript p Baseline left-parenthesis rho right-parenthesis">

inf

c
∈
C

{

∫

B
(

x
0

,
ρ
)
∩
Ω

|

f
(
x
)
−
c

|

p

d
x

}

≤
A

|

B
(

x
0

,
ρ
)

|

φ
p

(
ρ
)

{\inf _{c \in C}}\left \{ {\int \limits _{B({x_0},\rho ) \cap \Omega } {|f(x) - c{|^p}dx} } \right \} \leq A|B({x_0},\rho )|{\varphi ^p}(\rho )

\]
for every

x
0

∈
Ω

{x_0} \in \Omega

and

0
>
ρ
≤

ρ
0

0 > \rho \leq {\rho _0}

, where

|

B
(

x
0

,
ρ
)

|

|B({x_0},\rho )|

denotes the volume of the ball centered in

x
0

{x_0}

and radius

ρ
\rho

. The constant

A
>
0

A > 0

does not depend on

(

x
0

,
ρ
)

({x_0},\rho )

. (i) We list some results on the structure, regularity, and density properties of the space so defined. (ii)

M

φ
,
0

p

\mathcal {M}_{\varphi ,0}^p

is represented as the dual of an atomic space.
On the pointwise convergence of the eigenfunction expansion associated with some iterated boundary value problems
Authors: Das Jyoti , Sen Gupta Prabir Kumar
Published: June 2010

Given a boundary value problem consisting of a second-order differential equation and some boundary conditions, one can derive higher-order boundary value problems, called iterated boundary value problems, provided the coefficients in the second-order differential equation are sufficiently smooth. The problem of convergence of the eigenfunction expansions associated with boundary value problems of even order has been the central attraction for mathematicians since the beginning of this century. The idea of this paper is to single out some higher-order boundary value problems, for which the question of convergence of the said expansion is completely answered by the similar problem associated with the second-order boundary value problem responsible for the generation of the iterated boundary value problem.
A new proof of a weighted inequality for the ergodic maximal function
Authors: Andersen Kenneth F.
Published: June 2010

E. Atencia and A. de la Torre proved that the ergodic maximal function operator is bounded on

L
p

(
ω
)

{L^p}(\omega )

if

ω
\omega

satisfies an appropriate analogue of Muckenhoupt’s

A
p

{A_p}

condition. An alternate proof of this result is given.
The Borsuk-Ulam theorem and bisection of necklaces
Authors: Alon Noga , West Douglas B.
Published: June 2010

The Borsuk-Ulam theorem of topology is applied to a problem in discrete mathematics. A bisection of a necklace with

k
k

colors of beads is a collection of intervals whose union captures half the beads of each color. Every necklace with

k
k

colors has a bisection formed by at most

k
k

cuts. Higher-dimensional generalizations are considered.
On the topology of self-dual 4-manifolds
Authors: LeBrun Claude
Published: June 2010

Obstructions are obtained for the problem of finding a Riemannian metric with self-dual conformal curvature and nonnegative scalar curvature on a given smooth compact

4
4

-manifold. A list is given of those simply connected manifolds conceivably admitting such metrics.
A note on the unit group of 𝑍𝑆₃
Authors: Allen P. J. , Hobby C.
Published: June 2010

The group

V
(
Z

S
3

)

V(Z{S_3})

of units of augmentation 1 in

Z

S
3

Z{S_3}

is characterized as the group of all doubly stochastic matrices in

GL(

3
,
Z
)

{\text {GL(}}3,Z)

. Two normal complements are presented, one of which is torsion free while the other contains elements of order 2.
On the essential numerical range of a generalized derivation
Authors: Magajna Bojan
Published: June 2010

Let

A
A

and

B
B

be bounded operators on Hilbert spaces

H

\mathcal {H}

and

L

\mathcal {L}

, respectively. The essential numerical range of the operator

X
→
A
X
−
X
B

X \to AX - XB

, defined on the Hilbert-Schmidt class

C

2

(

L

,

H

)

{\mathcal {C}^2}\left ( {\mathcal {L},\mathcal {H}} \right )

is expressed in terms of the numerical and the essential numerical ranges of

A
A

and

B
B

.
A mean ergodic theorem in Banach spaces
Authors: Yoshimoto Takeshi
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma equals StartSet upper U Subscript t Baseline colon t element-of normal upper Lamda EndSet left-parenthesis normal upper Lamda equals bold upper Z Superscript plus Baseline minus StartSet 0 EndSet or bold upper R Superscript plus Baseline minus StartSet 0 EndSet right-parenthesis">

Γ
=

{

U
t

:
t
∈
Λ

}

(

Λ
=

Z

+

−

{
0
}

or 

R

+

−

{
0
}

)

\Gamma = \left \{ {{U_t}:t \in \Lambda } \right \}\left ( {\Lambda = {{\mathbf {Z}}^ + } - \left \{ 0 \right \}{\text {or }}{{\mathbf {R}}^ + } - \left \{ 0 \right \}} \right )

be a commuting family of nonexpansive affine operators in a Banach space

X
X

satisfying the following conditions: (i) there is a function

M

(

x

|
Γ

)

≥
0

M\left ( {x\left | \Gamma \right .} \right ) \geq 0

defined on

X
X

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper U Subscript t plus s Baseline x minus upper U Subscript t Baseline upper U Subscript s Baseline x double-vertical-bar less-than-or-equal-to upper M left-parenthesis x vertical-bar normal upper Gamma right-parenthesis left-parenthesis s comma t element-of normal upper Lamda comma x element-of upper X right-parenthesis comma">

‖

U

t
+
s

x
−

U
t

U
s

x

‖

≤
M

(

x

|
Γ

)

(

s
,
t
∈
Λ
,
x
∈
X

)

,

\left \| {{U_{t + s}}x - {U_t}{U_s}x} \right \| \leq M\left ( {x\left | \Gamma \right .} \right )\quad \left ( {s,t \in \Lambda ,x \in X} \right ),

\]
(ii)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace t Superscript negative 1 Baseline double-vertical-bar upper U Subscript t Baseline x double-vertical-bar colon t element-of normal upper Lamda comma t greater-than-or-equal-to 1 right-brace equals upper K left-parenthesis x vertical-bar normal upper Gamma right-parenthesis greater-than normal infinity left-parenthesis x element-of upper X right-parenthesis period">

sup

{

t

−
1

‖

U
t

x

‖

:
t
∈
Λ
,
t
≥
1

}

=
K

(

x

|
Γ

)

>
∞

(

x
∈
X

)

.

\sup \left \{ {{t^{ - 1}}\left \| {{U_t}x} \right \|:t \in \Lambda ,t \geq 1} \right \} = K\left ( {x\left | \Gamma \right .} \right ) > \infty \left ( {x \in X} \right ).

\]
Then it is proved that if

{

t

−
1

U
t

x
:
t
∈
Λ

}

\left \{ {{t^{ - 1}}{U_t}x:t \in \Lambda } \right \}

is relatively compact for

x
∈
X

x \in X

, the limit

X
−

lim

t
→
∞

t

−
1

U
t

x
=

x
¯

X -\lim _{t \to \infty }{t^{ - 1}}{U_t}x = \bar x

exists in

X
X

and

U
t

x

¯

=

x
¯

(

t
∈
Λ

)

\overline {{U_t}x} = \overline x \left ( {t \in \Lambda } \right )

.