bluuebunny/crossref_metadata_2025 huggingface dataset:
Infinitesimal characterization of homogeneous bundles
Authors: Mackenzie Kirill
Published: June 2010

Consider a principal bundle

Q
(
B
,
H
)

Q(B,H)

on a base

B
B

which is compact and has finite fundamental group. We give necessary and sufficient conditions, in terms of the Atiyah sequence of

Q
(
B
,
H
)

Q(B,H)

, for

Q
(
B
,
H
)

Q(B,H)

to be locally isomorphic to a bundle of the form

G
(
G

/

S
,
S
)

G(G/S,S)

for

G
G

a Lie group and

S
S

a closed subgroup of

G
G

. The proof involves the careful integration of certain infinitesimal actions of a Lie algebra on

Q
,
B

Q,B

and the universal cover of

B
B

.
A characterization of Lašnev spaces
Authors: Hung H. H.
Published: June 2010

We give here a characterization of closed images of metrizable spaces in terms of the primitive concept of an ever finer sequence of partitions and a requirement considerably weaker than that of a

k
k

-network.
Universal spaces for countable-dimensional metric spaces
Authors: Goto Tatsuo
Published: June 2010

Let

H
(
A
)

H(A)

be the Dowker’s generalized Hilbert space with weight

|

A

|

|A|

, where

A
A

is any infinite set, and

H
∞
(
A
)

H\infty (A)

its subspace consisting of all points which have only finitely many rational coordinates distinct from zero. Using a result of E. Pol, it will be shown that

H
∞
(
A
)

H\infty (A)

is a universal space for countable dimensional metric spaces with weight

≤

|

A

|

\leq |A|

.
Absolute endpoints of chainable continua
Authors: Rosenholtz Ira
Published: June 2010

An endpoint of chainable continuum is a point at which it is always possible to start chaining that continuum. Some endpoints appear to have the property that one is almost "forced" to start (or finish) the chaining at these points. This paper characterizes these "absolute endpoints", and this characterization is used to show that in a chainable continuum locally connected at

p
p

is equivalent to connected im kleinen at

p
p

.
Leopoldt’s conjecture in parameterized families
Authors: Buchmann Johannes , Sands Jonathan W.
Published: June 2010

For each fixed prime

p
≠
5

p \ne 5

, we prove Leopoldt’s conjecture in two infinite families of fields of degree five whose normal closure has Galois group over the rationals isomorphic to

S
5

{S_5}

. The units of these fields were determined by Maus [4]; we develop and apply a simple reformulation of Leopoldt’s conjecture to obtain the result. We also observe that Leopoldt’s conjecture in one field can imply the same in a second field related by congruence conditions.
A Banach space on which there are few operators
Authors: Shelah Saharon , Steprāns Juris
Published: June 2010

It is shown that there is a nonseparable Banach space on which every operator has the form

S
+
ρ
I

S + \rho I

where

ρ
\rho

is a scalar and

S
S

is an operator with separable range.
Dimension of the graph of Riemann integrable functions
Authors: Foran James
Published: June 2010

The Hausdorff

h
−
m

h - m

measure of the graph of a Riemann integrable function is shown to be finite provided

h
h

satisfies an inequality related to the rate of convergence of the upper and lower Riemann sums.
The Schottky problem on pants
Authors: Penner R. C.
Published: June 2010

In this note, we consider the classical problem of Schottky of characterizing the set of period matrices which arise from all possible conformal structures on a fixed topological surface. Restricting to a planar surface with Euler characteristic

−
1

- 1

, we find that a real symmetric

3
3

-by-

3
3

matrix arises as a period matrix if and only if the matrix has vanishing row sums, and the diagonal entries are positive and satisfy all three possible strict triangle inequalities. The technique of proof involves extremal and harmonic lengths of curve classes.
Stability of Anosov maps
Authors: Quandt Juergen
Published: June 2010

Anosov maps of Banach manifolds are shown to be inverse limit stable.
Undecidability of module homomorphisms
Authors: Kim K. H. , Roush F. W.
Published: June 2010

For finitely additively generated torsion free modules and rings, we show existence of module monomorphisms is decidable but of epimorphisms is undecidable.
Representation theorems for positive solutions of parabolic equations
Authors: Pinchover Yehuda
Published: June 2010

We determine all the minimal positive solutions of the parabolic equation

L
u
=
0

Lu = 0

in

R

n

×

R

−

{{\mathbf {R}}^n} \times {{\mathbf {R}}_ - }

, where

L
L

has time independent coefficients or

L
L

has periodic coefficients in

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

and

t
t

.
Pseudoprime 𝑙-ideals in a class of 𝑓-rings
Authors: Larson Suzanne
Published: June 2010

In a commutative

f
f

-ring, an

l
l

-ideal

I
I

is called pseudoprime if

a
b
=
0

ab = 0

implies

a
∈
I

a \in I

or

b
∈
I

b \in I

, and is called square dominated if for every

a
∈
I
,

|

a

|

≤

x
2

a \in I,|a| \leq {x^2}

for some

x
∈
A

x \in A

such that

x
2

∈
I

{x^2} \in I

. Several characterizations of pseudoprime

l
l

-ideals are given in the class of commutative semiprime

f
f

-rings in which minimal prime

l
l

-ideals are square dominated. It is shown that the hypothesis imposed on the

f
f

-rings, that minimal prime

l
l

-ideals are square dominated, cannot be omitted or generalized.
Separately subharmonic functions need not be subharmonic
Authors: Wiegerinck Jan
Published: June 2010

We give an example of a separately subharmonic function which is not subharmonic.
Pettis decomposition for universally scalarly measurable functions
Authors: Bator Elizabeth M.
Published: June 2010

It is shown that if

K
K

is a compact Hausdorff space,

X
X

is a Banach space, and

f
:
K
→

X
∗

f:K \to {X^ * }

is bounded and universally scalarly measurable, then

f
f

is

μ
\mu

-Pettis decomposable for every Radon measure

μ
\mu

on

K
K

.
Nonexpansive actions of topological semigroups on strictly convex Banach spaces and fixed points
Authors: Bartoszek Wojciech
Published: June 2010

Let

C
C

be a closed convex subset of a strictly convex Banach space

X
X

and

{

T
s

:
s
∈
S

}

\left \{ {{T_s}:s \in S} \right \}

be a continuous representation of a semitopological semigroup

S
S

as nonexpansive mappings of

C
C

into itself. The main result establishes the fact that if for some

x
∈
C

x \in C

the trajectory

{

T
s

x
:
s
∈
S

}

\left \{ {{T_s}x:s \in S} \right \}

is relatively compact and

A
P
(
S
)

AP(S)

has a left invariant mean then

K
=

conv
⁡
{

T
s

x
:
s
∈
S
}

¯

K = \overline {\operatorname {conv} \{ {T_s}x:s \in S\} }

contains a common fixed point for

{

T
s

}

s
∈
S

{\left \{ {{T_s}} \right \}_{s \in S}}

.
A lifting theorem and analytic operator algebras
Authors: Nakazi Takahiko
Published: June 2010

Let

K
K

be a complex Hilbert space and

H
H

a closed subspace. It is shown that if a

2
×
2

2 \times 2

selfadjoint operator matrix

T
T

with positive diagonals on

K
⊕
K

K \oplus K

is positive on

H
⊕

H
⊥

H \oplus {H^ \bot }

, then there exists a

2
×
2

2 \times 2

operator matrix

T
~

\tilde T

with the same diagonals such that

T
~

\tilde T

is positive on

K
⊕
K

K \oplus K

and

T
T

is the restriction of

T
~

\tilde T

to

H
⊕

H
⊥

H \oplus {H^ \bot }

. When

T
T

is in a von Neumann algebra, we consider the problems of finding

T
T

in the same algebra. This lifting theorem has applications to weighted norm inequalities for conjugation operators on analytic operator algebras.
Some cardinality properties of a hyperspace with the locally finite topology
Authors: Bella A.
Published: June 2010

Estimates of various cardinal functions defined on a hyperspace equipped with the locally finite topology are given.
On a domain characterization of Schrödinger operators with gradient magnetic vector potentials and singular potentials
Authors: Goldstein Jerome A. , Svirsky Roman
Published: June 2010

Of concern are the minimal and maximal operators on

L
2

(

R

n

)

{L^2}({{\mathbf {R}}^n})

associated with the differential expression
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="tau Subscript upper Q Baseline equals sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis i partial-differential slash partial-differential x Subscript j Baseline plus q Subscript j Baseline left-parenthesis x right-parenthesis right-parenthesis Superscript 2 Baseline plus upper W left-parenthesis x right-parenthesis">

τ
Q

=

∑

j
=
1

n

(
i
∂

/

∂

x
j

+

q
j

(
x
)

)
2

+
W
(
x
)

{\tau _Q} = \sum \limits _{j = 1}^n {(i\partial /\partial {x_j}} + {q_j}(x){)^2} + W(x)

\]
where

(
q
,
…
,

q
n

)
=
grad
⁡
Q

(q, \ldots ,{q_n}) = \operatorname {grad}Q

for some real function

W
W

on

R

n

{{\mathbf {R}}^n}

and

W
W

satisfies

c

|
x
|

−
2

≤
W
(
x
)
≤
C

|
x
|

−
2

c{\left | x \right |^{ - 2}} \leq W(x) \leq C{\left | x \right |^{ - 2}}

. In particular, for

Q
=
0

Q = 0

,

τ
Q

{\tau _Q}

reduces to the singular Schrödinger operator

−
Δ
+
W
(
x
)

- \Delta + W(x)

. Among other results, it is shown that the maximal operator (associated with the

τ
Q

{\tau _Q}

) is the closure of the minimal operator, and its domain is precisely
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D o m left-parenthesis sigma-summation Underscript j equals 1 Overscript n Endscripts left-parenthesis i partial-differential slash partial-differential x Subscript j Baseline plus q Subscript j Baseline left-parenthesis x right-parenthesis right-parenthesis squared right-parenthesis intersection upper D o m left-parenthesis upper W right-parenthesis comma">

Dom
⁡

(

∑

j
=
1

n

(
i
∂

/

∂

x
j

+

q
j

(
x
)
)

2

)

∩
Dom
⁡
(
W
)
,

\operatorname {Dom}\left ( {\sum \limits _{j = 1}^n {{{(i\partial /\partial {x_j} + {q_j}(x))}^2}} } \right ) \cap \operatorname {Dom}(W),

\]
provided that

C
≥
c
>
−
n
(
n
−
4
)

/

4

C \geq c > - n(n - 4)/4

.
A remark on the normality of infinite products
Authors: Chiba Keiko
Published: June 2010

In this note we shall prove the following: Suppose that all finite subproducts of a product space

X
=

∏

β
>
λ

X
β

X = \prod \nolimits _{\beta > \lambda } {{X_\beta }}

are normal. If

X
X

is

λ
\lambda

-paracompact, then

X
X

is normal. Here

λ
\lambda

stands for an infinite cardinal number.
On projections in power series spaces and the existence of bases
Authors: Krone Jörg
Published: June 2010

Mityagin posed the problem, whether complemented subspaces of nuclear infinite type power series spaces have a basis. A related more general question was asked by Pełczyński. It is well known for a complemented subspace

E
E

of a nuclear infinite type power series space, that its diametral dimension can be represented by

Δ
E
=
Δ

Λ
∞

(
α
)

\Delta E = \Delta {\Lambda _\infty }(\alpha )

for a suitable sequence

α
\alpha

with

α
j

≥
ln
⁡
(
j
+
1
)

{\alpha _j} \geq \ln (j + 1)

. In this article we prove the existence of a basis for

E
E

in case that

α
j

≥
j

{\alpha _j} \geq j

and

sup

α

2
j

α
j

>
∞

\sup \tfrac {{{\alpha _{2j}}}}{{{\alpha _j}}} > \infty

.
On compact and bounding holomorphic mappings
Authors: Lindström Mikael
Published: June 2010

Let

E
E

and

F
F

be complex Banach spaces. We say that a holomorphic mapping

f
f

from

E
E

into

F
F

is compact respectively bounding if

f
f

maps some neighbourhood of every point of

E
E

into a relatively compact respectively bounding subset of

F
F

. Recall that a subset of

E
E

is bounding if it is mapped onto a bounded set by every complex valued holomorphic mapping on

E
E

. Compact holomorphic mappings have been studied by R. Aron and M. Schottenloher in [1]. Since every relatively compact subset of a Banach space is trivially bounding it is clear that every compact holomorphic mapping is bounding. We show that the product of three bounding holomorphic mappings is compact.
A characterization of weak pseudoconvexity
Authors: Sakai Akira
Published: June 2010

It is proved that a smooth domain

D
D

of

C

n

{{\mathbf {C}}^n}

is weakly pseudoconvex, if, for every strongly pseudoconvex domain

D
′

D’

with

D
∩

D
′

=
∅

D \cap D’ = \emptyset

and

E
=

D
¯

∩

D
¯

′

≠
∅
,
E

E = \bar D \cap \bar D’ \ne \emptyset ,E

is totally real.
On the nonsimplicity of some convergence categories
Authors: Lowen E. , Lowen R.
Published: June 2010

The category TOP of topological spaces is known to be Convsimple in the sense that there exists a single object

E
E

in Top such that Top is the epireflective hull of

{
E
}

\left \{ E \right \}

in the category Conv of convergence spaces. Prtop, the category of pretopological spaces is also Conv-simple. We show that on the contrary the category Pstop of pseudo topological spaces and Conv itself are not Conv-simple. More specifically every epireflective subcategory of Conv which contains all Hausdorff

c
c

-embedded locally compact spaces is not Conv-simple.
Smooth 2-knots in 𝑆²×𝑆² with simply-connected complements are topologically unique
Authors: Sato Yoshihisa
Published: June 2010

For a given primitive homology class

ξ
\xi

of

H
2

(

S
2

×

S
2

;

Z

)

{H_2}({S^2} \times {S^2};{\mathbf {Z}})

, we show that there exists only one smoothly embedded

2
2

-sphere in

S
2

×

S
2

{S^2} \times {S^2}

, up to homeomorphism, which represents

ξ
\xi

and whose complement is simply connected.
One parameter submonoids in locally compact differentiable monoids
Authors: Anderson Mitch
Published: June 2010

Differentiable semigroups based on generalized manifolds were recently introduced by George Graham. We show that such locally compact monoids in which the multiplication is strongly differentiable at

(
1
,
1
)

(1,1)

must contain nontrivial one parameter submonoids.
Unitary representations of Lie groups and Gårding’s inequality
Authors: Bratteli Ola , Goodman Fred , Jorgensen Palle , Robinson Derek W.
Published: June 2010

We prove two versions of Gåding’s inequality for strongly elliptic operators in the enveloping Lie algebra associated with a unitary representation of a Lie group. We then deduce a characterization of the differential structure of the representation in terms of the elliptic operators.
Representation of an abstract measure using Borel-isomorphism types
Authors: Shortt R. M.
Published: June 2010

For

X
⊆
R

X \subseteq R

, the mapping

t
t

which assigns to each Borel subset of

X
X

its isomorphism type is an abstract measure. Given a monoid-valued measure

m
m

, we ask when there is an

X
⊆
R

X \subseteq R

such that

t
t

and

m
m

are isomorphic as measures.
The number of indecomposable sequences over an Artin algebra of finite type
Authors: Corwin Stephen P.
Published: June 2010

Let

Λ
\Lambda

be an artin algebra of finite representation type. For a finitely generated

Λ
\Lambda

-module

C
C

, there are only finitely many f.g. modules

A
A

such that

0
→
A
→
B
→
C
→
0

0 \to A \to B \to C \to 0

is indecomposable as a short exact sequence.
Second order differential equations with rational coefficients
Authors: Hinkkanen A. , Rossi John
Published: June 2010

The authors exhibit two linearly independent real solutions,

f
1

{f_1}

and

f
2

{f_2}

, to

w
+
H
w
=
0

w + Hw = 0

such that

f
1

f
2

{f_1}{f_2}

is transcendental,

f
1

{f_1}

and

f
2

{f_2}

have only real zeros and poles and

H
H

is a nonconstant rational function. This shows the sharpness of a recent result of Hellerstein and Rossi. Some related results are also proved.
Degrees of irreducible characters and normal 𝑝-complements
Authors: Berkovich Ya. G.
Published: June 2010

John Tate [1] proved that if

P
∈

S

y

l

p

(
G
)

P \in {\mathbf {S}}{\text {y}}{{\text {l}}_p}(G)

,

H
H

is a normal subgroup of a finite group

G
G

and

P
∩
H
≤
Φ
(
P
)

P \cap H \leq \Phi (P)

(

Φ
(
G
)

\Phi (G)

is the Frattini subgroup of

G
G

) then

H
H

has a normal

p
p

-complement. We prove in this note that Tate’s theorem has nice character-theoretic applications.
Surface subgroups of graph groups
Authors: Servatius Herman , Droms Carl , Servatius Brigitte
Published: June 2010

Given a graph

Γ
\Gamma

, define the group

F
Γ

{F_\Gamma }

to be that generated by the vertices of

Γ
\Gamma

, with a defining relation

x
y
=
y
x

xy = yx

for each pair

x
,
y

x,y

of adjacent vertices of

Γ
\Gamma

. In this article, we examine the groups

F
Γ

{F_\Gamma }

, where the graph

Γ
\Gamma

is an

n
n

-gon,

(
n
≥
4
)

(n \geq 4)

. We use a covering space argument to prove that in this case, the commutator subgroup

F
Γ
′

{F’_\Gamma }

contains the fundamental group of the orientable surface of genus

1
+
(
n
−
4
)

2

n
−
3

1 + (n - 4){2^{n - 3}}

. We then use this result to classify all finite graphs

Γ
\Gamma

for which

F
Γ
′

{F’_\Gamma }

is a free group.
Surjective isometries of weighted Bergman spaces
Authors: Kolaski Clinton J.
Published: June 2010

Let

Ω
\Omega

be a bounded, simply connected domain in

C

n

=

R

2
n

{{\mathbf {C}}^n} = {R^{2n}}

, let

F
∈

L
1

(

m
Ω

)

F \in {L^1}({m_\Omega })

be positive and continuo on

Ω
\Omega

, and let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript upper F Superscript upper P Baseline left-parenthesis normal upper Omega right-parenthesis equals upper L Superscript p Baseline left-parenthesis upper F d m right-parenthesis intersection upper H left-parenthesis normal upper Omega right-parenthesis left-parenthesis 0 greater-than p greater-than normal infinity right-parenthesis">

B
F
P

(
Ω
)
=

L
p

(
F
d
m
)
∩
H
(
Ω
)
(
0
>
p
>
∞
)

B_F^P(\Omega ) = {L^p}(Fdm) \cap H(\Omega )(0 > p > \infty )

denote the weighted Bergman space over

Ω
\Omega

. We characterize those automorphisms

Φ
\Phi

of

Ω
\Omega

such that the map

f
→
g
⋅
(
f
∘
Φ
)

f \to g \cdot (f \circ \Phi )

is a surjective isometry of

B
F
P

(
Ω
)

B_F^P(\Omega )

, including an explicit description of

|

g

|

|g|

.
Weakly dense subsets of the measure algebra
Authors: Burke Maxim R.
Published: June 2010

We introduce the notion of the weak density of a Boolean algebra and show that for homogeneous measure algebras it coincides with the density (=least size of a coinitial set). From this we obtain a partial lifting of the measure algebra of

[

0
,
1

]

\left [ {0,1} \right ]

of minimal size which does not extend to a lifting. It also follows that the

π
\pi

-character of each point and the

π
\pi

-weight are the same for the Stone space of a homogeneous measure algebra
Rescaling planar hyperbolic sectors
Authors: Barge Marcy
Published: June 2010

Constant rescaling of a planar hyperbolic sector produces a one parameter family of pairwise not locally topologically conjugate sectors if and only if the planar hyperbolic sector is not locally topologically conjugate to a linear hyperbolic sector.
A Hardy-Littlewood maximal inequality for Jacobi type hypergroups
Authors: Connett William C. , Schwartz Alan L.
Published: June 2010

A Hardy-Littlewood maximal inequality is proved for a class of probability preserving measure algebras on compact intervals.
Covers of Dehn fillings on once-punctured torus bundles
Authors: Baker Mark D.
Published: June 2010

Let

M
M

be a compact, orientable

3
3

-manifold that fibers over

S
1

{S^1}

with fiber a once-punctured torus,

T
0

{T_0}

, and characteristic homeomorphism

h
:

T
0

→

T
0

h:{T_0} \to {T_0}

. We prove that for certain characteristic homeomorphisms, most Dehn fillings on

M
M

yield manifolds with virturally

Z

{\mathbf {Z}}

-representable fundamental groups.
Injective hulls of simple 𝑠𝑙(2,𝒞) modules are locally Artinian
Authors: Dahlberg Randall P.
Published: June 2010

Let

L
L

denote the simple Lie algebra

sl(2,

C

)

{\text {sl(2,}}{\mathbf {C}}{\text {)}}

over the complex numbers

C

{\mathbf {C}}

. For any simple

L
L

-modules

S
S

, considered as a left unital module over the universal enveloping algebra of

L
,
U
(
L
)

L,U(L)

, the injective hull of

S
,

E
L

(
S
)

S,{E_L}(S)

, is a locally Artinian

U
(
L
)

U(L)

-module.
On topological entropy of group actions on 𝑆¹
Authors: Watanabe Nobuya
Published: June 2010

In this paper, we show that a surface group action on

S
1

{S^1}

with a non-zero Euler number has a positive topological entropy. We also show that if a surface group action on

S
1

{S^1}

has a Euler number which attains the maximal absolute value in the inequality of Milnor-Wood, then the topological entropy of the action equals the exponential growth rate of the group.
On the arithmetic of weight spaces in the root lattice
Authors: Deckhart Robert W.
Published: June 2010

Using the notion of

i
i

-blocks in a weight space of the positive cone of the root lattice we obtain, after suitable reduction to nontrivial cases, a surprising strongly inductive formula for Kostant’s partition function in terms of exactly 3 lower weight spaces. The proof is elementary, using arithmetic properties of

i
i

-blocks from [2], and avoiding any character formulas.
Upper semicontinuous differential inclusions without convexity
Authors: Bressan A. , Cellina A. , Colombo G.
Published: June 2010

We prove existence of solutions to the Cauchy problem for the differential inclusion

x
˙

∈
A
(
x
)

\dot x \in A(x)

, when

A
A

is cyclically monotone and upper semi-continuous.
Manifolds with (𝑍₂)^{𝑘}-action
Authors: Pergher Pedro L. Q.
Published: June 2010

Let

(

M
n

,

T
1

,
…
,

T
k

)

\left ( {{M^n},{T_1}, \ldots ,{T_k}} \right )

be a closed manifold with differentiable

(

Z

2

)

k

{\left ( {{{\mathbf {Z}}_2}} \right )^k}

-action. The purpose of this paper is to show how to construct the bordism class

[

M
n

]

[{M^n}]

in terms of the decomposed form of the fixed point data of this action.
Spheres with locally pinched metrics
Authors: Currier Robert J.
Published: June 2010

The purpose of this paper is the construction of metrics on spheres whose curvatures are locally almost constant but which have large variation globally. This construction also applies to spherical space forms.
Automorphisms of extended current algebras
Authors: Piazza Paolo , Wu Siye
Published: June 2010

We construct a (noncentral) extension of current algebras and study the adjoint action induced by the current group.
A strong Parrott theorem
Authors: Foias Ciprian , Tannenbaum Allen
Published: June 2010

In this note we discuss a strengthened version of a theorem due to Parrott [8] in operator dilation theory. We relate our result to the one-step extension procedure of Adamjan-Arov-Krein [1].
The Odlyzko conjecture and O’Hara’s unimodality proof
Authors: Stanton Dennis , Zeilberger Doron
Published: June 2010

We observe that Andrew Odlyzko’s conjecture that the Maclaurin coefficients of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 slash left-bracket left-parenthesis 1 plus q right-parenthesis left-parenthesis 1 plus q plus q squared right-parenthesis midline-horizontal-ellipsis left-parenthesis 1 plus q plus midline-horizontal-ellipsis plus q Superscript k minus 1 Baseline right-parenthesis right-bracket">

1

/

[
(
1
+
q
)
(
1
+
q
+

q
2

)
⋯
(
1
+
q
+
⋯
+

q

k
−
1

)
]

1/[(1 + q)(1 + q + {q^2}) \cdots (1 + q + \cdots + {q^{k - 1}})]

have alternating signs is an almost immediate consequence of an identity that is implied by Kathy O’Hara’s recent magnificent combinatorial proof of the unimodality of the Gaussian coefficients.
Fibrations that are cofibrations. II
Authors: Alonso Juan M.
Published: June 2010

We show that fibrations that are cofibrations can be described quite explicitly (in terms of localization) when the total space of the fibration is nilpotent and that, in the absence of nilpotency, no such simple characterization exists.
Affine invariant subspaces of 𝐶(𝐶)
Authors: Sternfeld Yaki , Weit Yitzhak
Published: June 2010

A linear subspace

A
A

of

C
(

C

)

C({\mathbf {C}})

is affine invariant if

f
(
z
)
∈
A

f(z) \in A

implies that

f
(
a
z
+
b
)
∈
A

f(az + b) \in A

for every

a
,
b
∈

C

a,b \in {\mathbf {C}}

. We present a classification of the affine invariant closed subspaces of

C
(

C

)

C({\mathbf {C}})

, and of those affine invariant subspaces which are also composition invariant (i.e.,

f
,
g
∈
A

f,g \in A

implies that

f
∘
g
∈
A
)

f \circ g \in A)

).
Matroids determine the embeddability of graphs in surfaces
Authors: Zaslavsky Thomas
Published: June 2010

The embeddability of a graph in a given surface is determined entirely by the polygon matroid of the graph. That is also true for cellular embeddability in nonorientable surfaces but not in orientable surfaces.
A simple proof of Koornwinder’s addition formula for the little 𝑞-Legendre polynomials
Authors: Rahman Mizan
Published: June 2010

Recently Koornwinder found an addition formula for the little

q
q

-Legendre polynomials by using Masuda et al.’s result that they are related to the matrix elements of the irreducible unitary representation of the twisted

SU(2)

{\text {SU(2)}}

quantum group. Here we give an alternate derivation of the addition formula by using some summation and transformation formulas of basic hypergeometric series.
Linear operators commuting with translations on 𝒟(𝐑) are continuous
Authors: Meisters Gary Hosler
Published: June 2010

Let

D

(

R

)

\mathcal {D}\left ( {\mathbf {R}} \right )

denote the Schwartz space of all

C
∞

{C^\infty }

-functions

f
:

R

→

C

f:{\mathbf {R}} \to {\mathbf {C}}

with compact supports in the real line

R

{\mathbf {R}}

. An earlier result of the author on the automatic continuity of translation-invariant linear functionals on

D

(

R

)

\mathcal {D}\left ( {\mathbf {R}} \right )

is combined with a general version of the Closed-Graph Theorem due to A. P. Robertson and W. J. Robertson in order to prove that every linear mapping

S
S

of

D

(

R

)

\mathcal {D}\left ( {\mathbf {R}} \right )

into itself, which commutes with translations, is automatically continuous.
Representatives for finite sets
Authors: Jia Xing De
Published: June 2010

This paper considers a combinatorial problem by M. B. Nathanson [1], concerning simultaneous systems of representatives for two families of finite sets.
Cyclic surgery on knots
Authors: Wang Shi Cheng
Published: June 2010

We get a necessary condition under which a nonsimple knot (i.e. a satellite knot) admits a nontrivial surgery producing a lens space. Interesting corollaries are: (1) if a lens space can be obtained from a nontrivial surgery on a nonsimple knot, then the order of its fundamental group is not smaller than 23; (2) any nonsimple knot admits at most one nontrivial surgery which produces a lens space.
Adherent compact spaces
Authors: Krystock Robert L.
Published: June 2010

A new class of

H
H

-closed spaces, herein referred to as the class of adherent compact spaces, is introduced—and shown to be properly contained in the class of

C
C

-compact spaces. It is also shown, using a construction of Goss and Viglino, that these spaces need not be seminormal.
Generic families and models of set theory with the axiom of choice
Authors: Ciesielski Krzysztof , Guzicki Wojciech
Published: June 2010

Let

M
M

be a countable transitive model of ZFC and

A
A

be a countable

M
M

-generic family of Cohen reals. We prove that there is no smallest transitive model

N
N

of ZFC that either

M
∪
A
⊆
N

M \cup A \subseteq N

or

M
∪
{
A
}
⊆
N

M \cup \{ A\} \subseteq N

. It is also proved that there is no smallest transitive model

N
N

of ZFC

−

^{-}

(ZFC theory without the power set axiom) such that

M
∪
{
A
}
⊆
N

M \cup \{ A\} \subseteq N

. It is also proved that certain classes of extensions of

M
M

obtained by Cohen generic reals have no minimal model.
A measuring argument for finite groups
Authors: Chermak Andrew , Delgado Alberto
Published: June 2010

We call attention to a simple measuring argument for finite groups. Direct applications of this argument lead to the construction of certain new characteristic subgroups of finite

p
p

-groups as well as an easy proof of a generalization, due to Timmesfeld, of the Thompson replacement lemma. Some applications to finite simple groups are also given.
Links not concordant to boundary links
Authors: Livingston Charles
Published: June 2010

Casson-Gordon invariants are used to prove that certain links in

S
3

{S^3}

are not concordant to boundary links. These examples were first described by Cochran and Orr.
Oscillation of neutral differential equations with periodic coefficients
Authors: Huang Qing Guang , Chen Shao Zhu
Published: June 2010

We establish a necessary and sufficient condition for the oscillation of all solutions to the neutral differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket x left-parenthesis t right-parenthesis minus p left-parenthesis x right-parenthesis left-parenthesis t minus r right-parenthesis right-bracket prime plus sigma-summation Underscript i equals 1 Overscript n Endscripts q Subscript i Baseline left-parenthesis t right-parenthesis x left-parenthesis t minus i r right-parenthesis equals 0 comma">

[
x
(
t
)
−
p
(
x
)
(
t
−
r
)

]
′

+

∑

i
=
1

n

q
i

(
t
)
x
(
t
−
i
r
)
=
0

,

[x(t) - p(x)(t - r)]’ + \sum \limits _{i = 1}^n {{q_i}(t)x(t - ir) = 0} ,

\]
where

0
≤
p
≤
1
,
r
>
0

0 \leq p \leq 1,r > 0

are constants and

q
i

(
t
)
≥
0
,
i
=
1
,
…
,
n

{q_i}(t) \geq 0,i = 1, \ldots ,n

, are continuous

r
r

-periodic functions.
Convexity properties of distinguished eigenvalues of certain classes of operators
Authors: Najman B.
Published: June 2010

We prove two convexity results: (1) Let

A
(
ε
)
=

A
0

+
ε

A
1

A(\varepsilon ) = {A_0} + \varepsilon {A_1}

be a family of selfadjoint operators in a Krein space with separated spectrum so that the maximum

λ
_
(
ε
)

\lambda \_(\varepsilon )

of the spectrum of negative type of

A
(
ε
)

A(\varepsilon )

is an isolated simple eigenvalue. Then

λ
_
(
ε
)

\lambda \_(\varepsilon )

is convex. (2) Let

λ
_
(
ε
)

\lambda \_(\varepsilon )

be the left distinguished eigenvalue of the generalized eigenvalue problem

(

A
0

+
ε

A
1

)
x
=
λ
B
x

({A_0} + \varepsilon {A_1})x = \lambda Bx

where

A
1

{A_1}

and

B
B

are real diagonal matrices and

A
0

{A_0}

is an irreducible essentially nonnegative matrix. Then

λ
_
(
ε
)

\lambda \_(\varepsilon )

is convex. In both cases

λ
_
(
ε
)

\lambda \_(\varepsilon )

is strictly convex unless it is linear.
Sufficiency and the separation of strongly convex sets of probability measures
Authors: Ramamoorthi R. V.
Published: June 2010

We show that the equivalence of two notions of sufficiency is related to the separation of two orthogonal, strongly convex sets of probability measures by a universally measurable set.
Rotundity, the C.S.R.P., and the 𝜆-property in Banach spaces
Authors: Aron Richard M. , Lohman Robert H. , Suárez Antonio
Published: June 2010

Two open questions stemming from the

λ
\lambda

-property in Banach spaces are solved. The following are shown to be equivalent in a Banach space

X
X

: (a)

X
X

has the

λ
\lambda

-property; (b) every vector in the closed unit ball of

X
X

is expressible as a convex series of extreme points of the unit ball of

X
X

. Also, by exhibiting a class of nonrotund Orlicz spaces for which the

λ
\lambda

-function is identically 1 on the unit spheres, we answer negatively the question of whether the

λ
\lambda

-function characterizes rotund Banach spaces.
Totally real sets in 𝐶²
Authors: Alexander H.
Published: June 2010

We establish the polynomial convexity of certain totally real disks and of annuli in the unit torus satisfying a topological condition.
Sharply transitive linear groups over algebraically closed fields
Authors: Cherlin Gregory , Grundhöfer Theo , Nesin Ali , Völklein Helmut
Published: June 2010

We classify the groups from the title. The result is interpreted in terms of nearfields, and applied to a problem in the model theory of permutation groups.
A characterization of the dual of the classical Lorentz sequence space 𝑑(𝑤,𝑞)
Authors: Ariño Miguel A. , Muckenhoupt Benjamin
Published: June 2010

A new proof is given that regularity of

w
w

implies that the dual of the classical Lorentz sequence space

d
(
w
,
q
)

d(w,q)

is the nonclassical

d
(

w

−

q
′

/

q

,

q
′

)

d({w^{ - q’/q}},q’)

, where

1

/

q
+
1

/

q
′

=
1

1/q + 1/q’ = 1

. It is also shown that regularity is necessary for this equality to hold.
Determinant of the Neumann operator on smooth Jordan curves
Authors: Edward Julian , Wu Siye
Published: June 2010

Using the method of heat kernel expansion, the determinant of the Neumann operator on an arbitrary smooth Jordan curve is shown to be equal to the circumference.
Twistor spaces with meromorphic functions
Authors: Poon Y. S.
Published: June 2010

Among the class of Kähler surfaces with zero scalar curvature, only the twistor space of those surfaces which are also Ricci-flat can admit nonconstant meromorphic functions. Moreover, the transcendental degree of the function field of the twistor space over such surfaces is equal to one. This result leads to a classification of twistor spaces with positive algebraic dimensions.
Maximum modulus algebras and analytic varieties
Authors: Kumagai Donna , Slodkowski Zbigniew
Published: June 2010

Let

A
A

be a maximum modulus algebra on

X
X

, and

V
V

a maximal open subset of

X
X

such that

V
V

has the structure of one-dimensional variety on which functions from

A
A

are analytic. Then, the restriction algebra

A

X
∖
V

{A_{X\backslash V}}

is again a maximum modulus algebra.
The stable rank of crossed products of sectional 𝐶*-algebras by compact Lie groups
Authors: Schulz Eckart
Published: June 2010

Let

X
X

be a second countable, locally compact Hausdorff space, and let a compact Lie group

D
D

act on a

C

∗

{{\text {C}}^ * }

-bundle

(
E
,
X
)

(E,X)

, the fibres of which are of bounded, finite dimension. Denote by

A
A

the

C

∗

{{\text {C}}^ * }

-algebra of sections vanishing at infinity, and by

α
^

\hat \alpha

the induced action of

D
D

on

A
A

. The stable ranks of both, the fixed point algebra

A

α
^

{A^{\hat \alpha }}

and the crossed product algebra

A

×

α
^

D

A{ \times _{\hat \alpha }}D

are determined.
Instability of nonnegative solutions for a class of semipositone problems
Authors: Brown K. J. , Shivaji R.
Published: June 2010

We consider the boundary value problem\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row minus normal upper Delta u left-parenthesis x right-parenthesis equals lamda f left-parenthesis u left-parenthesis x right-parenthesis right-parenthesis comma x element-of normal upper Omega 2nd Row upper B u left-parenthesis x right-parenthesis equals 0 comma x element-of partial-differential normal upper Omega EndLayout">−Δu(x)=λf(u(x)),x∈ΩBu(x)=0,x∈∂Ω\begin {gathered} - \Delta u(x) = \lambda f(u(x)),\quad x \in \Omega \hfill \\ Bu(x) = 0,\quad x \in \partial \Omega \hfill \\ \end {gathered}\]whereΩ\Omegais a bounded region inRN{R^N}with smooth boundary,Bu=αh(x)u+(1−α)∂u/∂nBu = \alpha h(x)u + (1 - \alpha )\partial u/\partial nwhereα∈[0,1]h:∂Ω→R+\alpha \in [0,1]h:\partial \Omega \to {R^ + }withh=1h = 1whenα=1\alpha = 1,λ>0,f\lambda > 0,fis a smooth function withf(0)>0f(0) > 0(semipositone),f′(u)>0f’(u) > 0foru>0u > 0andf(u)≥0f(u) \geq 0foru>0u > 0. We prove that every nonnegative solution isunstable.
The Rademacher cotype of operators from 𝑙^{𝑁}_{∞}
Authors: Montgomery-Smith S. J. , Talagrand M.
Published: June 2010

We show that for any operator

T
:

l
∞
N

→
Y

T:l_\infty ^N \to Y

, where

Y
Y

is a Banach space, its cotype 2 constant,

K

(
2
)

(
T
)

{K^{(2)}}(T)

, is related to its

(
2
,
1
)

(2,1)

-summing norm,

π

2
,
1

(
T
)

{\pi _{2,1}}(T)

, by
\[

K

(
2
)

(
T
)
≤
c
log
⁡
log
⁡
N

π

2
,
1

(
T
)
.

{K^{(2)}}(T) \leq c\operatorname {log} \operatorname {log} N{\pi _{2,1}}(T).

\]
Thus, we can show that there is an operator

T
:
C
(
K
)
→
Y

T:C(K) \to Y

that has cotype 2, but is not

2
2

-summing.
A lower bound for the spectrum of the Laplacian in terms of sectional and Ricci curvature
Authors: Setti Alberto G.
Published: June 2010

Let

M
M

be an

n
n

-dimensional, complete, simply connected Riemannian manifold. In this paper we show that if the sectional curvature is bounded above by

−
k
≤
0

- k \leq 0

and the Ricci curvature is bounded above by

−
α
≤
0

- \alpha \leq 0

, then the spectrum of the Laplacian on

M
M

is bounded below by

[
α
+
(
n
−
1
)
(
n
−
2
)
k
]

/

4

[\alpha + (n - 1)(n - 2)k]/4

. This improves a previous result due to H. P. McKean.
Spread and local properties
Authors: Coutant Basil
Published: June 2010

A technique for combining the spread of a space and several local properties is exploited to obtain decompositions of a space and cardinal function bounds on the size and weight. It is shown that a locally countable set is the union of

ω
1

{\omega _1}

discrete sets and that under

M
A

(

ω
1

)

MA\left ( {{\omega _1}} \right )

the weight of a manifold is equal to its spread.
Radii problems for generalized sections of convex functions
Authors: Fournier Richard , Silverman Herb
Published: June 2010

A classical theorem of Szëgo states that for functions

f
(
z
)
=
z
+

Σ

k
=
2

∞

a
k

z
k

f(z) = z + \Sigma _{k = 2}^\infty {a_k}{z^k}

convex in

|

z

|

>
1

|z| > 1

, the sequence of partial sums

f
n

(
z
)
=
z
+

Σ

k
=
2

n

a
k

z
k

{f_n}(z) = z + \Sigma _{k = 2}^n{a_k}{z^k}

must be convex in

|

z

|

>

1
4

|z| > \frac {1}{4}

. For the more general family consisting of functions of the form

z
+

Σ

k
=
2

∞

a

n
k

z

n
k

z + \Sigma _{k = 2}^\infty {a_{{n_k}}}{z^{{n_k}}}

, where

{

n
k

}

\left \{ {{n_k}} \right \}

denotes an increasing (finite or infinite) sequence of integers

(
≥
2
)

( \geq 2)

, we find the radius of convexity

(
≈
0.21
)

( \approx 0.21)

and the radius of starlikeness

(
≈
0.37
)

( \approx 0.37)

. The extremal function in both cases is

z
+

z
2

/

(
1
−

z
2

)
=
z
+

Σ

k
=
1

∞

z

2
k

z + {z^2}/(1 - {z^2}) = z + \Sigma _{k = 1}^\infty {z^{2k}}

associated with the convex function

z

/

(
1
−
z
)
=
z
+

Σ

k
=
2

∞

z
k

z/(1 - z) = z + \Sigma _{k = 2}^\infty {z^k}

.
A congruence for the signature of an embedded manifold
Authors: Little Robert D.
Published: June 2010

Let

M

2
n

{M^{2n}}

be a smooth, closed, orientable

2
n

2n

-manifold and suppose that

K
x

2
n
−
2

K_x^{2n - 2}

is an orientable submanifold of

M

2
n

{M^{2n}}

dual to a cohomology class

x
x

. If

d
d

is a positive integer, the signatures of

K

d
x

2
n
−
2

K_{dx}^{2n - 2}

and

K
x

2
n
−
2

K_x^{2n - 2}

are related by a numerical congruence. If

n
n

is odd, then any codimension 2 submanifold of

C

P
n

{\mathbf {C}}{P^n}

fixed by a diffeomorphism of odd prime order is dual to the generator of the cohomology algebra.
Some estimates for harmonic measures. II
Authors: Jenkins James A.
Published: June 2010

FitzGerald, Rodin, and Warschawski proved that, for a continuum of given diameter in the closed unit disc, the harmonic measure at the center is minimized when it is an arc on the circumference. A very simple proof of this result is given, using the method of the extremal metric.
A characterization of submetacompactness in terms of products
Authors: Yajima Yukinobu
Published: June 2010

A space

X
X

is said to be suborthocompact if for every open cover

U

\mathcal {U}

of

X
X

there is a sequence

{

V

n

}

\{ {\mathcal {V}_n}\}

of open refinements of

U

\mathcal {U}

such that for each

x
∈
X

x \in X

there is some

n
∈
ω

n \in \omega

such that

∩
{
V
∈

V

n

:
x
∈
V
}

\cap \{ V \in {\mathcal {V}_n}:x \in V\}

is a neighborhood of

x
x

in

X
X

. It is proved that a Tychonoff space

X
X

is submetacompact if and only if the product

X
×
β
X

X \times \beta X

is suborthocompact.
Weighted decay estimate for the wave equation
Authors: Covachev Valéry , Georgiev Vladimir
Published: June 2010

The work is devoted to the proof of a new

L
∞

−

L
2

{L^\infty } - {L^2}

weighted estimate for the solution to the nonhomogeneous wave equation in

(

3
+
1

)

\left ( {3 + 1} \right )

-dimensional space-time. The weighted Sobolev spaces are associated with the generators of the Poincaré group. The estimate obtained is applied to prove the global existence of a solution to a nonlinear system of wave and Klein-Gordon equations with small initial data.
On monotone trajectories
Authors: Mierczyński Janusz
Published: June 2010

In this paper

C
1

{C^1}

strongly monotone dynamical systems are investigated. It is proved that the set of points with precompact orbits which converge to a not unstable equilibrium but whose trajectories are not eventually strongly monotone is nowhere dense. This improves on and extends a recent result by P. Poláčik [13].
Rado’s theorem for the Loeb space of an internal *-finitely additive measure space
Authors: Živaljević Boško
Published: June 2010

A version of Rado’s theorem about the existence of a family of subsets of a given family of sets combinatorially similar to another family of sets is proved for the Loeb space of an internal

∗
*

-finitely additive measure space. As a corollary we obtain the Loeb measured case of the result of B. Bollobas and N. Th. Varopoulos about the existence of a family of mutually disjoint measurable subsets of the given family of measurable sets, having the prescribed measure.
Embedding Galois problems and reduced norms
Authors: Crespo Teresa
Published: June 2010

For certain embedding problems

G
~

→
G
≃

Gal

(

L

|
K

)

\tilde G \to G \simeq {\text {Gal}}\left ( {L\left | K \right .} \right )

associated to a representation

t
:
G
→

Aut

A

t:G \to {\text {Aut}}A

of the group

G
G

by automorphisms of a central simple

K
K

-algebra

A
A

of dimension

n
2

{n^2}

, we prove that the solutions are the fields

L

(

(

r
N

(
z
)

)

1

/

n

)

L\left ( {{{\left ( {rN\left ( z \right )} \right )}^{1/n}}} \right )

, with

r
r

running over

K
∗

/

K

∗
n

{K^ * }/{K^{ * n}}

and

N

(
z
)

N\left ( z \right )

the reduced norm of an invertible element

z
z

in the algebra

B
⊗
L

B \otimes L

, for

B
B

the twisted algebra of

A
A

by

t
t

.
Products of perfectly meagre sets
Authors: Recław Ireneusz
Published: June 2010

We show that there exists a perfect set

D
⊆

2
ω

×

2
ω

D \subseteq {2^\omega } \times {2^\omega }

such that for every Luzin set in

D
D

both projections of it are perfectly meagre. It follows (under CH) that the product of two perfectly meagre sets need not be perfectly meagre (or even have the Baire property in the restricted sense). This provides an answer to a 55-year-old question of Marczewski.
Harmonic two-forms in four dimensions
Authors: Seaman Walter
Published: June 2010

Conformal invariance of middle-dimensional harmonic forms is used to improve Kato’s inequality for four-manifolds. An application to positively curved four-manifolds is given.
Dimension of certain kernel spaces of linear operators
Authors: Shen Zuowei
Published: June 2010

Let

G
G

be a semigroup of linear operators on a vector space

S
S

into itself with the operation of composition. A subset of

G
G

may be associated with a matroid

X
X

. We discuss the dimension of the kernels of certain linear operators induced in a natural way by the matroid structure on

X
X

.
On the existence and uniqueness of fixed points for holomorphic maps in complex Banach spaces
Authors: Włodarczyk Kazimierz
Published: June 2010

We consider the problem of the existence and uniqueness of fixed points in

X
X

of holomorphic maps

F
:
X
→
X

F:X \to X

of bounded open convex sets

X
X

in complex Banach spaces

E
E

. As a result of the Earle-Hamilton theorem, the problem in the case where

F
(
X
)

F(X)

lies strictly inside

X
X

(i.e.,

dist
⁡
[
F
(
X
)
,
E
∖
X
]
>
0
)

\operatorname {dist}[F(X),E\backslash X] > 0)

has a solution. In this article we show that this problem is also solved in the case where

F
(
X
)

F(X)

does not lie strictly inside

X
X

(i.e.,

dist
⁡
[
F
(
X
)
,
E
∖
X
]
=
0
)

\operatorname {dist}[F(X),E\backslash X] = 0)

whenever: (i)

F
F

is compact; (ii)

F
F

is continuous on

X
¯

\bar {X}

and

F
(

X
¯

)
⊂

X
¯

F(\bar X) \subset \bar X

; (iii)

F
F

has no fixed points on

∂
X

\partial X

; and (iv) for each

x
∈
X

x \in X

, 1 is not contained in the spectrum of

D
F
(
x
)

DF(x)

.
Determinacy of complex analytic foliation germs without integrating factors
Authors: Cerveau Dominique , Suwa Tatsuo
Published: June 2010

It is shown that for a codimension 1 foliation germ without formal integrating factors, the finiteness of the dimension of the space of isomorphism classes of first order unfoldings implies its local finite determinacy. An effective estimate of the order of determinacy is given for general foliation germs in dimension 2. A simple proof for the stability of a foliation germ defined by a simple form in dimension 3 is also included.
L’espace des fonctions continues d’un espace métrique dénombrable
Authors: Cauty Robert
Published: June 2010

We prove that, for any countable nondiscrete metric space

X
X

, the space of continuous real-valued functions on

X
X

, with the topology of pointwise convergence, and its subspace of bounded functions, are both homeomorphic to

σ
ω

{\sigma _\omega }

, the countably infinite product of copies of

l
f
2

l_f^2

.
Spin characteristic classes and reduced 𝐾𝑆𝑝𝑖𝑛 group of a low-dimensional complex
Authors: Li Bang He , Duan Hai Bao
Published: June 2010

This note studies relations between Spin bundles, over a CW-complex of dimension

≤
9

\leq 9

, and their first two Spin characteristic classes. In particular by taking Spin characteristic classes, it is proved that the stable classes of Spin bundles over a manifold

M
M

with dimension

≤
8

\leq 8

are in one to one correspondence with the pairs of cohomology classes
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis q 1 comma q 2 right-parenthesis element-of upper H Superscript 4 Baseline left-parenthesis upper M semicolon double-struck upper Z right-parenthesis times upper H Superscript 8 Baseline left-parenthesis upper M semicolon double-struck upper Z right-parenthesis">

(

q
1

,

q
2

)
∈

H
4

(
M
;

Z

)
×

H
8

(
M
;

Z

)

({q_1},{q_2}) \in {H^4}(M;\mathbb {Z}) \times {H^8}(M;\mathbb {Z})

satisfying
\[

(

q
1

∪

q
2

+

q
2

)
mod
3
+

U
3
1

∪
(

q
1

mod
3
)
≡
0

({q_1} \cup {q_2} + {q_2})\bmod 3 + U_3^1 \cup ({q_1}\bmod 3) \equiv 0

\]
, where

U
3
1

∈

H
4

(
M
;

Z

3

)

U_3^1 \in {H^4}(M;{\mathbb {Z}_3})

is the indicated Wu-class of

M
M

. As an application a computation is made for

K
Spin

~

(
M
)

\widetilde {K\operatorname {Spin} }(M)

, where

M
M

is an eight-dimensional manifold with understood cohomology rings over

Z

,

Z

2

,

\mathbb {Z},{\mathbb {Z}_2},

, and

Z

3

{\mathbb {Z}_3}

.
On weak containment properties
Authors: Rindler Harald
Published: June 2010

We prove, that two concepts of weak containment do not coincide, contradicting results in [1, Lemma 3.3 and Proposition 3.4]. The statement of Theorem 3.5 remains valid. There exist infinite tall compact groups

G
G

(i.e. the set

{
σ
∈

G
^

,
dim
⁡
σ
=
n
}

\{ \sigma \in \hat G,\dim \sigma = n\}

is finite for each positive integer

n
n

) having the mean-zero weak containment property. Such groups do not have the dual Bohr approximation property or

A
P
(

G
^

)
≠

C
δ
∗

(
G
)

AP(\hat G) \ne C_\delta ^*(G)

.
On the mean curvature estimates for bounded submanifolds
Authors: Coghlan Leslie , Itokawa Yoe , Kosecki Roman
Published: June 2010

A Liouville-type theorem is proved for strongly subharmonic functions on complete riemannian manifolds of bounded curvature. We use this to give a simple proof of a theorem of Jorge. Koutroufiotis and Xavier, which gives an estimate for the exterior size of a submanifold in terms of the sup of the length of its mean curvature.
Orderings with 𝛼th jump degree 0^{(𝛼)}
Authors: Downey Rodney , Knight Julia F.
Published: June 2010

This paper completes an investigation of "jumps" of orderings. The last few cases are given in the proof that for each recursive ordinal

α
≥
1

\alpha \geq 1

and for each Turing degree

d

≥

0

(
α
)

{\mathbf {d}} \geq {{\mathbf {0}}^{(\alpha )}}

, there is a linear ordering

A

{\mathbf {A}}

such that

d

{\mathbf {d}}

is least among the

α
\alpha

th jumps of degrees of (open diagrams of) isomorphic copies of

A

{\mathbf {A}}

, and for

β
>
α

\beta > \alpha

, the set of

β
\beta

jumps of degrees of copies of

A

{\mathbf {A}}

has no least element.