bluuebunny/crossref_metadata_2025 huggingface dataset:
Unital multiplications on a Hilbert space
Authors: Froelich John
Published: July 2010

We provide a new proof of two theorems of Ingelstam that identify the Hilbert spaces with a unital multiplication satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue x y StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals StartAbsoluteValue EndAbsoluteValue x StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue EndAbsoluteValue y StartAbsoluteValue EndAbsoluteValue">

|

|

x
y

|

|

⩽

|

|

x

|

|

|

|

y

|

|

||xy|| \leqslant ||x||\,||y||

and

|

|

1

|

|

=
1

||1|| = 1

.
6-torsion and hyperbolic volume
Authors: Gehring F. W. , Martin G. J.
Published: July 2010

The Kleinian group

PGL
⁡
(
2
,

O
3

)

\operatorname {PGL} (2,\,{O_3})

is shown to have minimal covolume

(
≈
0.0846
…
)

( \approx 0.0846 \ldots )

among all Kleinian groups containing torsion of order

6
6

(the associated hyperbolic orbifold is also the minimal volume cusped orbifold). This follows from: Any cocompact Kleinian group with torsion of order

6
6

has covolume at least

1
9

\tfrac {1} {9}

. As a consequence, any compact hyperbolic manifold with a symmetry of order

6
6

(with fixed points) has volume at least

4
3

\tfrac {4} {3}

. These results follow from new collaring theorems for torsion in a Kleinian group arising from our generalizations of the Shimizu-Leutbecher inequality.
Symmetry-breakings for semilinear elliptic equations on finite cylindrical domains
Authors: Lin Song-Sun
Published: July 2010

We study the existence and multiplicity of asymmetric positive solutions of a semilinear elliptic equation on finite cylinders with mixed type boundary conditions. By using a Nehari-type variational method, we prove that the numbers of asymmetric positive solutions are increasing without bound when the lengths of cylinders are increasing. On the contrary, by using the blow up technique, we obtain an a priori bound for positive solutions and then prove that all positive solutions must be symmetric when the cylinders are short enough.
Remarks on the two well problem with rotations
Authors: Zhang Kewei
Published: July 2010

A conjecture of I. Fonseca on the form of minimizers of a double-well problem arising in the study of elastic crystal microstructure is answered to be true.
Lie and Jordan ideals in 𝐵(𝑐₀) and 𝐵(𝑙ᵨ)
Authors: Förster K.-H. , Nagy B.
Published: July 2010

It is shown that ideals with respect to the canonical Lie (commutator) product in these algebras are exactly the linear manifolds that contain the images of their elements under the action of inner automorphisms induced by invertible spectral operators of scalar type. Jordan ideals in these algebras are identical with two-sided associative ideals and are also applied to a characterization of Lie ideals.
A note on nonlinear Volterra operators
Authors: Mandhyan Indur
Published: July 2010

In this paper we discuss a class of nonlinear Volterra operators and show that they are globally invertible.
Chern-Simons-Maslov classes of some symplectic vector bundles
Authors: Suzuki Haruo
Published: July 2010

Let

E
0

,

J
0

{E_0},\;{J_0}

, and

L
0

{L_0}

be the symplectic

2
n

2n

-vector bundle, the compatible complex operator, and the Lagrangian subbundle that are determined by the

U
(
n
)

U(n)

-extension of the principal

O
(
n
)

O(n)

-bundle

U
(
n
)
→
U
(
n
)

/

O
(
n
)

U(n) \to U(n)/O(n)

. We compute the Chern-Simons-Maslov class

μ
1

(

E
0

,

J
0

,

L
0

)

{\mu ^1}({E_0},{J_0},{L_0})

. Then for a trivial symplectic

2
n

2n

-bundle

E
E

, a compatible complex operator

J
J

, and a Lagrangian subbundle

L
L

, we compute Chern-Simons-Maslov classes

μ
h

(
E
,
J
,
L
)

{\mu ^h}(E,J,L)

under some condition on the base space of

E
E

.
Phi-stable operators and inner approximation-solvability
Authors: Verma Ram U.
Published: July 2010

We extend, by applying a theorem of Petryshyn (1970), the approximation-solvability of the nonlinear functional equations involving strongly stable Hilbert space mappings to the case of strongly

ϕ
\phi

-stable mappings—a new and rather general class of mappings. These mappings constitute a generalization of monotone mappings. Finally, we upgrade the obtained results to the case of Banach space mappings.
A seminorm with square property on a Banach algebra is submultiplicative
Authors: Bhatt S. J.
Published: July 2010

The result stated in the title is proved in a Banach algebra and is used to discuss (i) commutativity criteria in normed algebras, (ii) uniqueness of the uniform norm in uniform Banach algebras, and (iii) existence of continuous multiplicative linear functionals on topological algebras together with a simple reduction of the Michael problem in Fréchet algebras. Submultiplicativity does not imply subadditivity in the presence of the square property.
Weakly confluent, 2-to-1 maps on hereditarily indecomposable continua
Authors: Heath Jo W.
Published: July 2010

One of the most important open questions today in the study of exactly

2
2

-to-

1
1

maps is whether or not such a map can be defined on the pseudoarc. We show that there is no weakly confluent

2
2

-to-

1
1

map defined on the pseudoarc. More precisely, it is shown that any reduced weakly confluent

2
2

-to-

1
1

map defined on a hereditarily indecomposable metric continuum must be a confluent local homeomorphism. It follows from this that if there is a weakly confluent

2
2

-to-

1
1

map from a hereditarily indecomposable continuum

X
X

onto a continuum

Y
Y

, then neither

X
X

nor

Y
Y

can be treelike.
A Volterra type derivative of the Lebesgue integral
Authors: Pfeffer Washek F.
Published: July 2010

Using functions of bounded variation, we define a Volterra type derivative of the linear functional associated with a Lebesgue integrable function and show that it is equal to this function almost everywhere.
Short chains and short cycles of modules
Authors: Reiten I. , Skowroński A. , Smalø S. O.
Published: July 2010

We show that for a large class of artin algebras including the algebras of finite representation type, an indecomposable module

M
M

is not the middle of a short chain if and only if there is no short cycle

M
→
N
→
M

M \to N \to M

of nonzero nonisomorphisms between indecomposable modules. We apply this to get sufficient conditions for modules to be determined by their composition factors. We also show that if for an algebra of finite representation type there is a sincere indecomposable

Λ
\Lambda

-module that is not the middle of a short chain, then

Λ
\Lambda

is a tilted algebra.
Positive harmonic majorization of the real part of a holomorphic function
Authors: Gardiner Stephen J.
Published: July 2010

Let

U
U

be the unit disc. This paper investigates which domains

D
D

in the complex plane have the property that

R

e
f

\mathcal {R}ef

belongs to

h
1

{h^1}

, or the more restrictive property that

e
f

{e^f}

belongs to the Smirnov class

N

+

{\mathcal {N}^ + }

, for every holomorphic function

f
:
U
→
D

f:U \to D

.
Distortion of sets by inner functions
Authors: Hamilton D. H.
Published: July 2010

For any inner function

f
f

with

f
(
0
)
=
0

f(0) = 0

and any Borel set

E
⊂

D

E \subset {\mathbf {D}}

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript alpha Baseline left-parenthesis z element-of bold upper D colon f left-parenthesis z right-parenthesis element-of upper E right-parenthesis greater-than-or-slanted-equals upper M Subscript alpha Baseline left-parenthesis upper E right-parenthesis comma 0 greater-than alpha less-than-or-slanted-equals 1 comma">

M
α

(
z
∈

D

:
f
(
z
)
∈
E
)
⩾

M
α

(
E
)
,

0
>
α
⩽
1
,

{M_\alpha }(z \in {\mathbf {D}}:f(z) \in E) \geqslant {M_\alpha }(E),\qquad 0 > \alpha \leqslant 1,

\]
where

M
α

{M_\alpha }

denotes

α
\alpha

-dimensional Hausdorff measure. In the case that

0
>

M
α

(
E
)
>
∞

0 > {M_\alpha }(E) > \infty

we have equality only for rotations of the identity.
The polyadic completion of a transformation algebra
Authors: Fleischer Isidore
Published: July 2010

The (faithful) polyadic completion was constructed in a special case using

{
0
,
1
}

\{ 0,1\}

-valued homomorphisms by Leblanc. Here a different method— adjoining extrema freely to the underlying Boolean algebra—succeeds in full generality. This is based on an apparently new axiomatization of locally finite polyadic algebras of infinite degree within the class of transformation algebras as those having certain extrema that the transformations preserve.
Embedding subspaces of 𝑙ⁿ_{∞} into spaces with Schauder basis
Authors: Mankiewicz Piotr , Tomczak-Jaegermann Nicole
Published: July 2010

It is proved that for sufficiently small

ε
>
0

\varepsilon > 0

and any

0
>
δ
>
1

/

2

0 > \delta > 1/2

, a random

n
n

-dimensional subspace

E
E

of

l
∞
N

l_\infty ^N

, where

N
=
(
1
+
ε
)
n

N = (1 + \varepsilon )n

, has the property: whenever

E
E

is embedded into any

(
1
+
γ
)
n

(1 + \gamma )n

-dimensional space with a basis, where

γ
=
c
δ
ε

\gamma = c\delta \varepsilon

, then the embedding constant exceeds

c
′

n

1

/

2
−
δ

{c’}{n^{1/2 - \delta }}

.
On the strong Parrott completion problem
Authors: Bakonyi Mihály , Woerdeman Hugo J.
Published: July 2010

In this paper we parametrize all solutions of the Strong Parrott problem and obtain necessary and sufficient conditions for existence of isometric, coisometric, and unitary solutions.
Some special cases of Bang’s inequality
Authors: Hunter Henry F.
Published: July 2010

Bang’s inequality concerning the relative widths of

n
n

planks covering a convex body is considered. Proofs are given that show just when equality occurs for two special cases in the plane, namely, two planks covering any convex body and three planks covering a triangle.
The total split curvatures of knotted space-like 2-spheres in Minkowski 4-space
Authors: Kossowski Marek
Published: July 2010

In this paper we establish an analogue of the Fary-Milnor Theorem for space-like

2
2

-spheres in Minkowski

4
4

-space.
Indecomposable continua and the Julia sets of polynomials
Authors: Mayer John C. , Rogers James T.
Published: July 2010

We find several necessary and sufficient conditions for the Julia set

J
J

of a polynomial of degree

d
⩾
2

d \geqslant 2

to be an indecomposable continuum. One condition that may be easier to check than others is the following: Suppose

J
J

is connected; then

J
J

is an indecomposable continuum iff the impression of some prime end of the unbounded complementary domain of

J
J

has interior in

J
J

.
Remarks on localization and standard modules: the duality theorem on a generalized flag variety
Authors: Chang Jen-Tseh
Published: July 2010

An amplification of the duality theorem of Hecht, Miličić, Schmid, and Wolf is given in the setting of a generalized flag variety. As an application, we give a different proof of the reducibility theorem for principal series by Speh and Vogan.
A remark on Hörmander’s uniqueness theorem
Authors: Del Santo D. , Saint Raymond X.
Published: July 2010

By using the paradifferential calculus, Hörmander’s classical uniqueness theorem for the Cauchy problem is shown to hold for operators with

C

2

{\mathcal {C}^2}

coefficients in the principal part, instead of

C

∞

{\mathcal {C}^\infty }

, under a special normality assumption.
Dense pure subgroups of locally compact groups
Authors: Kabenyuk M. I.
Published: July 2010

We prove that a locally compact abelian (LCA) group has no proper dense pure subgroups if and only if it does not have proper dense subgroups. This solves a problem of Armacost.
Some optimal bivariate Bonferroni-type bounds
Authors: Galambos Janos , Xu Yuan
Published: July 2010

Let

A
1

,

A
2

,
…
,

A
n

{A_1},{A_2}, \ldots ,{A_n}

and

B
1

,

B
2

,
…
,

B
m

{B_1},{B_2}, \ldots ,{B_m}

be two sets of events on a probability space. Let

X
n

{X_n}

and

Y
m

{Y_m}

be the number of those

A
j

{A_j}

and

B
s

{B_s}

, respectively, that occur. Let

S

k
,
t

{S_{k,t}}

be the

(
k
,
t
)

th

(k,t){\text {th}}

binomial moment of the vector

(

X
n

,

Y
m

)

({X_n},{Y_m})

. We establish optimal bounds on

P
(

X
n

⩾
1
,

Y
m

⩾
1
)

P({X_n} \geqslant 1,{Y_m} \geqslant 1)

by means of linear combinations of

S

1
,
1

,

S

2
,
1

,

S

1
,
2

{S_{1,1}},\;{S_{2,1}},\;{S_{1,2}}

and

S

2
,
2

{S_{2,2}}

. Optimal lower bounds are also determined when only

S

1
,
1

,

S

2
,
1

{S_{1,1}},\;{S_{2,1}}

and

S

1
,
2

{S_{1,2}}

are utilized.
Spaces whose 𝑛th power is weakly infinite-dimensional but whose (𝑛+1)th power is not
Authors: Pol Elżbieta
Published: July 2010

For every natural number

n
n

we construct a metrizable separable space

Y
Y

such that

Y
n

{Y^n}

is weakly infinite-dimensional (moreover, is a

C
C

-space) but

Y

n
+
1

{Y^{n + 1}}

is strongly infinite-dimensional.
Moment conditions for almost sure convergence of weakly correlated random variables
Authors: Bryc W. , Smoleński W.
Published: July 2010

For random sequences with unrestricted maximal correlation coefficient strictly less than

1
1

, sufficient moment conditions for almost sure convergence of a series and for the strong law of large numbers are given.
Asymptotics of the negative discrete spectrum of a class of Schrödinger operators with large coupling constant
Authors: Laptev Ari
Published: July 2010

We obtain the asymptotics of the negative discrete spectrum of the Schrödinger operator with a large coupling constant and potentials

V
∉

L

m

/

2

(

R
m

)
,

m
⩾
3

V \notin {L_{m/2}}({R^m}),\;m \geqslant 3

. The result is very sensitive to small perturbations of the potential and depends on the negative spectrum of some auxiliary differential problems on

S

m
−
1

{S^{m - 1}}

.
An analogue of Lyapunov’s criterion for (𝑚,𝑛-𝑚)-disfocality
Authors: Umamaheswaram S.
Published: July 2010

An integral criterion for the

(
m
,
n
−
m
)

(m,n - m)

-disfocality

(
1
⩽
m
⩽
n
−
1
)

(1 \leqslant m \leqslant n - 1)

on

[
a
,
b
]

[a,b]

of a linear nth order differential equation is given. The criterion is analogous to that of Lyapunov (1947) and Nehari (1962).
An example of Kac algebra actions on von Neumann algebras
Authors: Yamanouchi Takehiko
Published: July 2010

By using Majid’s bicrossproduct Kac algebra, a nontrivial example of an action of a Kac algebra on a von Neumann algebra is given. It is shown that the actions constructed are ergodic. The crossed products by the actions are examined.
Growth conditions for thin sets in Vilenkin groups of bounded order
Authors: Grubb D. J.
Published: July 2010

Let

G
G

be a Vilenkin group of bounded order and

H
n

{H_n}

a sequence of clopen subgroups of

G
G

forming a base at the identity. If

E
E

is a subset of

G
G

, let

N
n

(
E
)

{N_n}(E)

denote the number of cosets of

H
n

{H_n}

which intersect

E
E

. If
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript bar Endscripts StartFraction upper N Subscript n Baseline left-parenthesis upper E right-parenthesis Over log left-bracket upper G colon upper H Subscript n Baseline right-bracket EndFraction greater-than normal infinity comma">

lim
_

N
n

(
E
)

log
⁡
[
G
:

H
n

]

>
∞
,

\underline {\lim } \frac {{{N_n}(E)}} {{\log [G:{H_n}]}} > \infty ,

\]
then

E
E

is a U-set in the group

G
G

. It is also shown that for

G
G

satisfying a growth condition and

φ
(
n
)
→
∞

\varphi (n) \to \infty

, there is an M-set,

E
E

, with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N Subscript n Baseline left-parenthesis upper E right-parenthesis equals upper O left-parenthesis phi left-parenthesis n right-parenthesis log left-bracket upper G colon upper H Subscript n Baseline right-bracket right-parenthesis period">

N
n

(
E
)
=
O
(
φ
(
n
)

log
⁡
[
G
:

H
n

]
)
.

{N_n}(E) = O(\varphi (n)\,\log [G:{H_n}]).

\]
A note on medial division groupoids
Authors: Ježek J. , Kepka T.
Published: July 2010

In 1949 Sholander showed that every medial cancellation groupoid can be embedded into a medial quasigroup. In this note we prove the dual assertion that every medial division groupoid is a homomorphic image of a medial quasigroup.
Diagonalization in compact Lie algebras and a new proof of a theorem of Kostant
Authors: Wildberger N. J.
Published: July 2010

We exhibit a simple algorithmic procedure to show that any element of a compact Lie algebra is conjugate to an element of a fixed maximal abelian subalgebra. An estimate of the convergence of the algorithm is obtained. As an application, we provide a new proof of Kostant’s theorem on the projection of orbits onto a maximal abelian subalgebra.
A bridge between complex geometry and Riemannian geometry
Authors: Cassa Antonio
Published: July 2010

Every complex manifold

M
n

{M^n}

with holomorphic metric can be obtained (at least locally) from a complex manifold

E

2
∙
n
−
2

{E^{2 \bullet n - 2}}

and one of its

C

\mathbb {C}

principal bundles

L
L

. The manifold

E
E

is made of all (signed) complex geodesies of

M
M

and

L
L

is the bundle on

E
E

of all choices of "times" evolving along the geodesies.
The iterated weak Hilbert procedure
Authors: Defant Martin , Junge Marius
Published: July 2010

Applying Pisier’s concept of weak properties to weak Hilbert spaces we obtain the so-called weak weak Hilbert spaces. Our main result is that the classes of weak Hilbert spaces and of weak weak Hilbert spaces coincide. On the other hand we show that a generalization for operators does not hold.
Estimates for character sums
Authors: Friedlander J. , Iwaniec H.
Published: July 2010

We give a number of estimates for character sums
\[

∑

a
∈

A

∑

b
∈

B

χ
(
a
+
b
)

\sum \limits _{a \in \mathcal {A}} {\sum \limits _{b \in \mathcal {B}} {\chi (a + b)} }

\]
for rather general sets

A

,

B

\mathcal {A},\;\mathcal {B}

. These give, in particular, a modified proof of the inequalities of Pólya-Vinogradov and of Burgess, which displays the latter as a generalization of the former.
A general Hopf lemma and proper holomorphic mappings between convex domains in 𝐶ⁿ
Authors: Mercer Peter R.
Published: July 2010

We use a general version of the well-known Hopf lemma to study boundary regularity of proper holomorphic mappings between some bounded convex domains in

C

n

{\mathbb {C}^n}

which carry no boundary regularity assumption.
Conjugation and excess in the Steenrod algebra
Authors: Silverman Judith H.
Published: July 2010

In this paper we prove a formula involving the canonical antiautomorphism

χ
\chi

of the

mod

-

2

\bmod {\text {-}}2

Steenrod algebra

A

(
2
)

\mathcal {A}(2)

, namely,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row chi left-parenthesis upper S q Superscript 2 Super Superscript j Superscript left-parenthesis 2 Super Superscript i plus 1 Superscript minus 1 right-parenthesis Baseline upper S q Superscript 2 Super Superscript j minus 1 Superscript left-parenthesis 2 Super Superscript i plus 1 Superscript minus 1 right-parenthesis Baseline midline-horizontal-ellipsis upper S q Superscript left-parenthesis 2 Super Superscript i plus 1 Superscript minus 1 right-parenthesis Baseline right-parenthesis 2nd Row equals upper S q Superscript 2 Super Superscript i Superscript left-parenthesis 2 Super Superscript j plus 1 Superscript minus 1 right-parenthesis Baseline upper S q Superscript 2 Super Superscript i minus 1 Superscript left-parenthesis 2 Super Superscript j plus 1 Superscript minus 1 right-parenthesis Baseline midline-horizontal-ellipsis upper S q Superscript left-parenthesis 2 Super Superscript j plus 1 Superscript minus 1 right-parenthesis Baseline comma EndLayout">

χ
(
S

q

2
j

(

2

i
+
1

−
1
)

S

q

2

j
−
1

(

2

i
+
1

−
1
)

⋯
S

q

(

2

i
+
1

−
1
)

)

=
S

q

2
i

(

2

j
+
1

−
1
)

S

q

2

i
−
1

(

2

j
+
1

−
1
)

⋯
S

q

(

2

j
+
1

−
1
)

,

\begin {array}{*{20}{c}} {\chi (S{q^{{2^j}({2^{i + 1}} - 1)}}S{q^{{2^{j - 1}}({2^{i + 1}} - 1)}} \cdots S{q^{({2^{i + 1}} - 1)}})} \\ {\qquad \qquad = S{q^{{2^i}({2^{j + 1}} - 1)}}S{q^{{2^{i - 1}}({2^{j + 1}} - 1)}} \cdots S{q^{({2^{j + 1}} - 1)}},} \\ \end {array}

\]
and discuss its implications for the study of the image of the

A

(
2
)

\mathcal {A}(2)

-action on

F

2

[

x
1

,
⋯
,

x
s

]

{\mathbb {F}_2}[{x_1}, \cdots ,{x_s}]

.
The Sylow 𝑝-subgroups of semicomplete nilpotent groups
Authors: Dixon Martyn R. , Rigsby E. Myron
Published: July 2010

A nilpotent group whose group of outer automorphisms is trivial may contain elements of finite order. This paper is concerned with how large the Sylow

p
p

-subgroups of such a group can be. We show that in many cases the Sylow

p
p

-subgroups of such a semicomplete nilpotent group are always finite.
An example of a homeomorphism of the plane where the set of 𝜔-limit points is not a closed set
Authors: Hero Michael W.
Published: July 2010

Using a self-map of an interval and its associated inverse limit space we construct an example of a homeomorphism of the plane

h
h

such that the set of

ω
\omega

-limit points of

h
h

is not a closed set.
Reduction numbers and Rees algebras of powers of an ideal
Authors: Hoa Lê Tuan
Published: July 2010

Let

I
I

be an ideal in a Noetherian local ring

(
R
,

m

)

(R,\mathfrak {m})

. It is shown that for

n
≫
0

n \gg 0

the reduction number

r
J

(

I
n

)

{r_J}({I^n})

of

I
n

{I^n}

with respect to a minimal reduction

J
J

is not only independent from the choice of

J
J

but also is stable. If

I
I

is an

m

\mathfrak {m}

-primary ideal, we give a criterion for the Rees algebra

R
[

I
n

t
]

R[{I^n}t]

with

n
≫
0

n \gg 0

to be Cohen-Macaulay.
Composition operators on analytic Lipschitz spaces
Authors: Madigan Kevin M.
Published: July 2010

If

X
X

is a Banach space of functions analytic on the disk and

φ
:

D

→

D

\varphi :{\mathbf {D}} \to {\mathbf {D}}

is analytic, one can define the composition operator

C
φ

{C_\varphi }

on

X
X

by

C
φ

f
:=
f
∘
φ

{C_\varphi }f: = f \circ \varphi

. This paper discusses the boundedness and

w
w

-compactness of composition operators on the analytic Lipschitz spaces

A

α

,

0
>
α
>
1

{\mathcal {A}_\alpha },\;0 > \alpha > 1

.
Weakly convergent sequence coefficient of product space
Authors: Zhang Guang Lu
Published: July 2010

W. L. Bynum introduced the weakly convergent sequence coefficient

WCS
⁡
(
X
)

\operatorname {WCS} (X)

of the Banach space

X
X

as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W upper C upper S left-parenthesis upper X right-parenthesis equals sup left-brace upper M colon for each weakly convergent sequence left-brace x Subscript n Baseline right-brace in upper X comma there is some y element-of ModifyingAbove c o With bar left-parenthesis left-brace x Subscript n Baseline right-brace right-parenthesis such that upper M dot limit sup StartAbsoluteValue EndAbsoluteValue x Subscript n Baseline minus y StartAbsoluteValue EndAbsoluteValue less-than-or-slanted-equals upper A left-parenthesis left-brace x Subscript n Baseline right-brace right-parenthesis right-brace">

WCS
⁡
(
X
)
=

sup

{
M
:

for each weakly convergent sequence

{

x
n

}

in

X
,

there is some 

y
∈

co
¯

(
{

x
n

}
)

such that 

M
⋅
lim
sup

|

|

x
n

−
y

|

|

⩽
A
(
{

x
n

}
)
}

\operatorname {WCS} (X) = {\text {sup}}\{ M:{\text {for each weakly convergent sequence}}\;\{ {x_n}\} \;{\text {in}}\;X,\;{\text {there is some }}y \in \overline {\operatorname {co} } (\{ {x_n}\} )\;{\text {such that }}M \cdot \lim \sup ||{x_n} - y|| \leqslant A(\{ {x_n}\} )\}

. We consider the weakly convergent sequence coefficient of the

l
p

{l_p}

-product space

Z
=
(

∏

i
=
1

n

X
i

)

l
p

Z = (\prod \nolimits _{i = 1}^n {{X_i}{)_{lp}}}

of the finite non-Schur space

X
1

,
…
,

X
n

{X_1}, \ldots ,{X_n}

and show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W upper C upper S left-parenthesis upper Z right-parenthesis equals min left-brace right-brace colon of WCSWCS left-parenthesis right-parenthesis Xi colon less-than-or-slanted-equals less-than-or-slanted-equals less-than-or-slanted-equals 1 in">

WCS
⁡
(
Z
)
=
min
{
WCS
⁡
(

X
i

)
:
1
⩽
i
⩽
n
}

\operatorname {WCS} (Z) = \min \{ \operatorname {WCS} ({X_i}):1 \leqslant i \leqslant n\}

.
On the existence of periodic solutions for scalar second order differential equations when only the asymptotic behaviour of the potential is known
Authors: Fonda Alessandro
Published: July 2010

Assuming only asymptotic conditions on the potential function, we prove the existence of periodic solutions for equations whose nonlinearity stays below the first curve of Fučik’s spectrum.
Decomposition of Peano derivatives
Authors: Fejzić Hajrudin
Published: July 2010

Let

Δ
′

{\Delta ’}

be the class of all derivatives, and let

[

Δ
′

]

[{\Delta ’}]

be the vector space generated by

Δ
′

{\Delta ’}

and O’Malley’s class

B
1

∗

B_1^{\ast }

. In [1] it is shown that every function in

[

Δ
′

]

[{\Delta ’}]

is of the form

g
′

+
h

k
′

{g’} + h{k’}

, where

g
,
h

g,h

, and

k
k

are differentiable, and that

f
∈
[

Δ
′

]

f \in [{\Delta ’}]

if and only if there is a sequence of derivatives

v
n

{v_n}

and closed sets

A
n

{A_n}

such that

∪

n
=
1

∞

A
n

=

R

\cup _{n = 1}^\infty {A_n} = {\mathbf {R}}

and

f
=

v
n

f = {v_n}

on

A
n

{A_n}

. The sequence of sets

A
n

{A_n}

together with the corresponding functions

v
n

{v_n}

is called a decomposition of

f
f

. In this paper we show that every Peano derivative belongs to

[

Δ
′

]

[{\Delta ’}]

. Also we show that for Peano derivatives the sets

A
n

{A_n}

can be chosen to be perfect.
On the spectrum of the Neumann Laplacian of long-range horns: a note on the Davies-Simon theorem
Authors: Jakšić V.
Published: July 2010

For a class of regions with cusps (e.g.,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega equals left-brace left-parenthesis x comma y right-parenthesis colon x greater-than 1 comma StartAbsoluteValue y EndAbsoluteValue greater-than exp left-parenthesis minus x Superscript alpha Baseline right-parenthesis right-brace comma 0 greater-than alpha greater-than 1 right-parenthesis">

Ω
=
{
(
x
,
y
)
:
x
>
1
,

|

y

|

>
exp
⁡
(
−

x
α

)
}
,

0
>
α
>
1
)

\Omega = \{ (x,y):x > 1,|y| > \exp ( - {x^\alpha })\} ,\;0 > \alpha > 1)

) we show that

σ

ac

(
−

Δ
N
Ω

)
=
[
0
,
∞
)

{\sigma _{\operatorname {ac} }}( - \Delta _N^\Omega ) = [0,\infty )

of uniform multiplicity one,

σ

sing

(
−

Δ
N
Ω

)
=
∅

{\sigma _{\operatorname {sing} }}( - \Delta _N^\Omega ) = \emptyset

, and

σ

pp

(
−

Δ
N
Ω

)

{\sigma _{\operatorname {pp} }}( - \Delta _N^\Omega )

consists of a discrete set of embedded eigenvalues of finite multiplicity.
Multiplication operators and dynamical systems on weighted spaces of cross-sections
Authors: Singh R. K. , Manhas J. S.
Published: July 2010

Let

Y
Y

be a Hausdorff topological space, let

V
V

be a system of weights on

Y
Y

, and let

L

V
0

(
Y
)

L{V_0}(Y)

and

L

V
b

(
Y
)

L{V_b}(Y)

be the weighted locally convex spaces of cross-sections with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper, we characterise the multiplication operators on the spaces

L

V
0

(
Y
)

L{V_0}(Y)

and

L

V
b

(
Y
)

L{V_b}(Y)

induced by the scalar-valued, the vector-valued, and the operator-valued mappings. A (linear) dynamical system on the weighted spaces of cross-sections is obtained as an application of the theory of the multiplication operators. Many examples are given to illustrate the theory.
A note on boundary value problems for the heat equation in Lipschitz cylinders
Authors: Brown Russell M. , Shen Zhong Wei
Published: July 2010

We study the initial Dirichlet problem and the initial Neumann problem for the heat equation in Lipschitz cylinders, with boundary data in mixed norm spaces

L
q

(
0
,
T
,

L
p

(
∂
Ω
)
)

{L^q}(0,T,{L^p}(\partial \Omega ))

.
Block spaces on the unit sphere in 𝑅ⁿ
Authors: Keitoku Masahiro , Sato Enji
Published: July 2010

Let

B
q

μ
,
ν

B_q^{\mu ,\nu }

be the block space on the unit sphere introduced by S. Lu. We discuss the relation between

B
q

μ
,
ν

B_q^{\mu ,\nu }

and the

L
p

{L^p}

-space on the unit sphere. Then we give the characterization of

B
q

μ
,
ν

B_q^{\mu ,\nu }

and a simple proof of Theorem (12.11)(iii) of Spaces generated by blocks (Beijing Normal University Math. Ser., 1989).
The faithfulness question for the Burau representation
Authors: Moody John
Published: July 2010

We consider the Burau and Gassner representations of the braid groups

B
n

{B_n}

. A sufficient condition for faithfulness, involving just a pair of arcs, is shown to be necessary as well for all but at most two values of

n
n

. In the Burau case, this implies nonfaithfulness for

n
⩾
10

n \geqslant 10

.
On the abstract characterization of operator spaces
Authors: Effros Edward G. , Ruan Zhong-Jin
Published: July 2010

A direct proof is given for the matricial norm characterization of operator spaces.
Amenability and superharmonic functions
Authors: Northshield S.
Published: July 2010

Let

G
G

be a countable group and

μ
\mu

a symmetric and aperiodic probability measure on

G
G

. We show that

G
G

is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of

G
G

. We use this to show that if

G
G

is amenable, then the Martin boundary of

G
G

contains a fixed point. More generally, we show that

G
G

is amenable if and only if each member of a certain family of

G
G

-spaces contains a fixed point.
A double Stirling formula
Authors: Quine J. R. , Song Richard R.
Published: July 2010

We give an asymptotic expansion as

M
→
∞
,

N
→
∞

M \to \infty ,\;N \to \infty

of

∏

(
m
+
n
τ
)

2

\prod {{{(m + n\tau )}^2}}

, where the product is over
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-slanted-equals StartAbsoluteValue m EndAbsoluteValue less-than-or-slanted-equals upper M comma 1 less-than-or-slanted-equals StartAbsoluteValue n EndAbsoluteValue less-than-or-slanted-equals upper N comma left-parenthesis m comma n right-parenthesis not-equals left-parenthesis 0 comma 0 right-parenthesis">

1
⩽

|

m

|

⩽
M
,

1
⩽

|

n

|

⩽
N
,
(
m
,
n
)
≠
(
0
,
0
)

1 \leqslant |m| \leqslant M,\;1 \leqslant |n| \leqslant N,(m,n) \ne (0,0)

. The formula is analogous to the classical Stirling expansion on

M
!

M!

. Of special interest is the constant term in the expansion, which involves the Dedekind eta function

η
(
z
)

\eta (z)

. Finding this constant is related to the Kronecker limit formula for the derivative at

0
0

of the zeta function

Z
(
s
)
=

∑

m
,
n

′

|

m
+
n
τ

|

−
2
s

Z(s) = \sum \nolimits _{m,n}’{|m + n\tau {|^{ - 2s}}}

. We consider instead the zeta function without absolute values.
Shadowing property of continuous maps with zero topological entropy
Authors: Kuchta Milan
Published: July 2010

The study of the shadowing property has a long history but for interval maps it is rather new. Recent research in this direction is mainly focused on the positive entropy maps and work for zero entropy is still seldom to be found in the literature. In this paper we give a characterization of zero topological entropy maps which have the shadowing property. Moreover, our condition is necessary for any continuous function to have the shadowing property.
A generalization of the Aramata-Brauer theorem
Authors: Rhoades Sandra L.
Published: July 2010

The Aramata-Brauer Theorem says that the regular character minus the principal character of a finite group can be written as a positive rational linear combination of induced linear characters. In the language of Artin

L
L

-series this says that

ζ
E

(
s
)

/

ζ
F

(
s
)

{\zeta _E}(s)/{\zeta _F}(s)

is entire, where this is the quotient of the Dedekind

ζ
\zeta

-functions of a Galois extension

E

/

F

E/F

of number fields. Given any subset of characters of a finite group, we will give a necessary and sufficient condition for when a character is a positive rational linear combination of characters from this specified subset. This result implies that the regular character plus or minus any irreducible character can be written as a positive rational linear combination of induced linear characters. This both generalizes and gives a new proof of the Aramata-Brauer Theorem.
Arens regularity and the 𝐴_{𝑝}(𝐺) algebras
Authors: Forrest Brian
Published: July 2010

Let

G
G

be a locally compact group. Let

A
p

(
G
)

{A_p}(G)

denote the Herz algebra associated with

1
>
p
>
∞

1 > p > \infty

. We show that, for a large class of groups which contains every commutative group,

A
p

(
G
)

{A_p}(G)

is Arens regular if and only if

G
G

is finite.
An asymptotic stability and a uniform asymptotic stability for functional-differential equations
Authors: Ko Younhee
Published: July 2010

We consider a system of functional differential equation

x
′

(
t
)
=
F
(
t
,

x
t

)

{x’}(t) = F(t,{x_t})

and obtain conditions on a Liapunov functional to ensure the asymptotic stability and the uniform asymptotic stability of the zero solution.
Countable productivity of a class of pseudoradial spaces
Authors: Bella Angelo
Published: July 2010

It is shown that the class of compact

R
R

-monolithic spaces is countably productive.
Stabilization of solutions of weakly singular quenching problems
Authors: Fila Marek , Levine Howard A. , Vázquez Juan L.
Published: July 2010

In this paper we prove that if

0
>
β
>
1
,

D
⊂

R
N

0 > \beta > 1,\;D \subset {R^N}

is bounded, and

λ
>
0

\lambda > 0

, then every element of the

ω
\omega

-limit set of weak solutions of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row u Subscript t Baseline minus normal upper Delta u plus lamda u Superscript negative beta Baseline chi Subscript u greater-than 0 Baseline equals 0 in upper D times left-bracket 0 comma normal infinity right-parenthesis comma 2nd Row u equals StartLayout Enlarged left-brace 1st Row 1 on partial-differential upper D times left-parenthesis 0 comma normal infinity right-parenthesis comma 2nd Row u 0 greater-than 0 on upper D overbar times StartSet 0 EndSet EndLayout EndLayout">

u
t

−
Δ
u
+
λ

u

−
β

χ

u
>
0

=
0

in

D
×
[
0
,
∞
)
,

u
=

{

1

on

∂
D
×
(
0
,
∞
)
,

u
0

>
0

on

D
¯

×
{
0
}

\begin {array}{*{20}{c}} {{u_t} - \Delta u + \lambda {u^{ - \beta }}{\chi _{u > 0}} = 0\quad {\text {in}}\;D \times [0,\infty ),} \\ {u = \left \{ \begin {gathered} 1\qquad \quad {\text {on}}\;\partial D \times (0,\infty ), \hfill \\ {u_0} > 0\quad {\text {on}}\;\bar D \times \{ 0\} \hfill \\ \end {gathered} \right .} \\ \end {array}

\]
is a weak stationary solution of this problem. A consequence of this is that if

D
D

is a ball,

λ
\lambda

is sufficiently small, and

u
0

{u_0}

is a radial, then the set

{
(
x
,
t
)

|

u
=
0
}

\{ (x,t)|u = 0\}

is a bounded subset of

D
×
[
0
,
∞
)

D \times [0,\infty )

.
Asymptotic behavior of solutions of Poincaré difference equations
Authors: Trench William F.
Published: July 2010

It is shown that if the zeros

λ
1

,

λ
2

,
…
,

λ
n

{\lambda _1},{\lambda _2}, \ldots ,{\lambda _n}

of the polynomial
\[

q
(
λ
)
=

λ
n

+

a
1

λ

n
−
1

+
⋯
+

a
n

q(\lambda ) = {\lambda ^n} + {a_1}{\lambda ^{n - 1}} + \cdots + {a_n}

\]
are distinct and

r
r

is an integer in

{
1
,
2
,
…
,
n
}

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

such that

|

λ
s

|

≠

|

λ
r

|

|{\lambda _s}| \ne |{\lambda _r}|

if

s
≠
r

s \ne r

, then the Poincaré difference equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y left-parenthesis n plus m right-parenthesis plus left-parenthesis a 1 plus p 1 left-parenthesis m right-parenthesis right-parenthesis y left-parenthesis n plus m minus 1 right-parenthesis plus midline-horizontal-ellipsis plus left-parenthesis a Subscript n Baseline plus p Subscript n Baseline left-parenthesis m right-parenthesis right-parenthesis y left-parenthesis m right-parenthesis equals 0">

y
(
n
+
m
)
+
(

a
1

+

p
1

(
m
)
)
y
(
n
+
m
−
1
)
+
⋯
+
(

a
n

+

p
n

(
m
)
)
y
(
m
)
=
0

y(n + m) + ({a_1} + {p_1}(m))y(n + m - 1) + \cdots + ({a_n} + {p_n}(m))y(m) = 0

\]
has a solution

y
r

{y_r}

such that (A)

y
r

(
m
)
=

λ
r
m

(
1
+
o
(
1
)
)

{y_r}(m) = \lambda _r^m(1 + o(1))

as

m
→
∞

m \to \infty

, provided that the sums

∑

j
=
m

∞

p
i

(
j
)

(
1
⩽
i
⩽
n
)

\sum \nolimits _{j = m}^\infty {{p_i}(j)\;(1 \leqslant i \leqslant n)}

converge sufficiently rapidly. Our results improve over previous results in that these series may converge conditionally, and we give sharper estimates of the

o
(
1
)

o(1)

terms in (A).
On the Riemannian geometry of the nilpotent groups 𝐻(𝑝,𝑟)
Authors: Piu Paola , Goze Michel
Published: July 2010

We study some aspect of the left-invariant Riemannian geometry on a class of nilpotent Lie groups

H
(
p
,
r
)

H(p,r)

that generalize the Heisenberg group

H

2
p
+
1

{H_{2p + 1}}

. Let us prove that the groups of type

H
H

(or Kaplan’s spaces) and the

H
(
p
,
r
)

H(p,r)

groups have same common Riemannian properties but they are not the same spaces.
Fourier decompositions with positive coefficients in compact Gel′fand pairs
Authors: Blank Brian E.
Published: July 2010

For

G
G

a compact separable Hausdorff topological group and for

1
>
p
⩽
2

1 > p \leqslant 2

the finiteness of the Hausdorff-Young sequence operator is established for functions in

L
1

(
G
)

{L^1}(G)

with positive Fourier decompositions and which are

p
p

th-power integrable in a neighborhood of the identity. A similar result is established in the context of compact Gelfand pairs.
Every normal band with REP and REP^{𝑜𝑝} is an amalgamation base
Authors: Shoji Kunitaka
Published: July 2010

We shall prove that every normal band with the representation extension property and its dual is an amalgamation base in the class of all semigroups.
Graded permutation modules and a theorem of Quillen
Authors: Shalev Aner
Published: July 2010

A theorem of Quillen, which presents graded group algebras as certain enveloping algebras, is extended to permutation modules. In positive characteristic this provides a functorial version to Alperin’s recent extension of Jennings’s theory.
Vector bundles with no soul
Authors: Özaydin Murad , Walschap Gerard
Published: July 2010

We exhibit vector bundles over compact manifolds of nonnegative sectional curvature whose total spaces do not admit a complete metric of nonnegative sectional curvature.
Non-Cohen-Macaulay symbolic blow-ups for space monomial curves and counterexamples to Cowsik’s question
Authors: Goto Shiro , Nishida Koji , Watanabe Keiichi
Published: July 2010

Let

A
=
k
[
[
X
,
Y
,
Z
]
]

A = k[[X,Y,Z]]

and

k
[
[
T
]
]

k[[T]]

be formal power series rings over a field

k
k

, and let

n
⩾
4

n \geqslant 4

be an integer such that

n
≢
0

mod

3

n\not \equiv 0\;\bmod \;3

. Let

φ
:
A
→
k
[
[
T
]
]

\varphi :A \to k[[T]]

denote the homomorphism of

k
k

-algebras defined by

φ
(
X
)
=

T

7
n
−
3

,

φ
(
Y
)
=

T

(
5
n
−
2
)
n

\varphi (X) = {T^{7n - 3}},\;\varphi (Y) = {T^{(5n - 2)n}}

, and

φ
(
Z
)
=

T

8
n
−
3

\varphi (Z) = {T^{8n - 3}}

. We put

p

=
Ker

φ

{\mathbf {p}} = \operatorname {Ker} \,\varphi

. Then

R
s

(

p

)
=

⊕

i
⩾
0

p

(
i
)

{R_s}({\mathbf {p}}) = { \oplus _{i \geqslant 0}}{{\mathbf {p}}^{(i)}}

is a Noetherian ring if and only if

ch

k
>
0

\operatorname {ch} \,k > 0

. Hence on Cowsik’s question there are counterexamples among the prime ideals defining space monomial curves, too.
Module categories without short cycles are of finite type
Authors: Happel Dieter , Liu Shi Ping
Published: July 2010

Let

A
A

be an artin algebra. An indecomposable finitely generated

A
A

-module

X
X

is said to be on a short cycle if there exists an indecomposable finitely generated

A
A

-module

Y
Y

and two nonzero noninvertible maps

f
:
X
→
Y

f:X \to Y

and

g
:
Y
→
X

g:Y \to X

. If there are no short cycles we show that there exist only finitely many indecomposable

A
A

-modules up to isomorphism.
Multiple separation theorems
Authors: Maitra Ashok
Published: July 2010

We prove multiple separation theorems for duals of Spector pointclasses.
On the number of real curves associated to a complex algebraic curve
Authors: Bujalance Emilio , Gromadski Grzegorz , Singerman David
Published: July 2010

Using non-Euclidean crystallographic groups we give a short proof of a theorem of Natanzon that a complex algebraic curve of genus

g
⩾
2

g \geqslant 2

has at most

2
(

g

+
1
)

2(\sqrt g + 1)

real forms. We also describe the topological type of the real curves in the case when this bound is attained. This leads us to solve the following question: how many bordered Riemann surfaces can have a given compact Riemann surface of genus

g
g

as complex double?
Quantitative transcendence results for numbers associated with Liouville numbers
Authors: Caveny Deanna M.
Published: July 2010

In 1937, Franklin and Schneider generalized the Gelfond-Schneider result on the transcendence of

α
β

{\alpha ^\beta }

. They proved the following theorem: If

β
\beta

is an algebraic, irrational number and

α
\alpha

is "suitably well-approximated by algebraic numbers of bounded degree", then

α
β

{\alpha ^\beta }

is transcendental. In 1964, Feldman established the algebraic independence of

α
\alpha

and

α
β

{\alpha ^\beta }

under similar conditions. We use results concerning linear forms in logarithms to give quantitative versions of the Franklin-Schneider and Feldman results.
Construction of functors connecting homology and homotopy theories
Authors: Dragotti S. , Esposito R. , Magro G.
Published: July 2010

For each manifold class

F

\mathcal {F}

it is given a functor

Θ

F

{\Theta ^\mathcal {F}}

satisfying the Eilenberg and Steenrod axioms except the excision axiom. It provides a nice unification of geometric treatments of homology and homotopy theories.
A Π-algebra spectral sequence for function spaces
Authors: Dwyer W. G. , Kan D. M. , Smith J. H. , Stover C. R.
Published: July 2010

We construct a new kind of spectral sequence which is in some ways analogous to the unstable Adams spectral sequence. This new spectral sequence converts primary information about the homotopy groups of two spaces

K
K

and

L
L

into information about the homotopy groups of the function space of maps from

K
K

to

L
L

.
Ramsey theory and forcing extensions
Authors: Komjáth Péter
Published: July 2010

In every forcing extension of every model of set theory by a nontrivial (set) forcing there exist a graph Y and a cardinal

μ
\mu

, such that every graph has an edge coloring with

μ
\mu

. colors with no monochromatic induced copy of Y.
Lower bounds for relative class numbers of CM-fields
Authors: Louboutin Stéphane
Published: July 2010

Let

K

{\mathbf {K}}

be a CM-field that is a quadratic extension of a totally real number field

k

{\mathbf {k}}

. Under a technical assumption, we show that the relative class number of

K

{\mathbf {K}}

is large compared with the absolute value of the discriminant of

K

{\mathbf {K}}

, provided that the Dedekind zeta function of

k

{\mathbf {k}}

has a real zero

s
s

such that

0
>
s
>
1

0 > s > 1

. This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.
Cartan algebras and involutions
Authors: Spindler Karlheinz
Published: July 2010

We identify Cartan algebras in certain semidirect products and prove that every involution of a real Lie algebra leaves invariant some Cartan algebra. Moreover, we establish the adaptability of invariant Cartan algebras and invariant Levi decompositions and prove some conjugacy theorems for invariant Cartan algebras.
Parabolic constant mean curvature spacelike surfaces
Authors: Wan Tom Yau-Heng , Au Thomas Kwok-Keung
Published: July 2010

In this paper, we classify Lorentzian isometry classes of parabolic constant mean curvature cuts by conformal classes of nonzero holomorphic quadratic differentials on

C

\mathbb {C}

.
Comparison of the lengths of the continued fractions of √𝐷 and \frac12(1+√𝐷)
Authors: Williams Kenneth S. , Buck Nicholas
Published: July 2010

Let

D
D

denote a positive nonsquare integer such that

D
≡
1

(
mod
4
)

D \equiv 1\,(\bmod 4)

. Let

l
(

D

)

l(\sqrt D )

(resp.

l
(

1
2

(
1
+

D

)
)

l(\tfrac {1} {2}(1 + \sqrt D ))

) denote the length of the period of the continued fraction expansion of

D

\sqrt D

(resp.

1
2

(
1
+

D

)
)

\tfrac {1} {2}(1 + \sqrt D ))

). Recently Ishii, Kaplan, and Williams (On Eisenstein’s problem, Acta Arith. 54 (1990), 323-345) established inequalities between

l
(

D

)

l(\sqrt D )

and

l
(

1
2

(
1
+

D

)
)

l(\tfrac {1} {2}(1 + \sqrt D ))

. In this note it is shown that these inequalities are best possible in a strong sense.
On the isometries of 𝐻^{∞}_{𝐸}(𝐵)
Authors: Matsugu Yasuo , Yamada Takahiko
Published: July 2010

Let

E
E

be a complex Banach space on which all the multipliers are trivial. Let

H
E
∞

(
B
)

H_E^\infty (B)

denote the Banach space of

E
E

-valued bounded holomorphic functions on the open unit ball

B
B

of

C

n

{{\mathbf {C}}^n}

. In this paper we prove that every linear isometry

T
T

of

H
E
∞

(
B
)

H_E^\infty (B)

onto itself is of the form

(
T
F
)
(
z
)
=

T

F
(
φ
(
z
)
)

(TF)(z) = \mathfrak {T}F(\varphi (z))

for all

F
∈

H
E
∞

(
B
)
,

z
∈
B

F \in H_E^\infty (B),\;z \in B

, where

T

\mathfrak {T}

is a linear isometry of

E
E

onto itself and

φ
\varphi

is a biholomorphic map of

B
B

.
Countable metacompactness in Ψ-spaces
Authors: Szeptycki Paul
Published: July 2010

We prove under a variety of assumptions including

c

=

ℵ
2

\mathfrak {c} = {\aleph _2}

that, for every maximal almost disjoint family

A

\mathcal {A}

of countable subsets of

ω
1

,

Ψ
(

A

)

{\omega _1},\;\Psi (\mathcal {A})

is not countably metacompact. In addition, a first countable, countably metacompact, regular space with a closed discrete set which is not a

G
δ

{G_\delta }

is constructed from the mutually consistent assumptions that

b

=

ω
1

\mathfrak {b} = {\omega _1}

and there can exist a Q-set.