bluuebunny/crossref_metadata_2025 huggingface dataset:
Spaces whose closed images are 𝑀₁
Authors: Itō Munehiko , Tamano Ken-ichi
Published: June 2010

Almost local finiteness is introduced. The class of all spaces with a

σ
\sigma

-almost locally finite base is an intermediate class between that of free

L
L

-spaces and that of

M
1

{M_1}

-spaces. The class is countably productive, hereditary and the closed image of a space in the class is

M
1

{M_1}

.
Proof of Scott’s conjecture
Authors: Svrtan D.
Published: June 2010

We give a proof of Conjecture 7 in [2, p. 155] first stated in 1881 by R. F. Scott [4]. It reads as follows: Conjecture 7 (R. F. Scott). Let

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

and

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

be the distinct roots of

x
n

−
1
=
0

{x^n} - 1 = 0

and

y
n

+
1
=
0

{y^n} + 1 = 0

, respectively. Let

A
A

be the

n
×
n

n \times n

matrix whose

(
i
,
j
)

(i,j)

entry is

1

/

(

x
i

−

y
i

)
,
i
,
j
=
1
,
…
,
n

1/({x_i} - {y_i}),i,j = 1, \ldots ,n

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue p e r left-parenthesis upper A right-parenthesis EndAbsoluteValue equals StartLayout Enlarged left-brace 1st Row 1st Column n left-parenthesis 1 dot 3 dot 5 midline-horizontal-ellipsis left-parenthesis n minus 2 right-parenthesis right-parenthesis squared slash 2 Superscript n Baseline comma 2nd Column a m p semicolon if n is o d d comma 2nd Row 1st Column 0 comma 2nd Column a m p semicolon if n is e v e n period EndLayout">

|

per
⁡
(
A
)

|

=

{

n

(
1
⋅
3
⋅
5
⋯
(
n
−
2
)
)

2

/

2

n

,

a
m
p
;

if

n

is 

odd
,

0
,

a
m
p
;

if

n

is

even
⁡

.

\left | {\operatorname {per} (A)} \right | = \left \{ {\begin {array}{*{20}{c}} {n{{(1 \cdot 3 \cdot 5 \cdots (n - 2))}^2}/{2^{n}}}, & {{\text {if}}\;n\;{\text {is }}\operatorname {odd} ,} \\ {0,} & {{\text {if}}\;n\;{\text {is}}\;\operatorname {even} {\text {.}}} \\ \end {array} } \right .

\]
Actually, our proof gives more, namely an explicit expression for

per

(
A
)

{\operatorname {per}}(A)

(see Theorem 2.1).
Principal homogeneous spaces over Hensel rings
Authors: Strano Rosario
Published: June 2010

We prove that if

(
A
,

a
_

)

(A,\underline {a})

is a Hensel couple and

G
G

is an affine, smooth group scheme over

A
A

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript normal e normal t Superscript 1 Baseline left-parenthesis upper A comma upper G right-parenthesis equals upper H Subscript normal e normal t Superscript 1 Baseline left-parenthesis upper A slash a underbar comma upper G slash a underbar upper G right-parenthesis">

H

e
t

1

(
A
,
G
)
=

H

e
t

1

(
A

/

a
_

,
G

/

a
_

G
)

H_{\mathrm {et}}^1 (A, G) = H_{\mathrm {et}}^1 (A/\underline {a}, G/\underline {a} G)

.
On the uniqueness of generalized matrix functions
Authors: Beasley L. B. , Cummings L. J.
Published: June 2010

We consider the sets of "singular" matrices corresponding to generalized matrix functions. The latter are generalizations of the determinant function. It is shown that two generalized matrix functions determine the same set of "singular" matrices if and only if they differ by a scalar multiple.
Length of ray-images under conformal maps
Authors: Karunakaran V.
Published: June 2010

Let

w
=
f
(
z
)

w = f(z)

be regular and univalent in

|

z

|

>
1

|z| > 1

with

f
(
0
)
=
0

f(0) = 0

. Suppose that

f
f

maps the unit disc onto a domain

D
D

. Let

l
(
r
,
θ
)

l(r,\theta )

be the length of the image curve of the ray joining

z
=
0

z = 0

to

z
=
r

e

i
θ

z = r{e^{i\theta }}

in

D
D

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis r right-parenthesis equals upper S u p left-bracket StartAbsoluteValue f left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue Superscript negative 1 Baseline l left-parenthesis r comma theta right-parenthesis right-bracket">

A
(
r
)
=
Sup
⁡
[

|

f
(
r

e

i
θ

)

|

−
1

l
(
r
,
θ
)
]

A(r) = \operatorname {Sup}[{\left | {f(r{e^{i\theta }})} \right |^{ - 1}}l(r,\theta )]

where the supremum is taken over all starlike functions. In this paper we show that

A
(
r
)
⩽
(
1
+
r
)

A(r) \leqslant (1 + r)

.
Spatial theory for algebras of unbounded operators. II
Authors: Inoue A. , Takesue K.
Published: June 2010

In the previous paper [6], we have studied the spatial theory of

O
p
∗

O_p^ *

-algebras with a strongly cyclic vector. In this paper, we will investigate the spatial theory between

O
p
∗

O_p^ *

-algebras induced by a positive invariant sesquilinear form, which contains the former result.
On stability of unitary matrix spaces
Authors: Arazy Jonathan
Published: June 2010

Stability (in the sense of Krivine-Maurey) of a symmetric sequence

E
E

is equivalent to stability of the associated unitary matrix space

C
E

{C_E}

.
On the spectrum of towers
Authors: Donnelly Harold
Published: June 2010

A new relation is derived between the spectrum of a noncompact Riemannian manifold

M
M

and certain sequences of compact quotients of

M
M

.
Nilpotency of homotopy pushouts
Authors: Rao Vidhyanath
Published: June 2010

Precise and effectively verifiable conditions are obtained for the pushout of a nilpotent space along a cofibration inducing a surjection of fundamental groups to be nilpotent. As an application we show that homotopical localization preserves cofiber sequences of nilpotent spaces.
Concerning exactly (𝑛,1) images of continua
Authors: Nadler Sam B. , Ward L. E.
Published: June 2010

A surjective mapping

f
:
X
→
Y

f:X \to Y

is exactly

(
n
,
1
)

(n,1)

if

f

−
1

(
y
)

{f^{ - 1}}(y)

contains exactly

n
n

points for each

y
∈
Y

y \in Y

. We show that if

Y
Y

is a continuum such that each nondegenerate subcontinuum of

Y
Y

has an endpoint, and if

2
⩽
n
>
∞

2 \leqslant n > \infty

, then there is no exactly

(
n
,
1
)

(n,1)

mapping from any continuum onto

Y
Y

. However, if

Y
Y

is a continuum which contains a nonunicoherent subcontinuum, then such an

(
n
,
1
)

(n,1)

mapping exists. Therefore, a Peano continuum is a dendrite if and only if for each

n
n

(
2
⩽
n
>
∞
)

(2 \leqslant n > \infty )

there is no exactly

(
n
,
1
)

(n,1)

mapping from any continuum onto

Y
Y

. We also show that for each positive integer

n
n

there is an exactly

(
n
,
1
)

(n,1)

mapping from the Hilbert cube onto itself.
On the relative de Rham sequence
Authors: Buchdahl N.
Published: June 2010

The classical de Rham sequence on a (smooth, paracompact) manifold provides a connection between solutions of certain differential equations and the topology of the manifold. This paper shows how the relative de Rham sequence for a mapping between manifolds gives a connection between solutions of differential equations and the topology of the fibres of the mapping.
Cell-like decompositions of homogeneous continua
Authors: Rogers James T.
Published: June 2010

Certain decompositions of homogeneous continua are shown to be cell-like. In particular, the aposyndetic decomposition described by F. B. Jones of a homogeneous, decomposable continuum is cell-like, and we prove that any homogeneous decomposable continuum admits a continuous decomposition into mutually homeomorphic, indecomposable, homogeneous, cell-like terminal continua so that the quotient space is an aposyndetic homogeneous continuum.
Fixed subalgebra of a Frobenius algebra
Authors: Wang Kai
Published: June 2010

In this note, we study the fixed subalgebra of a Frobenius algebra under a finite group of automorphism. If the automorphisms preserve the nondegenerate bilinear form, then the fixed subalgebra is also Frobenius.
Effective determination of the decomposition of the rational primes in a cubic field
Authors: Llorente Pascual , Nart Enric
Published: June 2010

The decomposition of the rational primes in a cubic field

K
K

is determined in terms of the coefficients of a defining polynomial of

K
K

. As a consequence, the discriminant

D
D

of

K
K

is straightforwardly computed and the cubic fields with index

i
(
K
)
=
2

i(K) = 2

are easily characterized.
A peculiar unmixed domain
Authors: Brodmann Markus , Rotthaus Christel
Published: June 2010

We construct a local noetherian domain

R
R

containing a nonzero prime element

U
U

and satisfying: (i) The completion

R
^

\hat R

of

R
R

is a domain. (ii)

U

R
^

U\hat R

has an embedded prime divisor.
A simple construction for rigid and weakly homogeneous Boolean algebras answering a question of Rubin
Authors: Brenner Gary
Published: June 2010

We introduce a method for constructing Boolean algebras from trees which preserves some of the trees’ properties. The method is used to produce a very simple construction for rigid Boolean algebras and to construct a weakly homogeneous Boolean algebra without homogeneous factors.
Conservative and divergence free algebraic vector fields
Authors: Connell E. , Drost J.
Published: June 2010

Suppose

k
k

is a field of characteristic 0 and

k

[
n
]

=
k
[

x
1

,
…
,

x
n

]

{k^{[n]}} = k[{x_1}, \ldots ,{x_n}]

. If

u
i

{u_i}

,

f
j

∈

k

[
n
]

{f_j} \in {k^{[n]}}

for

1
⩽
i
⩽
n

1 \leqslant i \leqslant n

,

1
⩽
j
⩽
m

1 \leqslant j \leqslant m

,

u
=
(

u
1

,
…
,

u
n

)

u = ({u_1}, \ldots ,{u_n})

, the

f
j

{f_j}

are relatively prime, and each

f
j

u

{f_j}u

is conservative, then

u
u

is conservative and

(

f
1

,
…
,

f
m

)

({f_1}, \ldots ,{f_m})

is unimodular. Given any

u
u

with

|

J
(
u
)

|

=
1

\left | {J(u)} \right | = 1

, then each derivation

∂

/

∂

u
i

\partial /\partial {u_i}

, has divergence 0. If

D
:

k

[
n
]

→

k

[
n
]

D:{k^{[n]}} \to {k^{[n]}}

is a

k
k

-derivation with kernel of dimension

n
n

-

−
1

- 1

, then there exists a

g
g

so that

g
D

gD

has divergence 0.
Totally positive units and squares
Authors: Hughes I. , Mollin R.
Published: June 2010

Let

K
K

be a finite cyclic extension of the rational number field

Q
Q

, with Galois group

G
(
K

/

Q
)

G(K/Q)

of order

p
a

{p^a}

for an odd prime

p
p

. Armitage and Fröhlich [1] proved that if the order of 2 modulo

p
p

is even and the class number

h
K

{h_K}

of

K
K

is odd then

U
K
+

=

U
K
2

U_K^ + = U_K^2

, where

U
K

{U_K}

is the group of units of the ring of integers

C

K

{\mathcal {C}_K}

of

K
K

,

U
K
+

U_K^ +

is the group of totally positive units, and

U
K
2

U_K^2

is the group of unit squares. The purpose of this paper is to provide a generalization of this result to a larger class of abelian extensions of

Q

.
2

{Q.^2}
𝐹_{𝑝} classes and hypergeometric series
Authors: Moak Daniel S.
Published: June 2010

For

Re
⁡
(
p
)
⩾
0

\operatorname {Re} (p) \geqslant 0

, let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript p Baseline equals left-brace f colon f left-parenthesis z right-parenthesis equals f left-parenthesis 1 minus x z right-parenthesis Superscript negative p Baseline d mu left-parenthesis x right-parenthesis comma StartAbsoluteValue z EndAbsoluteValue greater-than 1 comma mu">

F
p

=

 { 

f
:
f
(
z
)
=
f

(
1
−
x
z

)

−
p

d
μ
(
x
)
,

|
z
|

>
1
,

μ

{F_p} = {\text { \{ }}f:f(z) = f{(1 - xz)^{ - p}}d\mu (x),\;\left | z \right | > 1,\;\mu

a probability measure on

|
x
|

=
1
}

\left | x \right | = 1\}

, and let

F
p

⋅

F
q

=

{

f
⋅
g
:
f

in
⁡

F
p

,
g

in

F
q

}

{F_p} \cdot {F_q} = \left \{ {f \cdot g:f\;\operatorname {in} {F_p},g\;{\text {in}}\;{F_q}} \right \}

. Brickman, Hallenbeck, MacGregor and Wilken proved that

p
>
0

p > 0

and

q
>
0

q > 0

, then

F
p

⋅

F
p

⊆

F

p
+
q

{F_p} \cdot {F_p} \subseteq {F_{p + q}}

. Kent Pearce recently proved a converse result: if

F
p

⋅

F
q

⊆

F

p
+
q

{F_p} \cdot {F_q} \subseteq {F_{p + q}}

, then

p
>
0

p > 0

and

q
>
0

q > 0

, or

p
=
q
=
1
+
i
t

p = q = 1 + it

for some real

t
t

. The case

p
=
q
=
1
+
i
t

p = q = 1 + it

,

t
≠
0

t \ne 0

, will be excluded. Consequently a full converse of the above theorem holds.
Approximation order from bivariate 𝐶¹-cubics: a counterexample
Authors: de Boor C. , Höllig K.
Published: June 2010

It is shown that the space of bivariate

C
1

{C^1}

piecewise cubic functions on a hexagonal mesh of size

h
h

approximates to certain smooth functions only to within

O
(

h
3

)

O({h^3})

even though it contains a local partition of every cubic polynomial.
On Sturmian theory for second order systems
Authors: Ahmad Shair
Published: June 2010

A generalization of the Sturm Comparison Theorem is given to nonselfadjoint second order linear systems. In addition, a theorem, involving the existence of a solution with strictly positive components, is proven. Counterexamples are given to show that the theorems are false without the stated assumptions.
An alternate variational principle for Δ𝑢-𝑢+∣𝑢∣^{𝑟-1}𝑠𝑔𝑛𝑢=0
Authors: Coffman Charles V.
Published: June 2010

An alternate variational principle for the equation in the title has been proposed by H. A. Levine. We analyse the relation between this principle and the Rayleigh quotient that has been used previously for the variational study of this problem in

R
N

{R^N}

. The main result is an existence theorem for

W

1
,
2

(

R
N

)

{W^{1,2}}({R^N})

-solutions of the variational problem posed by Levine.
Contractions without cyclic vectors
Authors: Sz.-Nagy Béla , Foiaş Ciprian
Published: June 2010

It is proved that if

T
T

is a completely nonunitary contraction on Hilbert space such that

T

∗
n

{T^{*n}}

does not converge strongly to 0 as

n
→
∞

n \to \infty

, there is an integer

N
>
0

N > 0

so that none of the powers

T

∗
m

{T^{*m}}

with

m
⩾
N

m \geqslant N

has a cyclic vector. Both conditions on

T
T

are essential, and the integer

N
N

is not universal, i.e., it depends on

T
T

.
New proofs for the maximal ergodic theorem and the Hardy-Littlewood maximal theorem
Authors: Jones Roger L.
Published: June 2010

A new proof of the maximal ergodic theorem is presented. The same idea used in this proof is then used to show that the Hardy-Littlewood maximal function is weak type

(
1
,
1
)

(1,1)

.
Fourier coefficients of continuous functions on compact groups
Authors: Heiman Barbara
Published: June 2010

Let

G
G

be an infinite compact group with dual object

Σ
\Sigma

. Letting

K

σ

{\mathcal {K}_\sigma }

be the representation space for

σ
∈
Σ

\sigma \in \Sigma

,

E

2

(
Σ
)

{\mathcal {E}^2}(\Sigma )

is the set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet upper A equals left-parenthesis upper A Superscript sigma Baseline right-parenthesis element-of normal upper Pi script upper B left-parenthesis script upper K Subscript sigma Baseline right-parenthesis colon double-vertical-bar upper A double-vertical-bar Subscript 2 Superscript 2 Baseline equals sigma-summation Underscript sigma Endscripts d Subscript sigma Baseline trace left-parenthesis upper A Superscript sigma Baseline upper A Superscript sigma asterisk Baseline right-parenthesis greater-than normal infinity EndSet">

{
A
=
(

A
σ

)
∈
Π

B

(

K

σ

)
:

‖
A
‖

2
2

=

∑
σ

d
σ

Tr
⁡
(

A
σ

A

σ
∗

)
>
∞
}

\{ A = ({A^\sigma }) \in \Pi \mathcal {B}({\mathcal {K}_\sigma }):\left \| A \right \|_2^2 = {\sum _\sigma }{d_\sigma }\operatorname {Tr} ({A^\sigma }{A^{\sigma *}}) > \infty \}

. For

A
∈

E

2

(
Σ
)

A \in {\mathcal {E}^2}(\Sigma )

, we show that there is a function

f
f

in

C
(
G
)

C(G)

such that

‖
f
‖

∞

⩽
C

‖
A
‖

2

{\left \| f \right \|_\infty } \leqslant C{\left \| A \right \|_2}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="trace left-parenthesis ModifyingAbove f With caret left-parenthesis sigma right-parenthesis ModifyingAbove f With caret left-parenthesis sigma right-parenthesis Superscript asterisk Baseline right-parenthesis greater-than-or-slanted-equals trace left-parenthesis upper A Superscript sigma Baseline upper A Superscript sigma asterisk Baseline right-parenthesis">

Tr
⁡
(

f
^

(
σ
)

f
^

(
σ

)
∗

)
⩾
Tr
⁡
(

A
σ

A

σ
∗

)

\operatorname {Tr} (\hat f(\sigma )\hat f{(\sigma )^*}) \geqslant \operatorname {Tr} ({A^\sigma }{A^{\sigma *}})

for every

σ
∈
Σ

\sigma \in \Sigma

.
Meromorphic mappings into compact complex manifolds with a Grauert positive bundle of 𝑞-forms
Authors: Kwack Myung H.
Published: June 2010

It is shown that a Schwarz lemma, a Picard extension theorem, and a Bloch theorem are valid for meromorphic mappings into compact complex manifolds with Grauert-positive bundle of

q
q

-forms.
A theorem of Cramér and Wold revisited
Authors: Sitaram Alladi
Published: June 2010

Let

H
=
{
(
x
,
y
)
;
x
>
0
}
⊆

R

2

H = \{ (x,y);x > 0\} \subseteq {{\mathbf {R}}^2}

and let

E
E

be a Borel subset of

H
H

of positive Lebesgue measure. We prove that if

μ
\mu

and

υ
\upsilon

are two probability measures on

R

2

{{\mathbf {R}}^2}

such that

μ
(
σ
(
E
)
)
=
υ
(
σ
(
E
)
)

\mu (\sigma (E)) = \upsilon (\sigma (E))

for all rigid motions

σ
\sigma

of

R

2

{{\mathbf {R}}^2}

, then

μ
=
υ

\mu = \upsilon

This generalizes a well-known theorem of Cramér and Wold.
Another interesting property concerning the probability measures on the rationals
Authors: Lane K. W.
Published: June 2010

Let

X
X

be a perfect, complete, separable metric space and

P
(
X
)

P(X)

denote the space of Borel probability measures on

X
X

equipped with the topology of weak convergence. If

Y
Y

is a countable dense subset of

X
X

then

P
(
Y
)

P(Y)

is not a

G

δ
σ

{G_{\delta \sigma }}

subset of

P
(
X
)

P(X)

. Furthermore if

X
X

is separable, complete and metric, and

Y
⊆
X

Y \subseteq X

, and

P
(
Y
)

P(Y)

is a

G

δ
σ

{G_{\delta \sigma }}

subset of

P
(
X
)

P(X)

, then

P
(
Y
)

P(Y)

is in fact a

G
δ

{G_\delta }

subset of

P
(
X
)

P(X)

.
The chain recurrent set for maps of the interval
Authors: Block Louis , Franke John E.
Published: June 2010

Let

f
f

be a continuous map of a compact interval into itself. We show that if the set of periodic points of

f
f

is a closed set then every chain recurrent point is periodic.
Elementary abelian 2-groups that act freely on products of real projective spaces
Authors: Cusick Larry W.
Published: June 2010

For a natural number

N
N

let

N
¯

\bar N

be 0 if

N
N

is even and 1 if

N
N

is odd. We prove that if

(

Z
2

)
l

{({Z_2})^l}

acts freely on

∏
l
k

=
1

R

P

N
l

\prod _l^k{ = 1}{\mathbf {R}}{P^{{N_l}}}

in such a way that the induced action on

mod

2

\mod 2

cohomology is trivial, then

l
⩽
2
(

N
¯

1

+
⋯
+

N
k

)

l \leqslant 2({\bar N_1} + \cdots + {N_k})

. If no

N
l

{N_l}

is congruent to

3

mod

4

3\mod 4

then

l
⩽

N
¯

1

+
⋯
+

N
¯

k

l \leqslant {\bar N_1} + \cdots + {\bar N_k}

.
Compact dispersed spaces and the 𝛼-left property
Authors: De Marco G. , Le Donne A. , Wilson R. G.
Published: June 2010

The concept of an

α
\alpha

-left space was introduced by Arhangel’skij in [A

1

_{1}

], where, among other results, it was shown that every

T
2

{T_2}

compact

α
\alpha

-left space is dispersed. In [A

2

_{2}

] an example was given to show that, assuming the continuum hypothesis CH, not every

T
2

{T_2}

compact dispersed space is

α
\alpha

-left. The aim of this article is to obtain some sufficient conditions for

T
2

{T_2}

space to be

α
\alpha

-left and to construct a large class (which contains all products of uncountable ordinals) of compact

T
2

{T_2}

dispersed spaces which are not

α
\alpha

-left.
Pseudo-arcs and connectedness in homeomorphism groups
Authors: Lewis Wayne
Published: June 2010

We prove that the homeomorphism group of the pseudo-arc contains no nondegenerate continua. An equivalent result holds for any continuum all of whose small subcontinua are pseudo-arcs, e.g. for the pseudo-circle and the circle of pseudo-arcs. Related questions are briefly mentioned.
Fibrations that are cofibrations
Authors: Alonso Juan M.
Published: June 2010

We give necessary and sufficient conditions for a homotopy cartesian square to be homotopy cocartesian. Specializing, we obtain a necessary and sufficient condition for a fibration to be a cofibration. We apply the above to localization of spaces and to acyclic maps.
A transcendence measure for some special values of elliptic functions
Authors: Tubbs Robert
Published: June 2010

Among T. Schneider’s results is the following: Let

℘
(
z
)

\wp (z)

be the Weierstrass elliptic function with algebraic invariants. If

℘
(
u
)

\wp (u)

and

β
\beta

are both algebraic,

β
∉

K
τ

\beta \notin {K_\tau }

, then

℘
(
β
u
)

\wp (\beta u)

is transcendental. In this paper we provide a transcendence measure for this value. Let

P
(
X
)

P(X)

be a nonzero polynomial, with integral coefficients, of degree

d
d

and height

h
h

, and put

t
=
d
+
log
⁡
h

t = d + \log h

. Then there is an effectively computable constant

C
C

, which does not depend on

P
(
X
)

P(X)

. such that: (A) If

℘
(
z
)

\wp (z)

has complex multiplication then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log StartAbsoluteValue upper P left-parenthesis normal script upper P left-parenthesis beta u right-parenthesis right-parenthesis EndAbsoluteValue greater-than minus upper C d squared t squared left-parenthesis log t right-parenthesis Superscript 4">

log
⁡

|

P
(
℘
(
β
u
)
)

|

>
−
C

d
2

t
2

(
log
⁡
t

)
4

\log \left | {P(\wp (\beta u))} \right | > - C{d^2}{t^2}{(\log t)^4}

. (B) If

℘
(
z
)

\wp (z)

does not have complex multiplication then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="log StartAbsoluteValue upper P left-parenthesis normal script upper P left-parenthesis beta u right-parenthesis right-parenthesis EndAbsoluteValue greater-than minus upper C d Superscript 6 Baseline t squared left-parenthesis log t right-parenthesis Superscript 14">

log
⁡

|

P
(
℘
(
β
u
)
)

|

>
−
C

d
6

t
2

(
log
⁡
t

)

14

\log \left | {P(\wp (\beta u))} \right | > - C{d^6}{t^2}{(\log t)^{14}}

.
On relative normal complements in finite groups. II
Authors: Leonard Henry S.
Published: June 2010

Given a finite group

G
G

and subgroups

H
H

and

H
0

{H_0}

with

H
0

◃
H

{H_0} \triangleleft H

, we let

π
\pi

denote the set of prime divisors of

(
H
:

H
0

)

(H:{H_0})

, and we denote this configuration by

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

. Pamela Ferguson has shown that if

H

/

H
0

H/{H_0}

is solvable, then under certain conditions there exists a unique relative normal complement

G
0

{G_0}

of

H
H

over

H
0

{H_0}

. In this paper we give alternative proofs of her two theorems.
On tensor products and extended centroids
Authors: Nicholson W. K. , Watters J. F.
Published: June 2010

For prime algebras

R
R

and

S
S

over a field

F
F

it is shown that each nonzero ideal of

R
⊗
S

R \otimes S

contains a nonzero element

r
⊗
s

r \otimes s

,

r
∈
R

r \in R

,

s
∈
S

s \in S

, if and only if

C
(
R
)
⊗
C
(
S
)

C(R) \otimes C(S)

is a field, where

C
(
R
)

C(R)

(respectively

C
(
S
)

C(S)

) is the extended centroid of

R
R

(respectively

S
S

).
Ultimately closed projective resolutions and rationality of Poincaré-Betti series
Authors: Wilson George V.
Published: June 2010

A condition on the syzygies of a module is given which implies the rationality of certain Poincaré-Betti series.
On the automorphism group of a linear algebraic monoid
Authors: Putcha Mohan S.
Published: June 2010

Let

S
S

be a connected regular monoid with zero. It is shown that an automorphism of

S
S

is inner if and only if it sends each idempotent of

S
S

to a conjugate idempotent. In the language of semigroup theory, the automorphism group of

S
S

maps homomorphically into the automorphism group of the finite lattice of

G

\mathcal {G}

-classes of

S
S

, and the kernel of this homomorphism is the group of inner automorphisms of

S
S

. In particular, if the

G

\mathcal {G}

-classes of

S
S

are linearly ordered, then every automorphism of

S
S

is inner.
On Fourier transforms of distributions with one-sided bounded support
Authors: Shambayati R. , Zieleźny Z.
Published: June 2010

Fourier transforms of distributions of finite order with left-sided bounded support are characterized. Furthermore, a product of these Fourier transforms is defined which corresponds to the convolution of the original distributions.
Disappearance of extreme points
Authors: Morris Peter
Published: June 2010

It is shown that every separable Banach space which contains an isomorphic copy of

c
0

{c_0}

is isomorphic to a strictly convex space

E
E

such that no point of

E
E

is an extreme point of the unit ball of

E

∗
∗

{E^{ * * }}

.
An ergodic theorem for semigroups of contractions
Authors: Gutman S. , Pazy A.
Published: June 2010

An ergodic theorem for semigroups of nonlinear contractions having precompact trajectories in a Banach space is proved.
Analytic generators and the KMS condition
Authors: De Cannière Jean
Published: June 2010

The KMS condition at arbitrary

β
\beta

for a state

ω
\omega

of a

C
∗

{C^ * }

-dynamical system

(
A
,

R

,
α
)

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

is stated in terms of the analytic continuation of the correlation functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t right-arrow from bar omega left-parenthesis x Superscript asterisk Baseline alpha Subscript t Baseline left-parenthesis x right-parenthesis right-parenthesis left-parenthesis x element-of upper A right-parenthesis in minus i left-parenthesis or plus i right-parenthesis">

t
↦
ω
(

x
∗

α
t

(
x
)
)
(
x
∈
A
)

in

−
i
(

or

+
i
)

t \mapsto \omega ({x^ * }{\alpha _t}(x))(x \in A){\text {in}} - i({\text {or}} + i)

. The precise formulation involves the analytic generator of

α
\alpha

(or its inverse).
On generalized Fuglede-Putnam theorems of Hilbert-Schmidt type
Authors: Kittaneh Fuad
Published: June 2010

We prove the following statements about the bounded linear operators on a separable, complex Hilbert space: (1) If

A
A

and

B
∗

{B^ * }

are subnormal operators, and

X
X

is an invertible operator such that

A
X
−
X
B
∈

C
2

AX - XB \in {C_2}

, then there exists a unitary operator

U
U

such that

A
U
−
U
B
∈

C
2

AU - UB \in {C_2}

. Moreover,

A
∗

A
−
A

A
∗

{A^ * }A - A{A^ * }

and

B
∗

B
−
B

B
∗

{B^ * }B - B{B^ * }

are in

C
1

{C_1}

. (2) If

A
A

is a subnormal operator with

A
∗

A
−
A

A
∗

∈

C
1

{A^ * }A - A{A^ * } \in {C_1}

, then for any operator

X
X

,

A
X
−
X
A
∈

C
2

AX - XA \in {C_2}

implies

A
∗

X
−
X

A
∗

∈

C
2

{A^ * }X - X{A^ * } \in {C_2}

. (3) If

A
A

is a hyponormal contraction with

1
−
A

A
∗

∈

C
1

1 - A{A^ * } \in {C_1}

, then for any operator

X
X

,

A
X
−
X
A
∈

C
2

AX - XA \in {C_2}

implies

A
∗

X
−
X

A
∗

∈

C
2

{A^ * }X - X{A^ * } \in {C_2}

. (4) For every operator

T
T

for which

T
2

{T^2}

is normal and

T
∗

T
−
T

T
∗

∈

C
1

{T^ * }T - T{T^ * } \in {C_1}

,

T
X
−
X
T
∈

C
2

TX - XT \in {C_2}

implies

T
∗

X
−
X

T
∗

∈

C
2

{T^ * }X - X{T^ * } \in {C_2}

for any operator

X
X

. Applications of a recent result of Moore, Rogers and Trent [8] are also given.
Zero-free parabolic regions for polynomials with complex coefficients
Authors: Runckel Hans-J.
Published: June 2010

Recent results by P. Henrici, E. B. Saff and R. S. Varga on zero-free parabolic regions for sequences of polynomials generated from three-term recurrence relations with real coefficients are generalized to complex coefficients by continued fraction methods. Especially, it is shown that all zeros of the generalized Bessel polynomials

Y
n

(
δ
)

Y_n^{(\delta )}

for complex

δ
\delta

are contained in a cardioid region, which generalizes a result of E. B. Saff and R. S. Varga for real

δ
\delta

.
Rigid 3-dimensional compacta whose squares are manifolds
Authors: Ancel Fredric D. , Duvall Paul F. , Singh S.
Published: June 2010

A space is rigid if its only self-homeomorphism is the identity. In response to a question of Jan van Mill, Ancel and Singh have given examples of rigid

n
n

-dimensional compacta, for each

n
⩾
4

n \geqslant 4

, whose squares are manifolds. We construct a rigid

3
3

-dimensional compactum whose square is the manifold

S
3

×

S
3

{S^3} \times {S^3}

. In fact, we construct uncountably many topologically distinct compacta with these properties.
A generalization of the Poincaré-Birkhoff theorem
Authors: Ding Wei Yue
Published: June 2010

A generalized form of the Poincaré-Birkhoff theorem is proved. The generalization is useful for the further applications of this famous fixed point theorem.
Equivariant covering spaces and cohomology
Authors: Waner Stefan
Published: June 2010

The Bredon cohomology of classifying spaces for categories of equivariant covering spaces is considered and shown to correspond to derived functors for the coefficient systems of the Bredon theory.
Schrödinger operator methods in the study of a certain nonlinear P.D.E
Authors: Harrell E. M. , Simon B.
Published: June 2010

We prove that

Δ
u
+
h

u
α

=
0

\Delta u + h{u^\alpha } = 0

has no positive solutions for certain

h
h

,

α
\alpha

by studying the linearized equation

(
Δ
+
h

u

α
−
1

)
ψ
=
e
ψ

(\Delta + h{u^{\alpha - 1}})\psi = e\psi

.
Picone-type theorems for semidiscrete hyperbolic equations
Authors: Kreith Kurt
Published: June 2010

A comparison theorem of Picone-type is established for hyperbolic boundary value problems by means of a semidiscrete approximation.
A matrix inverse
Authors: Bressoud D. M.
Published: June 2010

George Andrews has demonstrated that the Bailey transform is equivalent to the inversion of an infinite-dimensional matrix whose entires are rational functions in

q
q

. We generalize this inversion by introducing an extra parameter which brings much greater symmetry.
Essentially subnormal operators and 𝐾-spectral sets
Authors: Lange Ridgley
Published: June 2010

Let

T
T

be an essentially subnormal operator. We give six conditions which are equivalent to the spectrum of

T
T

being a

K
K

-spectral set. From this follow two corollaries which give sufficient conditions for invariant subspaces of essentially subnormal operators. Several examples are given that show that some essentially subnormal operators are not essentially normal nor perturbations of subnormal operators.
Shift basic sequences in the Wiener disc algebra
Authors: Holub J. R.
Published: June 2010

Let

W
(
D
)

W(D)

denote the set of functions

f
(
z
)
=

∑

n
=
0

∞

a
n

z
n

f(z) = \sum \nolimits _{n = 0}^\infty {{a_n}{z^n}}

for which

∑

n
=
0

∞

|

a
n

|

>
+
∞

\sum \nolimits _{n = 0}^\infty {\left | {{a_n}} \right | > + \infty }

. It is shown that for any positive integer

k
k

the

k
k

-shifted sequence

{

z

k
n

⋅
f
(
z
)

}

n
=
0

∞

\left \{ {{z^{kn}} \cdot f(z)} \right \}_{n = 0}^\infty

is a basic sequence in

W
(
D
)

W(D)

equivalent to the basis

{

z
n

}

n
=
0

∞

\left \{ {{z^n}} \right \}_{n = 0}^\infty

if and only if

f
(
z
)

f(z)

has no set of

k
k

symmetrically distributed zeros on the circle

|
z
|

=
1

\left | z \right | = 1

.
Geometric conditions for interpolation
Authors: Belna C. L. , Obaid S. A. , Rung D. C.
Published: June 2010

Let

{

z
n

}

\left \{ {{z_n}} \right \}

be a sequence lying in either the upper half-plane or the unit disc in the complex plane. If

{

z
n

}

\left \{ {{z_n}} \right \}

is a separated sequence we give a simple geometric condition that implies the sequence is an interpolating sequence for the algebra of bounded holomorphic functions. This result contains most of the known results of this type.
Iteration of holomorphic maps of the unit ball into itself
Authors: Kubota Yoshihisa
Published: June 2010

Let

Ω
\Omega

be a plane disc and let

f
f

be a holomorphic map of

Ω
\Omega

into itself. It is known that the iterates

f
n

{f_n}

of

f
f

converge to a constant

ζ
∈

Ω
¯

\zeta \in \bar \Omega

as

n
→
∞

n \to \infty

unless

f
f

is a conformal map of

Ω
\Omega

onto itself. In the present paper it is shown that a more complicated statement of this kind is true in the unit ball of

C

N

{{\mathbf {C}}^N}

.
On the mapping torus of an automorphism
Authors: Paschke William L.
Published: June 2010

Let

ρ
\rho

be an automorphism of a

C
∗

{C^ * }

-algebra

A
A

. The mapping torus

T
ρ

(
A
)

{T_\rho }(A)

is the

C
∗

{C^ * }

-algebra of

A
A

-valued continuous functions

x
x

on

[
0
,
1
]

[0,1]

satisfying

x
(
1
)
=
ρ
(
x
(
0
)
)

x(1) = \rho (x(0))

. Using his Thom isomorphism theorem, A. Connes has shown that the

K
K

-groups of

T
ρ

(
A
)

{T_\rho }(A)

, with indices reversed, are isomorphic to those of the crossed product

A

×
ρ

Z

A{ \times _\rho }Z

. We provide here an alternative proof of this fact which gives an explicit description of the isomorphism.
A relation between pointwise convergence of functions and convergence of functionals
Authors: Brézis Haïm , Lieb Elliott
Published: June 2010

We show that if

{

f
n

}

\left \{ {{f_n}} \right \}

is a sequence of uniformly

L
p

{L^p}

-bounded functions on a measure space, and if

f
n

→
f

{f_n} \to f

pointwise a.e., then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts left-brace double-vertical-bar f Subscript n Baseline double-vertical-bar Subscript p Superscript p Baseline minus double-vertical-bar f Subscript n Baseline minus f double-vertical-bar Subscript p Superscript p Baseline right-brace equals double-vertical-bar f double-vertical-bar Subscript p Superscript p">

lim

n
→
∞

{

‖

f
n

‖

p
p

−

‖

f
n

−
f

‖

p
p

}

=

‖
f
‖

p
p

{\lim _{n \to \infty }}\left \{ {\left \| {{f_n}} \right \|_p^p - \left \| {{f_n} - f} \right \|_p^p} \right \} = \left \| f \right \|_p^p

for all

0
>
p
>
∞

0 > p > \infty

. This result is also generalized in Theorem 2 to some functionals other than the

L
p

{L^p}

norm, namely
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue j left-parenthesis f Subscript n Baseline right-parenthesis minus j left-parenthesis f Subscript n Baseline minus f right-parenthesis minus j left-parenthesis f right-parenthesis EndAbsoluteValue right-arrow 0">

∫

|

j
(

f
n

)
−
j
(

f
n

−
f
)
−
j
(
f
)

|

→
0

\int \left | {j({f_n}) - j({f_n} - f) - j(f)} \right | \to 0

for suitable

j
:

C

→

C

j:{\mathbf {C}} \to {\mathbf {C}}

and a suitable sequence

{

f
n

}

\left \{ {{f_n}} \right \}

. A brief discussion is given of the usefulness of this result in variational problems.
A remark on Bony maximum principle
Authors: Lions P.-L.
Published: June 2010

We extend a result due to J. M. Bony concerning a form of the classical maximum principle adapted to Sobolev spaces. We treat the case of the limiting exponent and show that the result is optimal. We give various applications to nonlinear elliptic partial differential equations.
Nonuniqueness of Teichmüller extremal mappings
Authors: Reich Edgar
Published: June 2010

A boundary correspondence for the unit disk is found with the following property. Among the extremal quasiconformal mappings of the disk with the specified boundary correspondence there are at least two Teichmüller mappings.
Geometrical properties of cones and the comparison of solutions of differential equations
Authors: Saet Yuly A.
Published: June 2010

We introduce here certain geometrical concepts related to the properties of cones in Banach spaces and show that these definitions are realizable. Then the results on the comparison of solutions are formulated where the ordering is introduced by means of a cone which is not required to be solid.
New examples of strictly almost Kähler manifolds
Authors: Watson Bill
Published: June 2010

A parametrized family of non-Kähler almost Kähler manifolds is constructed as the product of solvable Lie groups with almost cosymplectic structures. A family of compact strictly almost Kähler manifolds whose cohomology is consistent with that of Kähler manifolds is similarly obtained.
An Archimedian property for groups with polynomial growth
Authors: Ganesan S. , Jenkins J. W.
Published: June 2010

The notion of Archimedian group is introduced. It is shown that if

G
G

is either a finitely generated, solvable group or a connected, locally compact group, then

G
G

is Archimedian if it has polynomial growth. A partial converse is also proven.
Types of almost continuity
Authors: Garrett B. D. , Kellum K. R.
Published: June 2010

The differences for properties of the range space

f
(
X
)

f(X)

are considered for functions

f
:
X
→
Y

f:X \to Y

, where

f
f

is almost continuous relative to

X
×
Z

X \times Z

, and

f
(
X
)
⊂
Z
⊂
Y

f(X) \subset Z \subset Y

. It is shown that if

f
f

is allowed to be almost continuous relative to

X
×
Z

X \times Z

, where

X
X

is a Peano continuum and

Z
Z

is a locally connected metric space, then

f
(
X
)

f(X)

can be any type of subcontinuum of

Z
Z

. This contrasts the known results for the case where

Z
=
f
(
X
)

Z = f(X)

and almost continuity is relative to

X
×
f
(
X
)

X \times f(X)

. Outer almost continuous retracts (

f
:
X
→
X

f:X \to X

is almost continuous relative to

X
×
X

X \times X

) and inner almost continuous retracts (

f
:
X
→
X

f:X \to X

is almost continuous relative to

X
×
f
(
X
)
)

X \times f(X))

) are defined. Properties of outer almost continuous retracts, including the existence of an outer almost continuous retract

M
M

of a fixed point space

X
X

, where

M
M

does not have the fixed point property, are found.
Subrepresentations of direct integrals and finite volume homogeneous spaces
Authors: Gootman Elliot C.
Published: June 2010

We prove a result on representations of separable

C
∗

{C^*}

-algebras which has application to, and was in fact motivated by, a problem concerning relations between unitary representations of a second countable locally compact group

G
G

and those of a closed subgroup

K
K

, when

G

/

K

G/K

is of finite volume. The result is that if an irreducible representation

π
\pi

is contained in

∫
X

π
x

d
μ
(
x
)

\int _X {{\pi _x}} d\mu (x)

, then

π
⊆

π
x

\pi \subseteq {\pi _x}

for all

x
x

in a set of positive measure. With

G
G

and

K
K

as above, it follows that for each

π
∈

G
^

\pi \in \hat G

there exists

σ
∈

K
^

\sigma \in \hat K

with

π
⊆

U
σ

\pi \subseteq {U^\sigma }

, the induced representation. Frobenius reciprocity type results are derived as further consequences.
On the product of class 𝐴-semigroups of linear operators
Authors: Abdelaziz Nazar Hussein
Published: June 2010

The present paper extends a result of Trotter concerning the product of

c
0

{c_0}

semigroups. We show that the product of two commuting semigroups of class

A
A

is again a semigroup of class

A
A

and that its generator is the sum (or its closure) of the generators of the factor semigroups. This provides a partial answer in the affirmative to a question of Hille and Phillips concerning the sum of unbounded operators [3, p. 417]. Sufficient conditions are also given in terms of the generators and their domains so that the semigroups commute.
A proof of a formula in Fourier analysis on the sphere
Authors: Yang Jin Gen
Published: June 2010

A short and elementary proof of a useful formula in spherical harmonic analysis is provided.
A family of polynomials with concyclic zeros
Authors: Stolarsky Kenneth B.
Published: June 2010

Expand

E

(
z
)

=

(

e
z

−
1

)
m

E\left ( z \right ) = {({e^z} - 1)^m}

by the binomial theorem, and replace every

e
x
p

(

k
z

)

exp\left ( {{k_z}} \right )

by its approximation

(

1
+
k

n

−
1

z

)

n

{\left ( {1 + k{n^{ - 1}}z} \right )^n}

. The resulting polynomial has all of its zeros on a circle of radius

r
r

centered at

−
r

- r

, where

r
=
n

/

m

r = n/m

.
A remark on the image of the Ahlfors function
Authors: Yamada Akira
Published: June 2010

Let

Ω
\Omega

denote a planar maximal region for bounded holomorphic functions and

p
∈
Ω

p \in \Omega

. By an example we show that the complement in the unit disc of the image of the Ahlfors function for

Ω
\Omega

and

p
p

can be a fairly general set of logarithmic capacity zero.
On functions that approximate relations
Authors: Beer Gerald
Published: June 2010

Let

X
X

be a metric space and let

Y
Y

be a separable metric space. Suppose

R
R

is a relation in

X
×
Y

X \times Y

. The following are equivalent: (a) for each

ε
>
0

\varepsilon > 0

there exists

f
:
X
→
Y

f:X \to Y

such that the Hausdorff distance from

f
f

to

R
R

is at most

ε
\varepsilon

; (b) the domain of

R
R

is a dense subset of

X
X

, and for each isolated point

x
x

of the domain the vertical section of

R
R

at

x
x

is a singleton; (c) for each

ε
>
0

\varepsilon > 0

there exists

f
:
X
→
Y

f:X \to Y

of Baire class one such that the Hausdorff distance from

f
f

to

R
R

is at most

ε
\varepsilon

.
On isoclinal sequences of spheres
Authors: Weiss Asia Ivić
Published: June 2010

In

n
n

-dimensional inversive geometry it is possible to construct an infinite sequence of

(
n
−
1
)

(n - 1)

-spheres with the property that every

n
+
2

n + 2

consecutive numbers are isoclinal. For every such sequence there is a Möbius transformation which advances the spheres of the sequence by one.
Tall 𝛼-recursive structures
Authors: Friedman Sy D. , Shelah Saharon
Published: June 2010

The Scott rank of a structure

M
M

,

sr
⁡
(
M
)

\operatorname {sr}(M)

, is a useful measure of its model-theoretic complexity. Another useful invariant is

o

(
M
)

{\text {o}}(M)

, the ordinal height of the least admissible set above

M
M

, defined by Barwise. Nadel showed that

sr

(
M
)
⩽

o

(
M
)

{\operatorname {sr}}(M) \leqslant {\text {o}}(M)

and defined

M
M

to be tall if equality holds. For any admissible ordinal

α
\alpha

there exists a tall structure

M
M

such that

o

(
M
)
=
α

{\text {o}}(M) = \alpha

. We show that if

α
=

β
+

\alpha = {\beta ^ + }

, the least admissible ordinal greater than

β
\beta

, then

M
M

can be chosen to have a

β
\beta

-recursive presentation. A natural example of such a structure is given when

β
=

ω
1
L

\beta = \omega _1^L

and then using similar ideas we compute the supremum of the levels at which

Π
1

(

L

ω
1
L

)

{\Pi _1}({L_{\omega _1^L}})

singletons appear in

L
L

.
On the total variation and Hellinger distance between signed measures; an application to product measures
Authors: Steerneman Ton
Published: June 2010

Firstly, the Hellinger metric on the set of probability measures on a measurable space is extended to the set of signed measures. An inequality between total variation and Hellinger metric due to Kraft is generalized to the case of signed measures. The inequality is used in order to derive a lower estimate concerning the total variation distance between products of signed measures. The lower bound depends on the total variation norms of the signed measures and the total variation distances between the total variation measures of the single components.
Correction to: “Zero-one laws for stable measures” [Proc. Amer. Math. Soc. 45 (1974), 245–252; MR 51 #6901]
Authors: Dudley R. M. , Kanter Marek
Published: June 2010

A law on a vector space (e.g. the plane) with all one-dimensional marginals stable (of index less than 1) need not be stable, by examples of David Marcus. An error in the authors’ proof to the contrary is noted.
Immersions of real flag manifolds
Authors: Stong R. E.
Published: June 2010

This paper shows that, in many cases, Lam’s immersions of real flag manifolds are best possible.
Geometric automorphisms of a free group of rank at least three are rare
Authors: Gersten S. M.
Published: June 2010

The characteristic polynomials of geometric automorphisms of a free group of finite rank at least three form a nowhere dense set in the Zariski topology.
Geodesics and Jacobi fields in bounded homogeneous domains
Authors: D’Atri J. E. , Zhao Yan Da
Published: June 2010

We examine geodesics in bounded homogeneous domains equipped with an admissible Kähler metric. As in the symmetric case, there are no selfintersecting geodesics but, in contrast with the symmetric case, focal points can exist even in the Bergman metric.
Archimedean, semiperfect and 𝜋-regular lattice-ordered algebras with polynomial constraints are 𝑓-algebras
Authors: Steinberg Stuart A.
Published: June 2010

It is shown that a lattice-ordered algebra is embeddable in a product of totally ordered algebras provided (i) it is archimedean, contains a left superunit which is an

f
f

-element, and satisfies a polynomial identity

p
(
x
)
⩾
0

p(x) \geqslant 0

or

f
(
x
,
y
)
⩾
0

f(x,y) \geqslant 0

(for suitable

f
(
x
,
y
)

f(x,y)

); or (ii) it is unital, and semiperfect,

π
\pi

-regular, or left

π
\pi

-regular, and some power of each element is positive.
On spectral synthesis for sets of the form 𝐸=\overline{𝑖𝑛𝑡(𝐸)}
Authors: Colella David
Published: June 2010

The existence of a Helson set disobeying spectral synthesis is combined with the modified Herz criterion to construct a subset

E
E

of the circle such that spectral synthesis holds for

E
E

and fails for

∂
E

\partial E

.
Inequalities for the derivative of a polynomial
Authors: Aziz Abdul
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis z right-parenthesis equals sigma-summation Underscript j equals 0 Overscript n Endscripts a Subscript j Baseline z Superscript j Baseline equals c product Underscript j equals 1 Overscript n Endscripts left-parenthesis z minus z Subscript j Baseline right-parenthesis">

P
(
z
)
=

∑

j
=
0

n

a
j

z
j

=
c

∏

j
=
1

n

(
z
−

z
j

)

P(z) = \sum \nolimits _{j = 0}^n {{a_j}{z^j} = c\prod \nolimits _{j = 1}^n {(z - {z_j})} }

be a polynomial of degree

n
n

and

P
′

(
z
)

P’(z)

its derivative. In this paper we consider the problem of estimating the maximum of

|

P
′

(
z
)

|

\left | {P’(z)} \right |

on

|
z
|

=
1

\left | z \right | = 1

under some assumptions on the zeros or on the coefficients of

P
(
z
)

P(z)

and obtain certain sharp results.
Crossed products of 𝐶*-dynamical systems with ground states
Authors: Kusuda Masaharu
Published: June 2010

Let

C
∗

(
A
,

R

)

{C^*}(A,{\mathbf {R}})

be the crossed product associated with a

C
∗

{C^*}

-dynamical system

(
A
,

R

,
α
)

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

. We show that if there is a ground state for

A
A

. then there is an ideal

I
I

in

C
∗

(
A
,

R

)

{C^*}(A,{\mathbf {R}})

such that if

λ
1

>

λ
2

\lambda _1 > \lambda _2

, then

α
^

λ
1

(
I
)
⫋

α
^

λ
2

(
I
)

\hat {\alpha }_{\lambda _1}(I) \subsetneqq \hat {\alpha }_{\lambda _2}(I)

. where

α
^

\hat \alpha

is the dual automorphism group on

C
∗

(
A
,
R
)

{C^*}(A,R)

, and such that

C
∗

(
A
,

R

)
=

∪

λ
⩾
0

α
^

λ

(
I
)

{C^*}(A,{\mathbf {R}}) = { \cup _{\lambda \geqslant 0}}{\hat \alpha _\lambda }(I)

. Conversely, we show that if

A
A

is unital, and if

C
∗

(
A
,

R

)

{C^*}(A,{\mathbf {R}})

has such a decomposition for some proper ideal

I
I

. then

A
A

has a ground state.
A note on 𝛼-compact spaces
Authors: Retta Teklehaimanot
Published: June 2010

For an infinite cardinal

α
\alpha

,

m
(
α
)

m(\alpha )

denotes the least measurable cardinal, if one exists, not less than

α
\alpha

. We give easy proofs of generalizations of some results on realcompact spaces. Among these we prove the following generalization of a theorem of A. Kato. Let

{

X
i

:
i
∈
I
}

\{ {X_i}:i \in I\}

be a collection of spaces each having at least two elements. Then the

k
k

-box product

(
∏

X
i

)
k

{(\prod {X_i})_k}

is

α
\alpha

-compact if and only if either

X
i

{X_i}

is

α
\alpha

-compact for each

i
∈
I

i \in I

and

k
⩽
m
(
α
)

k \leqslant m(\alpha )

or

|
I
|

>
m
(
α
)

\left | I \right | > m(\alpha )

.
Connecting locally compact abelian groups
Authors: Enochs Ed , Gerlach Walt
Published: June 2010

Those locally compact abelian groups having connected envelopes are characterized as those

G
G

such that the dimension of

Hom

(
G
,
R
)

{\operatorname {Hom }}(G,R)

over

R
R

is finite (where

R
R

is the field of real numbers).
The asymptotic norming property and martingale convergence
Authors: van Dulst D.
Published: June 2010

A martingale proof is given of the result of R. G. James and A. Ho in [3] that the asymptotic norming property implies the Radon-Nikodym property.