bluuebunny/crossref_metadata_2025 huggingface dataset:
Properties of the solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimension
Authors: Glassey Robert T. , Chadam John M.
Published: June 2010

Solutions of the Maxwell-Dirac equations coupled through the standard electromagnetic interaction are shown to blow up at each spatial point for large times. This is used to show that these solutions do not tend asymptotically to free solutions. In addition it is used to prove that these equations do not admit a nontrivial stationary solution.
Two characteristic properties of the sphere
Authors: Koutroufiotis Dimitri
Published: June 2010

Let

S
S

be a closed, convex surface in

E
3

{E^3}

with positive Gaussian curvature

K
K

and let

K

II

{K_{{\text {II}}}}

be the curvature of its second fundamental form. It is shown that

S
S

is a sphere if

K

II

≡
c
K

{K_{{\text {II}}}} \equiv cK

for some constant

c
c

or if

K

II

≡
√
K

{K_{{\text {II}}}} \equiv \surd K

.
Wallman-type compactifications on 0-dimensional spaces
Authors: Su Li Pi
Published: June 2010

Let

E
E

be Hausdorff

0
0

-dimensional,

D

\mathcal {D}

the discrete space

{
0
,
1
}

\{0, 1\}

, and

N

\mathcal {N}

the discrete space of all nonnegative integers. Every

E
E

-completely regular space

X
X

has a clopen normal base

F

\mathcal {F}

with

X
∖
F
∈

F

X\backslash F \in \mathcal {F}

for each

F
∈

F

F \in \mathcal {F}

. The Wallman compactification

ω
(

F

)

\omega (\mathcal {F})

is

D

\mathcal {D}

-compact. Moreover, if an

E
E

-completely regular space

X
X

has a countably productive clopen normal base

F

\mathcal {F}

with

X
∖
F
∈

F

X\backslash F \in \mathcal {F}

for each

F
∈

F

F \in \mathcal {F}

, then the Wallman space

η
(

F

)

\eta (\mathcal {F})

is

N

\mathcal {N}

-compact. Hence, if

X
X

has such an

F

\mathcal {F}

, and is an

F

\mathcal {F}

-realcompact space, then

X
X

is

N

\mathcal {N}

-compact.
An endomorphism ring which is not Ore
Authors: Blair William D.
Published: June 2010

An example is given of a finitely generated module over a commutative ring whose endomorphism ring does not have a classical ring of quotients.
The radius of starlikeness of certain analytic functions
Authors: Bajpai P. L. , Singh Prem
Published: June 2010

Let

f
(
z
)
=

1
2

[
z
F
(
z
)

]
′

f(z) = \tfrac {1}{2}[zF(z)]’

, where

F
(
z
)

F(z)

is starlike of order

α
,
0
≦
α
>
1

\alpha ,0 \leqq \alpha > 1

in

D
=
{
z
:

|

z

|

>
1
}

D = \{ z:|z| > 1\}

. The purpose of this paper is to find out the disc in which

f
(
z
)

f(z)

is starlike of order

β
,
0
≦
β
>
1

\beta ,0 \leqq \beta > 1

. Results are best possible.
Coefficients associated with the expansion of certain products
Authors: Roselle D. P.
Published: June 2010

Simple combinatorial interpretations are given for the coefficients of the polynomials

H
n

(
x
,
y
)

{H_n}(x,y)

and

G
n

(
x
,
y
)

{G_n}(x,y)

defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product left-parenthesis 1 plus x Superscript n Baseline y Superscript m Baseline t right-parenthesis equals normal upper Sigma upper G Subscript n Baseline left-parenthesis x comma y right-parenthesis t Superscript n Baseline slash left-parenthesis x right-parenthesis Subscript n Baseline left-parenthesis y right-parenthesis Subscript n">

∏
(
1
+

x
n

y
m

t
)
=
Σ

G
n

(
x
,
y
)

t
n

/

(
x

)
n

(
y

)
n

\prod (1 + {x^n}{y^m}t) = \Sigma {G_n}(x,y){t^n}/{(x)_n}{(y)_n}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product left-parenthesis 1 minus x Superscript n Baseline y Superscript m Baseline t right-parenthesis Superscript negative 1 Baseline equals normal upper Sigma upper H Subscript n Baseline left-parenthesis x comma y right-parenthesis t Superscript n Baseline slash left-parenthesis x right-parenthesis Subscript n Baseline left-parenthesis y right-parenthesis Subscript n">

∏

(
1
−

x
n

y
m

t

)

−
1

=
Σ

H
n

(
x
,
y
)

t
n

/

(
x

)
n

(
y

)
n

\prod {(1 - {x^n}{y^m}t)^{ - 1}} = \Sigma {H_n}(x,y){t^n}/{(x)_n}{(y)_n}

.
Prime von Neumann regular rings and primitive group algebras
Authors: Fisher Joe W. , Snider Robert L.
Published: June 2010

Kaplansky posed the following question: Are prime von Neumann regular rings primitive? We show that with a certain countability condition the answer is affirmative. This is used to simplify and clarify earlier work on the existence of primitive group algebras.
Solution of a nonlinear partial differential equation with initial conditions
Authors: Reid James L. , Pritchard W. M.
Published: June 2010

The exact solution

ϕ
\phi

of a particular nonlinear partial differential equation is obtained in terms of solution u of a related linear partial differential equation. It is noted that solution

ϕ
\phi

may be found subject to initial conditions if certain initial conditions can be determined for solution u. Two examples are solved explicitly.
On regions of 𝛼-convexity for starlike functions
Authors: Bajpai S. K.
Published: June 2010

In this note an alternative proof to a result of Mocanu and Reade [see Notices Amer. Math. Soc. 20 (1973), A-107] on

α
\alpha

-convexity region for starlike functions has been obtained.
The difference between consecutive members of a difference set (𝑚𝑜𝑑 𝑣)
Authors: Spence E.
Published: June 2010

Let

D

=
{

d
1

,

d
2

,
⋯
,

d
k

}

\mathcal {D} = \{ {d_1},{d_2}, \cdots ,{d_k}\}

be a difference set

(
mod

v
)

\pmod v

so that for any

d
≢
0

(
mod

v
)

d\not \equiv 0\pmod v

there are exactly

λ
\lambda

pairs

(

d
i

,

d
j

)
,

d
i

,

d
j

∈

D

({d_i},{d_j}),{d_i},{d_j}\in \mathcal {D}

such that

d
i

−

d
j

≡
d

(
mod

v
)

{d_i} - {d_j} \equiv d\pmod v

. Suppose further that

0
≤

d
1

>

d
2

>
⋯
>

d
k

>
v

0 \leq {d_1} > {d_2} > \cdots > {d_k} > v

and write

d

k
+
1

=
v
+

d
1

{d_{k + 1}} = v+ {d_1}

. The following two results are proved: (i)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma Subscript i equals 1 Superscript k Baseline left-parenthesis d Subscript i plus 1 Baseline minus d Subscript i Baseline right-parenthesis squared equals upper O left-parenthesis left-parenthesis v squared slash k right-parenthesis log k right-parenthesis comma">

Σ

i
=
1

k

(

d

i
+
1

−

d
i

)
2

=
O
(
(

v
2

/

k
)
log
⁡
k
)
,

\Sigma _{i = 1}^k{({d_{i + 1}} - {d_i})^2} = O(({v^2}/k)\log k),

(ii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max Underscript 1 less-than-or-slanted-equals i less-than-or-slanted-equals k Endscripts left-parenthesis d Subscript i plus 1 Baseline minus d Subscript i Baseline right-parenthesis equals upper O left-parenthesis v slash square-root k right-parenthesis period">

max

1
⩽
i
⩽
k

(

d

i
+
1

−

d
i

)
=
O
(
v

/

√
k
)
.

{\max _{1 \leqslant i \leqslant k}}({d_{i + 1}} - {d_i}) = O(v/\surd k).
Almost 𝜎-Dedekind complete Riesz spaces and the main inclusion theorem
Authors: Aliprantis C. D. , Langford Eric
Published: June 2010

A Riesz space is almost

σ
\sigma

-Dedekind complete if it can be embedded as a super order dense Riesz subspace of a

σ
\sigma

-Dedekind complete space. The location of this concept in the Main Inclusion Theorem of Luxemburg and Zaanen is investigated; it is shown that this concept is implied by

σ
\sigma

-Dedekind completeness and by the projection property, that it implies the property of being Archimedean, and that it is independent of the principal projection property and the property of having sufficiently many projections.
Products of 𝑀-spaces
Authors: Bandy C.
Published: June 2010

The author associates with each pair

X
,
Y

X,Y

of

M
M

-spaces such that

X
×
Y

X \times Y

is not an

M
M

-space, a pair of countably compact closed subspaces

A
⊂
X
,
B
⊂
Y

A \subset X,B \subset Y

such that

A
×
B

A \times B

is not countably compact, and for each pair

A
,
B

A,B

of countably compact spaces whose product is not countably compact, there is a pair of

M
M

-spaces

S
,
T

S,T

(in fact,

S
S

and

T
T

ate countably compact) such that

S
×
T

S \times T

is not an

M
M

-space and such that

A
A

and

B
B

are closed subspaces of

S
S

and

T
T

respectively.
Two unrelated results involving Baire spaces
Authors: White H. E.
Published: June 2010

Two results are obtained in this paper. The first is a generalization of J. C. Oxtoby’s category analogue of the Kolmogoroff zero-one law. The second result: every dense

G
δ

{G_\delta }

subset of a quasi-regular

α
\alpha

-favorable space is

α
\alpha

-favorable.
Holomorphic mappings of bounded distortion
Authors: Marden A. , Rickman S.
Published: June 2010

Nonelementary holomorphic mappings

C

n

→

C

n

,
n
≥
2

{{\mathbf {C}}^n} \to {{\mathbf {C}}^n},n \geq 2

, cause severe geometric distortion near

∞
\infty

in sharp contrast to the case

n
=
1

n = 1

when there is no distortion.
Compact and quasinormal composition operators
Authors: Singh Raj Kishor
Published: June 2010

Let

C
ϕ

{C_\phi }

be a composition operator on

L
2

(
λ
)

{L^2}(\lambda )

, where

λ
\lambda

is a

σ
\sigma

-finite measure on a set

X
X

. If

X
X

is nonatomic, then Ridge proved that no one-to-one composition operator

C
ϕ

{C_\phi }

, with dense range is compact. This result is generalized in the paper by removing one-to-one and dense range conditions. The quasinormal composition operators are also characterized in terms of commutativity with the multiplication operator induced by the Radon-Nikodym derivative of the measure

λ

ϕ

−
1

\lambda {\phi ^{ - 1}}

with respect to

λ
\lambda

.
Sylowizers in locally finite groups
Authors: Tomkinson M. J.
Published: June 2010

A result of W. Gaschütz for finite soluble groups is extended to two classes of locally finite, locally soluble groups.
Some properties of self-inversive polynomials
Authors: O’Hara P. J. , Rodriguez R. S.
Published: June 2010

A complex polynomial is called self-inversive [5, p. 201] if its set of zeros (listing multiple zeros as many times as their multiplicity indicates) is symmetric with respect to the unit circle. We prove that if

P
P

is self-inversive and of degree

n
n

then

|

|

P
′

|

|

=

1
2

n

|

|

P

|

|

||P’|| = \tfrac {1}{2}n||P||

where

|

|

P
′

|

|

||P’||

and

|

|

P

|

|

||P||

denote the maximum modulus of

P
′

P’

and

P
P

, respectively, on the unit circle. This extends a theorem of P. Lax [4]. We also show that if

P
(
z
)
=

Σ

j
=
0

n

a
j

z
j

P(z) = \Sigma _{j = 0}^n{a_j}{z^j}

has all its zeros on

|

z

|

=
1

|z| = 1

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 normal upper Sigma Subscript j equals 0 Superscript n Baseline StartAbsoluteValue a Subscript j Baseline EndAbsoluteValue squared less-than-over-equals StartAbsoluteValue EndAbsoluteValue upper P StartAbsoluteValue EndAbsoluteValue squared">

2

Σ

j
=
0

n

|

a
j

|

2

≦

|

|

P

|

|

2

2\Sigma _{j = 0}^n|{a_j}{|^2} \leqq ||P|{|^2}

. Finally, as a consequence of this inequality, we show that when

P
P

has all its zeros on

|

z

|

=
1

|z| = 1

then

2

1

/

2

|

a

n

/

2

|

≦

|

|

P

|

|

{2^{1/2}}|{a_{n/2}}| \leqq ||P||

and

2

|

a
j

|

≦

|

|

P

|

|

2|{a_j}| \leqq ||P||

for

j
≠
n

/

2

j \ne n/2

. This answers in part a question presented in [3, p. 24].
Coefficient estimates for exponential series
Authors: Sledd W. T.
Published: June 2010

Results are obtained which relate the size of the coefficients

a
n

{a_n}

of an exponential series
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals normal upper Sigma Subscript n equals 0 Superscript normal infinity Baseline a Subscript n Baseline epsilon Superscript minus normal upper Lamda Super Subscript n Superscript x Baseline comma x greater-than 0 comma upper R e normal upper Lamda Subscript n Baseline greater-than 0">

f
(
x
)
=

Σ

n
=
0

∞

a
n

ε

−

Λ
n

x

,
x
>
0
,
Re
⁡

Λ
n

>
0

f(x) = \Sigma _{n = 0}^\infty {a_n}{\varepsilon ^{ - {\Lambda _n}x}},x > 0,\operatorname {Re} {\Lambda _n} > 0

, to the function

f
f

. These results involve comparisons between weighted

l
p

{l^p}

sums of the sequence

(

a
n

)

({a_n})

and weighted

L
p

{L^p}

integrals of

f
f

on

[
0
,
∞
)

[0,\infty )

.
On the a.e. convergence of Walsh-Kaczmarz-Fourier series
Authors: Young Wo Sang
Published: June 2010

It is shown that partial sums of Walsh-Kaczmarz-Fourier series of functions in the Orlicz class

L

(

log
+

L

)
2

L{({\log ^ + }L)^2}

converge a.e. The proof utilizes an estimate of P. Sjölin on the partial sums of the usual Walsh-Fourier series.
Note on a family of Volterra equations
Authors: Hannsgen Kenneth B.
Published: June 2010

We prove that the solutions of a certain family of Volterra integrodifferential equations are uniformly bounded. We use this result to determine the asymptotic behavior of the solution of a Volterra equation in Hilbert space.
Representation theorems for unconditionally converging operators
Authors: Howard Joe , Melendez Kenneth
Published: June 2010

Let

N
(
X
)

N(X)

be the set

{
F
ε
X

\{ F\varepsilon X

: there exists a weakly unconditionally converging series

Σ

x
n

\Sigma {x_n}

in

X
X

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F equals sigma left-parenthesis upper X comma upper X Superscript prime Baseline right-parenthesis minus limit Underscript n Endscripts normal upper Sigma Subscript i equals 1 Superscript n Baseline upper J x Subscript i Baseline right-brace">

F
=
σ
(
X
,

X
′

)
−

lim
n

Σ

i
=
1

n

J

x
i

}

F = \sigma (X,X’) - {\lim _n}\Sigma _{i = 1}^nJ{x_i}\}

. Representation theorems for the unconditionally converging operators (map weakly unconditionally converging series into unconditionally converging series) are developed by using the

σ
(

X
′

,
N
(
X
)
)

\sigma (X’,N(X))

topology of

X
′

X’

.
Characterization of the flip operator
Authors: Seid H. A.
Published: June 2010

The flip operator

F
F

on

L
p

(
[
0
,
1
]
)

{L_p}([0,1])

, defined by

F
(
f
)
(
t
)
=
f
(
1
−
t
)

F(f)(t) = f(1 - t)

, for

f
f

in

L
p

(
[
0
,
1
]
)

{L_p}([0,1])

is characterized up to isometric transformation by means of its induced

σ
\sigma

-isomorphism on the Borel sets of

[
0
,
1
]

[0,1]

.
Linear superposition of smooth functions
Authors: Kaufman Robert
Published: June 2010

We give a simple proof of the impossibility of representing an arbitrary continuous function as a superposition (1), when

F
1

,
⋯
,

F
N

{F_1}, \cdots ,{F_N}

are smooth mappings of

R

n
+
1

{R^{n + 1}}

to

R
n

{R^n}

. The main tool is the Riemann-Lebesgue lemma.
Coefficient differences and Hankel determinants of areally mean 𝑝-valent functions
Authors: Noonan James W.
Published: June 2010

With

p
>
0

p > 0

, denote by

S
p

{S_p}

the class of functions analytic and areally mean

p
p

-valent in the open unit disc. If

f
∈

S
p

f \in {S_p}

, it is well known that

α
(
f
)
=

lim

r
→
1

(
1
−
r

)

2
p

M
(
r
,
f
)

\alpha (f) = {\lim _{r \to 1}}{(1 - r)^{2p}}M(r,f)

exists and is finite. If

q
≥
1

q \geq 1

is an integer, denote the

q
q

Hankel determinant of

f
f

by

H
q

(
n
,
f
)

{H_q}(n,f)

. In this paper the asymptotic behavior of

H
q

(
n
,
f
)

{H_q}(n,f)

, as

n
→
∞

n \to \infty

, is related to

α
(
f
)

\alpha (f)

. A typical result is: if

α
(
f
)
>
0

\alpha (f) > 0

, and if

p
>
q
−
3

/

4

p > q - 3/4

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper H Subscript q Baseline left-parenthesis n comma f right-parenthesis EndAbsoluteValue slash n Superscript 2 p q minus q squared Baseline tilde StartAbsoluteValue upper Q Subscript q Baseline left-parenthesis p right-parenthesis EndAbsoluteValue left-parenthesis alpha left-parenthesis f right-parenthesis slash normal upper Gamma left-parenthesis 2 p right-parenthesis right-parenthesis Superscript q Baseline comma">

|

H
q

(
n
,
f
)

|

/

n

2
p
q
−

q
2

∼

|

Q
q

(
p
)

|

(
α
(
f
)

/

Γ
(
2
p
)

)
q

,

|{H_q}(n,f)|/{n^{2pq - {q^2}}} \sim |{Q_q}(p)|{(\alpha (f)/\Gamma (2p))^q},

\]
where

Q
q

{Q_q}

is a polynomial of degree at most

q
2

−
q

{q^2} - q

. In the course of the proof, asymptotic results are proved concerning certain coefficient differences, and in particular concerning

|

a
n

|

−

|

a

n
−
1

|

|{a_n}| - |{a_{n - 1}}|

.
The DAD theorem for arbitrary row sums
Authors: Brualdi Richard A.
Published: June 2010

Given an

m
×
m

m \times m

symmetric nonnegative matrix

A
A

and a positive vector

R
=
(

r
1

,
⋯
,

r
m

)

R = ({r_1}, \cdots ,{r_m})

, necessary and sufficient conditions are obtained in order that there exist a diagonal matrix

D
D

with positive main diagonal such that DAD has row sum vector

R
R

.
Iterated Littlewood-Paley functions and a multiplier theorem
Authors: Madych W. R.
Published: June 2010

A sufficient condition for a bounded function to be a multiplier of Fourier transforms on

L
p

(

R
n

)
,
1
>
p
>
∞

{L^p}({R^n}),1 > p > \infty

, is established. The classical case of Marcinkiewicz is properly included. The main tools used in obtaining this result are iterated variants of the classical LittlewoodPaley functions together with an

L
p

{L^p}

estimate on certain maximal functions closely related to strong differentiability of multiple integrals.
Maximal sublattices of finite distributive lattices. II
Authors: Rival Ivan
Published: June 2010

Let

L
L

be a lattice,

J
(
L
)
=
{
x
∈
L

|

x

J(L) = \{ x \in L|x

join-irreducible in

L
}

L\}

and

M
(
L
)
=
{
x
∈
L

|

x

M(L) = \{ x \in L|x

meet-irreducible in

L
}

L\}

. As is well known the sets

J
(
L
)

J(L)

and

M
(
L
)

M(L)

play a central role in the arithmetic of a lattice

L
L

of finite length and particularly, in the case that

L
L

is distributive. It is shown that the “quotient set”
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q left-parenthesis upper L right-parenthesis equals left-brace b slash a vertical-bar a element-of upper J left-parenthesis upper L right-parenthesis comma b element-of upper M left-parenthesis upper L right-parenthesis comma a less-than-over-equals b right-brace">

Q
(
L
)
=
{
b

/

a

|

a
∈
J
(
L
)
,
b
∈
M
(
L
)
,
a
≦
b
}

Q(L) = \{ b/a|a \in J(L),b \in M(L),a \leqq b\}

plays a somewhat analogous role in the study of the sublattices of a lattice

L
L

of finite length. If

L
L

is a finite distributive lattice, its quotient set

Q
(
L
)

Q(L)

) in a natural way determines the lattice of all sublattices of

L
L

. By examining the connection between

J
(
K
)

J(K)

and

J
(
L
)

J(L)

, where

K
K

is a maximal proper sublattice of a finite distributive lattice

L
L

, the following is proven: every finite distributive lattice of order

n
≧
3

n \geqq 3

which contains a maximal proper sublattice of order

m
m

also contains sublattices of orders

n
−
m
,
2
(
n
−
m
)

n - m,2(n - m)

, and

3
(
n
−
m
)

3(n - m)

; and, every finite distributive lattice

L
L

contains a maximal proper sublattice

K
K

such that either

|

K

|

=

|

L

|

−
1

|K| = |L| - 1

or

|

K

|

≧
2
l
(
L
)

|K| \geqq 2l(L)

, where

l
(
L
)

l(L)

denotes the length of

L
L

.
Lattice points and Lie groups. III
Authors: Cahn Robert S.
Published: June 2010

If a compact, simply connected, semisimple Lie group is considered as a Riemannian manifold with metric arising from the negative of the Killing form it is shown that its volume is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 4 pi right-parenthesis Superscript dimension upper G slash 2 Baseline normal upper Gamma left-parenthesis dimension upper G slash 2 plus 1 right-parenthesis left-parenthesis 1 slash StartAbsoluteValue w EndAbsoluteValue right-parenthesis integral Underscript StartAbsoluteValue normal upper Lamda EndAbsoluteValue less-than-or-slanted-equals 1 Endscripts f Superscript 2 Super Subscript left-parenthesis normal upper Lamda right-parenthesis d normal upper Lamda Superscript Baseline period">

(
4
π

)

dim
⁡
G

/

2

Γ
(
dim
⁡
G

/

2
+
1
)
(
1

/

|

w

|

)

∫

|

Λ

|

⩽
1

f

2

(
Λ
)
d
Λ

.

{(4\pi )^{\dim G/2}}\Gamma (\dim G/2 + 1)(1/|w|)\int _{|\Lambda | \leqslant 1} {{f^{{2_{(\Lambda )d\Lambda }}}}.}

\]
Characterization of Mergelyan sets
Authors: Stray A.
Published: June 2010

Necessary and sufficient conditions on a relatively closed subset

F
F

of

D
=
{
z
:

|

z

|

>
1
}

D = \{ z:|z| > 1\}

are given such that each analytic function in

D
D

which is uniformly continuous on

F
F

can be uniformly approximated by polynomials on

K
∪
F

K \cup F

for each compact subset

K
K

of

D
D

.
Note on a Volterra integro-differential equation system
Authors: Weis Dennis G.
Published: June 2010

This note concerns a type of nonlinear Volterra integro-differential equation system which has only asymptotically zero solutions. It is a simple generalization of a one-dimensional result of J. J. Levin. It is a correction to another such note and gives a counterexample to that note.
A Galois connection for reduced incidence algebras
Authors: Davis Robert L.
Published: June 2010

If

N
=
{
1
,
⋯
,
n
}
,
D
⊂
N
×
N

N = \{ 1, \cdots ,n\} ,D \subset N \times N

, and

F
F

is an equivalence relation on the “entries” of

D
D

the reduced incidence space

g
(
F
)

g(F)

is the set of all real matrices

A
A

with support in

D
D

and such that

a

i
j

=

a

r
s

{a_{ij}} = {a_{rs}}

whenever

(
i
,
j
)
F
(
r
,
s
)

(i,j)F(r,s)

. Let

L

(
D
)

\mathcal {L}(D)

be the lattice of all subspaces of

R
n

{R_n}

having support contained in

D
D

, and

E

(
D
)

\mathcal {E}(D)

that of all equivalences on

D
D

. Then the map

g
g

defined above is Galois connected with a map

f
f

which sends a subspace

S
S

into the equivalence

f
(
S
)

f(S)

having

(
i
,
j
)
[
f
(
S
)
]
(
r
,
s
)

(i,j)[f(S)](r,s)

whenever all

A
A

in

S
S

have

a

i
j

=

a

r
s

{a_{ij}} = {a_{rs}}

. The Galois closed subspaces (i.e. reduced incidence spaces) are shown to be just those subspaces which are closed under Hadamard multiplication, and if

S
=
g
(
F
)

S = g(F)

is also a subalgebra then its support

D
D

must be a transitive relation. Consequences include not only pinpointing the role of Hadamard multiplication in characterizing reduced incidence algebras, but methods for constructing interesting new types of algebras of matrices.
A theorem in the representation theory of totally disconnected groups
Authors: Kallman Robert R.
Published: June 2010

We prove that certain von Neumann algebras naturally associated with specific direct summands of the left regular representation of a unimodular totally disconnected group are finite von Neumann algebras. The Lie group analogue of this result is false.
Some nonexistence theorems for initial-boundary value problems with nonlinear boundary constraints
Authors: Levine H. A. , Payne L. E.
Published: June 2010

The present work extends previous results of the authors [4] on nonexistence of solutions of classes of second order equations subject to certain nonlinear boundary constraints and appropriate hypotheses on the data. Similar results are derived here for more general classes of higher order equations.
ℒ-realcompactifications as epireflections
Authors: Bentley H. L. , Naimpally S. A.
Published: June 2010

If

L

\mathcal {L}

is a countably productive normal base on a Tychonoff space

X
X

, then

η
(
X
,

L

)

\eta (X,\mathcal {L})

is an

L

∗

{\mathcal {L}_ \ast }

-realcompact extension of

X
X

. R. A. Alo and H. L. Shapiro thus generalized the Hewitt realcompactification of

X
X

. In the following paper, we extend this construction to

T
1

{T_1}

-spaces and show that it is an epireflection functor on an appropriate category. We are thus concerned with the question of the extendibility of a continuous map

f
:
X
→
Y

f:X \to Y

to a continuous map

g
:
η
(
X
,

L

X

)
→
η
(
Y
,

L

Y

)

g:\eta (X,{\mathcal {L}_X}) \to \eta (Y,{\mathcal {L}_Y})

. We derive necessary and sufficient conditions therefor in the case when

L

Y

{\mathcal {L}_Y}

is a nest generated intersection ring on

Y
Y

.
Topologies on the quasi-spectrum of a 𝐶*-algebra
Authors: Ching Wai Mee
Published: June 2010

We study two topologies, the Gelfand topology and the Jacobson topology, on the quasi-spectrum of a

C
∗

{C^ \ast }

-algebra. We prove that the Gelfand topology is finer than the Jacobson topology.
On the distribution of zeros of entire functions
Authors: Reddy A. R.
Published: June 2010

Let

f
(
z
)

f(z)

be any transcendental entire function. Let

r
k

{r_k}

denote the absolute value of the zero

z
k

{z_k}

of

f

(
k
)

(
z
)

{f^{(k)}}(z)

which is nearest to the origin. Ålander, Erdös and Rényi, and Pólya have investigated the relation between

r
k

{r_k}

and the growth of the function

f
(
z
)

f(z)

. Let

s
k

{s_k}

denote the largest disk about the origin where

f

(
k
)

(
z
)

{f^{(k)}}(z)

is univalent. Boas, Levinson, and Pólya have obtained some relations between the growth of the function

f
(
z
)

f(z)

and

s
k

{s_k}

. Recently Shah and Trimble have sharpened the results of Boas and Pólya. We present here results in a different direction, generalizing the above quoted results. We also present results connecting the zero-free disks and the univalent disks about the origin of the normalized remainders of

f
(
z
)

f(z)

with the growth of

f
(
z
)

f(z)

.
Positive linear operators continuous for strict topologies
Authors: Crenshaw J. A.
Published: June 2010

If

A
A

is an

S
W

SW

algebra of real-valued functions on a set

X
X

equipped with the weak topology for

A
A

, and if

A
A

separates its zero sets, then many results valued for

C
b

(
X
)

{C^b}(X)

equipped with a strict topology remain true when

A
A

is equipped with a strict topology. The concepts of

α
\alpha

-additivity and tight positive linear operators are introduced. It is shown that if

T
T

is a positive linear map on

A
A

into

z
z

-separating

S
W

SW

algebra

B
B

and if

T
(

1
A

)
=

1
B

T({1_A}) = {1_B}

, then there exists a continuous function

ϕ
\phi

on

Y
Y

(the domain of elements in

B
B

) into

X
X

such that

T
f
(
y
)
=
f
(
ϕ
(
y
)
)

Tf(y) = f(\phi (y))

if and only if

T
T

is an algebraic homomorphism and

τ
\tau

-additive.
A note on 𝑇₁ topologies
Authors: Crisler Wilson R. , Hicks Troy L.
Published: June 2010

Let

t
t

be a

T
1

{T_1}

topology for a set

X
X

. The problem of representing

t
t

as the lattice product (intersection) of stronger topologies is considered.
On fundamental transversal matroids
Authors: Brualdi R. A.
Published: June 2010

Unified proofs of two theorems on fundamental transversal matroids are presented. A necessary condition for a matroid to be a fundamental transversal matroid with respect to a given basis is given.
Bordism of manifolds with oriented boundaries
Authors: Mitchell G. E.
Published: June 2010

A bordism theory is defined for manifolds with oriented boundaries. The relation of this theory with the ordinary bordism theories is shown. These bordism classes are then characterized via characteristic numbers.
Harrison’s Witt ring of a generalized valuation ring
Authors: Chang Chan Nan
Published: June 2010

The purpose of this paper is to determine the relations between

H
(
R
)

H(R)

(Harrison’s Witt ring given by generators and relations over a generalized valuation ring

R
R

) and

W
(
K
)

W(K)

(classical Witt ring of inner product spaces over the field

K
K

). Using the main results, we are able to establish some theorems on the structure of

H
(
R
)

H(R)

analogous to the field case.
Spectral synthesis of functions of bounded variation
Authors: Atzmon Aharon
Published: June 2010

It is proved that every bounded measurable function on

(
−
∞
,
∞
)

( - \infty ,\infty )

which for some constant

a
>
0

a > 0

is of bounded variation on

(
−
∞
,
−
a
)

( - \infty , - a)

and on

(
a
,
∞
)

(a,\infty )

, admits spectral synthesis in the weak-star topology of

L
∞

(
−
∞
,
∞
)

{L^\infty }( - \infty ,\infty )

.
Mapping properties of relatively regular operators
Authors: Caradus S. R.
Published: June 2010

A relatively regular operator is one with closed complemented range and nullspace. It is shown that if

T
T

is relatively regular and

f
f

is univalent on the spectrum of

T
T

with

f
(
0
)
=
0

f(0) = 0

, then

f
(
T
)

f(T)

is also relatively regular.
Nonexpansive mappings on compact subsets of metric linear spaces
Authors: Guseman L. F. , Peters B. C.
Published: June 2010

Results are presented concerning the existence of fixed points of nonexpansive selfmaps on a compact starshaped subset of a metric linear space. The results are based on a general argument of D. R. Smart and extend recent results of W. G. Dotson, Jr.
The regularization of nonlinear electrical circuits
Authors: Ihrig E.
Published: June 2010

Recently Smale [1] proposed the question ’under what conditions can an electrical circuit be regularized?’ This note gives a solution to the problem.
Absolute zero divisors and local nilpotence in alternative algebras
Authors: McCrimmon Kevin
Published: June 2010

It has been conjectured that absolute zero divisors generate locally nilpotent ideals in Jordan and alternative algebras. A. M. Slin’ko has recently established this result for special Jordan algebras; in this note we show how his method can be modified to establish the result for alternative algebras.
Representations of almost connected groups
Authors: Miličić Dragan
Published: June 2010

A simple proof is given that an irreducible representation of an almost connected group on a quasi-complete locally convex space is essentially a Lie group representation.
Convergence sets in reflexive Banach spaces
Authors: Calvert Bruce
Published: June 2010

A closed linear subspace

M
M

of a reflexive Banach space

X
X

with

X
X

and

X
∗

{X^ \ast }

strictly convex is the range of a linear contractive projection iff

J
(
M
)

J(M)

is a linear subspace of

X
∗

{X^ \ast }

. Hence the convergence set of a net of linear contractions is the range of a contractive projection if

X
X

and

X
∗

{X^ \ast }

are locally uniformly convex.
The dual of a theorem of Bishop and Phelps
Authors: Luna George
Published: June 2010

We dualize a theorem of Bishop and Phelps by showing that in the dual of a Banach space the intersection of a weak* closed finite codimensional linear variety and a weak* closed convex subset

C
C

contains a norm dense set of weak* support points of

C
C

. We use this theorem to obtain a result which is related to an abstract approximation problem of Deutsch and Morris.
Normal and Hermitian composition operators
Authors: Singh Raj Kishor
Published: June 2010

Let

C
ϕ

{C_\phi }

be a composition operator on

L
2

(
λ
)

{L^2}(\lambda )

. Some conditions under which

C
ϕ

{C_\phi }

is an isometry and Hermitian are investigated in this paper. Some study of normal composition operators is also made.
Equidistant sets and their connectivity properties
Authors: Wilker J. B.
Published: June 2010

If

A
A

and

B
B

are nonvoid subsets of a metric space

(
X
,
d
)

(X,d)

, the set of points

x
∈
X

x \in X

for which

d
(
x
,
A
)
=
d
(
x
,
B
)

d(x,A) = d(x,B)

is called the equidistant set determined by

A
A

and

B
B

. Among other results, it is shown that if

A
A

and

B
B

are connected and

X
X

is Euclidean

n
n

-space, then the equidistant set determined by

A
A

and

B
B

is connected.
A spectral condition determining the Kaehler property
Authors: Donnelly Harold
Published: June 2010

We prove that the spectrum of the reduced complex Laplacian determines if a Hermitian manifold is Kaehler.
The measurability of a stochastic process of second order and its linear space
Authors: Cambanis Stamatis
Published: June 2010

It is of considerable theoretical and practical interest to know whether a stochastic process has a measurable modification. For the important class of second order processes, simple necessary and sufficient conditions for the existence of a measurable modification are given in terms of the autocorrelation of the process and the separability of its reproducing kernel Hilbert space or its linear space. It is shown that weakly continuous processes, processes with orthogonal increments and second order martingales always have measurable modifications. Also necessary and sufficient conditions are given in terms of integral representations for the linear space of a second order process to be separable. As a consequence it is shown that a second order process is oscillatory if and only if its linear space is separable.
On sequences spanning a complex 𝑙₁ space
Authors: Dor Leonard E.
Published: June 2010

If

(

f
n

)

({f_n})

is a bounded sequence in a complex Banach space

B
B

, and no subsequence of

(

f
n

)

({f_n})

is weakly Cauchy, then a subsequence of

(

f
n

)

({f_n})

is equivalent to the unit vector basis of the complex

l
1

{l_1}

space.
Zeros of polynomials over finite principal ideal rings
Authors: Marshall Murray , Ramage Garry
Published: June 2010

Let

R
R

be a finite commutative ring with identity. For

f
∈
R
[

X
1

,
⋯
,

X
n

]

f \in R[{X_1}, \cdots ,{X_n}]

denote by

N
(
f
)

N(f)

the number of zeros of

f
f

in

R

(
n
)

{R^{(n)}}

. For integers

n
,
d
≥
1

n,d \geq 1

denote by

A

n
,
d

{A_{n,d}}

the greatest common divisor of the integers

N
(
f
)
;
f
∈
R
[

X
1

,
⋯
,

X
n

]
,
deg
⁡
f
=
d

N(f);f \in R[{X_1}, \cdots ,{X_n}],\deg f = d

. J. Ax has shown that if

R
R

is a field, then

A

n
,
d

=

|

R

|

α

{A_{n,d}} = |R{|^\alpha }

where

α
\alpha

is the integer satisfying

α
>
n

/

d
≤
α
+
1

\alpha > n/d \leq \alpha + 1

. In this paper,

A

n
,
d

{A_{n,d}}

is computed in the case that

R
R

is a principal ideal ring.
The relative completion of an 𝐴-Segal algebra is closed
Authors: Burnham James T.
Published: June 2010

The main result is this theorem: If the Banach algebra

A
A

has bounded approximate right units and

B
B

is an

A
A

-Segal algebra, then the relative completion of

B
B

with respect to

A
A

is an

A
A

-Segal algebra. Furthermore,

B
B

is a closed ideal of its relative completion with respect to

A
A

.
A biorthogonal expansion related to the zeros of an entire function
Authors: Benzinger Harold E.
Published: June 2010

Let

D
(
λ
)
=

a
0

+

Σ

k
=
1

∞

a
k

e

λ

/

k

D(\lambda ) = {a_0} + \Sigma _{k = 1}^\infty {a_k}{e^{\lambda /k}}

, where the

a
k

{a_k}

’s are given complex constants, with

a
0

a
1

≠
0
,

Σ

k
=
1

∞

|

a
k

|

>
∞

{a_0}{a_1} \ne 0,\Sigma _{k = 1}^\infty |{a_k}| > \infty

. Let

{

λ
k

}
,
k
=
1
,
2
,
⋯

\{ {\lambda _k}\} ,k = 1,2, \cdots

, denote the sequence of distinct zeros of

D
(
λ
)

D(\lambda )

, labeled in order of increasing modulus, and with multiplicities

m
k

≥
1

{m_k} \geq 1

. Let

{

ϕ
l

(
x
)
}

\{ {\phi _l}(x)\}

denote the sequence of functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x Superscript j Baseline exp left-parenthesis lamda Subscript k Baseline x right-parenthesis colon j equals 0 comma midline-horizontal-ellipsis comma m Subscript k Baseline minus 1 semicolon k equals 1 comma 2 comma midline-horizontal-ellipsis semicolon 0 greater-than x less-than-or-equal-to 1 right-brace">

{

x
j

exp
⁡
(

λ
k

x
)
:
j
=
0
,
⋯
,

m
k

−
1
;
k
=
1
,
2
,
⋯
;
0
>
x
≤
1
}

\{ {x^j}\exp ({\lambda _k}x):j = 0, \cdots ,{m_k} - 1;k = 1,2, \cdots ;0 > x \leq 1\}

. We show that for each

p
.1
≤
p
>
∞

p.1 \leq p > \infty

, there is a sequence

{

ψ
l

(
x
)
}

\{ {\psi _l}(x)\}

in

L
q

(
0
,
1
)

(
p
q
=
p
+
q
)

{L^q}(0,1)\;(pq = p + q)

such that

(

ϕ
l

,

ψ
m

)
=

δ

l
m

({\phi _l},{\psi _m}) = {\delta _{lm}}

. Then we show that

{

ϕ
l

}

\{ {\phi _l}\}

is complete in

L
p

(
0
,
1
)
,
1
≤
p
>
∞

{L^p}(0,1),1 \leq p > \infty

, and for

1
>
p
>
∞

1 > p > \infty

, we find a subspace of

L
p

(
0
,
1
)

{L^p}(0,1)

such that the biorthogonal expansion

f
=

Σ

k
=
1

∞

(
f
,

ψ
k

)

ϕ
k

f = \Sigma _{k = 1}^\infty (f,{\psi _k}){\phi _k}

is valid in the norm of

L
p

(
0
,
1
)

{L^p}(0,1)

.
New criteria for univalent functions
Authors: Ruscheweyh Stephan
Published: June 2010

The classes

K
n

{K_n}

of functions

f
(
z
)

f(z)

regular in the unit disc

U

\mathfrak {U}

with

f
(
0
)
=
0
,

f
′

(
0
)
=
1

f(0) = 0,f’(0) = 1

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e left-bracket left-parenthesis z Superscript n Baseline f right-parenthesis Superscript left-parenthesis n plus 1 right-parenthesis Baseline slash left-parenthesis z Superscript n minus 1 Baseline f right-parenthesis Superscript left-parenthesis n right-parenthesis Baseline right-bracket greater-than left-parenthesis n plus 1 right-parenthesis slash 2">

Re
⁡
[

(

z
n

f

)

(
n
+
1
)

/

(

z

n
−
1

f

)

(
n
)

]
>
(
n
+
1
)

/

2

\operatorname {Re} [{({z^n}f)^{(n + 1)}}/{({z^{n - 1}}f)^{(n)}}] > (n + 1)/2

\]
in

U

\mathfrak {U}

are considered and

K

n
+
1

⊂

K
n

,
n
=
0
,
1
,
⋯

{K_{n + 1}} \subset {K_n},n = 0,1, \cdots

, is proved. Since

K
0

{K_0}

is the class of functions starlike of order 1/2 all functions in

K
n

{K_n}

are univalent. Some coefficient estimates are given and special elements of

K
n

{K_n}

are determined.
Toeplitz operators associated with isometries
Authors: Moore B. , Rosenblum M. , Rovnyak J.
Published: June 2010

Shift analysis includes abstract treatments of inner-outer factorization problems, the factorization problem for nonnegative functions on a circle, and Szegö’s infimum problem (for scalar or operator valued functions). These problems are here generalized to a setting where the shift operator is replaced by a pair of isometries.
On the existence of 𝜅-free abelian groups
Authors: Eklof Paul C.
Published: June 2010

It is proved that if

ℵ
α

{\aleph _\alpha }

is a regular cardinal such that there is an

ℵ
α

{\aleph _\alpha }

-free abelian group which is not

ℵ

α
+
1

{\aleph _{\alpha + 1}}

-free, then for every positive integer

n
n

there is an

ℵ

α
+
n

{\aleph _{\alpha + n}}

-free abelian group which is not

ℵ

α
+
n
+
1

{\aleph _{\alpha + n + 1}}

-free. A corollary is that for each positive integer

n
n

there is a group of cardinality

ℵ
n

{\aleph _n}

which is

ℵ
n

{\aleph _n}

-free but not free. Some results on

κ
\kappa

-free abelian groups which involve notions from logic are also proved.
Sums of powers of conjugates of algebraic numbers
Authors: Blanksby P. E.
Published: June 2010

This note is primarily concerned with algebraic numbers which have at least one conjugate with modulus exceeding one. Theorems which give lower bounds for the maximum of the moduli of the conjugates of such algebraic numbers are connected with equivalent theorems giving lower bounds for the sums of powers of the conjugates. The results relate to some earlier work of S. Chowla, and the method used depends on one of P. Turán’s main theorems on lower bounds for sums of powers of complex numbers.
Induced representations of discrete groups
Authors: Corwin Lawrence
Published: June 2010

We determine necessary and sufficient conditions for a unitary representation of a discrete group induced from a finite-dimensional representation to be irreducible, and also briefly examine the question of when these induced representations are Type II.
Fixed point theorems for mappings with a contractive iterate at a point
Authors: Matkowski Janusz
Published: June 2010

Let (X,d) be a complete metric space,

T
:
X
→
X

T:X \to X

, and

α
:
[
0
,
∞

)
5

→
[
0
,
∞
)

\alpha :[0,\infty )^5 \to [0,\infty )

be nondecreasing with respect to each variable. Suppose that for the function

γ
(
t
)
=
α
(
t
,
t
,
t
,
2
t
,
2
t
)

\gamma (t) = \alpha (t,t,t,2t,2t)

, the sequence of iterates

γ
n

{\gamma ^n}

tends to 0 in

[
0
,
∞
)

[0,\infty )

and

lim

t
→
∞

(
t
−
γ
(
t
)
)
=
∞

{\lim _{t \to \infty }}(t - \gamma (t)) = \infty

. Furthermore, suppose that for each

x
∈
X

x \in X

there exists a positive integer

n
=
n
(
x
)

n = n(x)

such that for all

y
∈
X

y \in X

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d left-parenthesis upper T Superscript n Baseline x comma upper T Superscript n Baseline y right-parenthesis less-than-or-slanted-equals alpha left-parenthesis d left-parenthesis x comma upper T Superscript n Baseline x right-parenthesis comma d left-parenthesis x comma upper T Superscript n Baseline y right-parenthesis comma d left-parenthesis x comma y right-parenthesis comma d left-parenthesis upper T Superscript n Baseline x comma y right-parenthesis comma d left-parenthesis upper T Superscript n Baseline y comma y right-parenthesis right-parenthesis period">

d
(

T
n

x
,

T
n

y
)
⩽
α
(
d
(
x
,

T
n

x
)
,
d
(
x
,

T
n

y
)
,
d
(
x
,
y
)
,
d
(

T
n

x
,
y
)
,
d
(

T
n

y
,
y
)
)
.

d({T^n}x,{T^n}y) \leqslant \alpha (d(x,{T^n}x),d(x,{T^n}y),d(x,y),d({T^n}x,y),d({T^n}y,y)).

\]
Under these assumptions our main result states that T has a unique fixed point. This generalizes an earlier result of V. M. Sehgal and some recent results of L. Khazanchi and K. Iseki.
Inverse-cycles in weak-inverse loops
Authors: Artzy R.
Published: June 2010

In a loop let J be the map taking every element into its right inverse. If the order of J is n, then

{
x
,
x
J
,
…
,
x

J

n
−
1

}

\{ x,xJ, \ldots ,x{J^{n - 1}}\}

is called an inverse-cycle of length n. If a weak-inverse loop consists only of the unit element and m inverse-cycles of length n, then n divides 2m.
Elementary surgery manifolds and the elementary ideals
Authors: Neuzil J. P.
Published: June 2010

We prove the following: If

M
3

{M^3}

is a closed 3-manifold obtained by elementary surgery on a knot K in

S
3

{S^3}

and

H
1

(

M
3

)

{H_1}({M^3})

is a nontrivial cyclic group, then the first elementary ideal

π
1

(

M
3

)

{\pi _1}({M^3})

in the integral group ring of

H
1

(

M
3

)

{H_1}({M^3})

is the principal ideal generated by the polynomial of K.
Free Lie algebras as modules over their enveloping algebras
Authors: Labute John P.
Published: June 2010

In this paper we determine the linear relations that exist between the free generators of a free Lie algebra L when it is viewed as a module over its enveloping algebra via the adjoint representation. As an application, the annihilator of a homogeneous element of L is determined.
Noncrossed products of small exponent
Authors: Saltman David J.
Published: June 2010

Using the generic division algebras, we construct new division algebras and prove the following. Theorem. Let

m
m

,

n
n

be positive integers,

m

|

n

m|n

, such that

m
m

is divisible by every prime which divides

n
n

. If

k
k

is any field, and there is a prime

p
p

such that

p
3

|

m

{p^3}|m

, then there is a noncrossed product division

k
k

-algebra of exponent 

m
m

and degree 

n
n

. If

k
k

is a global field of characteristic 0,

p
p

is an odd prime, and

k
k

does not contain a primitive

p
p

th root of 1, we need only assume

p
2

|

m

p^2|m

.
The Knapp-Stein dimension theorem for 𝑝-adic groups
Authors: Silberger Allan J.
Published: June 2010

Knapp and Stein have proved for semisimple Lie groups that the dimension of the commuting algebra of an induced tempered representation equals the index of a certain reflection group in a larger group. A precise analogue of their result is stated and proved in this paper for p-adic groups.
On density of algebras with minimal invariant operator ranges
Authors: Radjavi Heydar
Published: June 2010

Let

A

\mathfrak {A}

be an arbitrary subalgebra of

B

(

H

)

\mathcal {B}(\mathcal {H})

and let

M

\mathfrak {M}

be a dense operator range invariant under

A

\mathfrak {A}

such that every nonzero operator range invariant under

A

\mathfrak {A}

contains

M

\mathfrak {M}

. Then the closure of

A

\mathfrak {A}

in the strong operator topology is

B

(

H

)

\mathcal {B}(\mathcal {H})

.
Triangulation of stratified objects
Authors: Goresky R. Mark
Published: June 2010

A procedure for triangulating an abstract prestratified set is outlined.
Factorization in codimension two ideals of group algebras
Authors: Willis G. A.
Published: June 2010

Let

G
G

be a finitely generated group and

I
I

be a closed, two-sided ideal with codimension two in

L
1

(
G
)

{L^1}(G)

. Then the linear span of the set of all products in

I
I

is equal to

I
I

.
Square integrable representations of unimodular groups
Authors: Shucker David S.
Published: June 2010

An elementary proof is given showing that if a continuous irreducible unitary representation of a locally compact unimodular group

G
G

has one nontrivial square integrable matrix entry, then all its matrix entries are square integrable. This result was first proved by R. Godement.
A note on compact groups
Authors: Wu Ta Sun
Published: June 2010

We show that the product of certain subsets in a compact connected topological group is the group itself.
An extension of the Hausdorff-Toeplitz theorem on the numerical range
Authors: Au-Yeung Yik Hoi , Tsing Nam-Kiu
Published: June 2010

Let

H

n

{\mathcal {H}_n}

be the set of all

n
×
n

n \times n

hermitian matrices and

U

n

{\mathcal {U}_n}

the set of all

n
×
n

n \times n

unitary matrices. For any

c
=
(

c
1

,
…
,

c
n

)
∈

R

n

c = ({c_1}, \ldots ,{c_n}) \in {{\mathbf {R}}^n}

and

A
1

{A_1}

,

A
2

{A_2}

,

A
3

∈

H

n

{A_3} \in {\mathcal {H}_n}

, let

W
(

A
1

,

A
2

,

A
3

)

W({A_1},{A_2},{A_3})

denote the set
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet left-parenthesis trace left-bracket c right-bracket upper U upper A 1 upper U Superscript asterisk Baseline comma trace left-bracket c right-bracket upper U upper A 2 upper U Superscript asterisk Baseline comma trace left-bracket c right-bracket upper U upper A 3 upper U Superscript asterisk Baseline right-parenthesis colon upper U element-of script upper U Subscript n Baseline EndSet comma">

{
(

tr

[
c
]
U

A
1

U
∗

,

tr

[
c
]
U

A
2

U
∗

,

tr

[
c
]
U

A
3

U
∗

)
:
U
∈

U

n

}
,

\{ ({\operatorname {tr}}[c]U{A_1}{U^*},{\operatorname {tr}}[c]U{A_2}{U^*},{\operatorname {tr}}[c]U{A_3}{U^*}):U \in {\mathcal {U}_n}\} ,

\]
where

[
c
]

[c]

is the diagonal matrix with

c
1

,
…
,

c
n

{c_1}, \ldots ,{c_n}

as diagonal entries. In this present note, the authors prove that if

n
>
2

n > 2

, then

W
c

(

A
1

,

A
2

,

A
3

)

{W_c}({A_1},{A_2},{A_3})

is always convex. Equivalent statements of this result, in terms of definiteness and inclusion relations, are also given. These results extend the theorems of Hausdorff-Toeplitz, Finsler and Westwick on numerical ranges, respectively.
Fourier type and complex interpolation
Authors: Milman Mario
Published: June 2010

Using Fourier type arguments we provide a very simple proof of recent results on complex interpolation of

H
p

{H^p}

spaces and martingale

H
p

{H^p}

spaces. The same method gives a new result on complex interpolation of Sobolev spaces.
Mapping theorems on mesocompact spaces
Authors: Kao Kuo Shih , Wu Li Sheng
Published: June 2010

In this paper we prove two mapping theorems on mesocompact spaces: (1) Perfect mappings preserve mesocompactness; (2) Closed mappings preserve normal mesocompactness.
A family of semistable elliptic curves with large Tate-Shafarevitch groups
Authors: Kramer Kenneth
Published: June 2010

We present a family of elliptic curves defined over the rationals

Q

{\mathbf {Q}}

such that each curve admits only good or multiplicative reduction and for every integer

n
n

there is a curve whose Tate-Shafarevitch group over

Q

{\mathbf {Q}}

has more than

n
n

elements of order 2. Previously known examples of large Tate-Shafarevitch groups were constructed by forcing many places of additive reduction.
Representing a differentiable function as a Cartesian product
Authors: Colvin Michael R.
Published: June 2010

This article produces an elementary proof of a result originally stated without proof by J. Leray. The main result gives conditions so that a continuously differentiable map from a product neighborhood of the origin in

R

n

{{\mathbf {R}}^n}

into

R

n

{{\mathbf {R}}^n}

can be homotoped to a cartesian product of maps on intervals. The resulting product function preserves properties of the original map near the origin.
Decompositions of rigid spaces
Authors: van Engelen Fons , van Mill Jan
Published: June 2010

We give an example of a rigid subspace of

R

{\mathbf {R}}

which can be decomposed into two homeomorphic homogeneous parts, and of a rigid subspace of

R

{\mathbf {R}}

which can be decomposed into two homeomorphic rigid parts.