bluuebunny/crossref_metadata_2025 huggingface dataset:
The operator 𝑖𝑑/𝑑𝑥, on 𝐶[0,1], is 𝐶¹-scalar
Authors: deLaubenfels Ralph
Published: July 2010

We consider closed, unbounded linear operators

A
A

, on a Banach space,

X
X

, with discrete real spectrum, with corresponding eigenvectors,

{

x
k

}

−
∞

∞

\{ {x_k}\} _{ - \infty }^\infty

, such that

A

x
k

=

a
k

x
k

,

a
k

≤

a

k
+
1

A{x_k} = {a_k}{x_k},{a_k} \leq {a_{k + 1}}

, for all

k
k

, and

lim

k
→
±
∞

|

a
k

|

=
∞

{\lim _{k \to \pm \infty }}\left | {{a_k}} \right | = \infty

. For arbitrary

n
n

, we present necessary and sufficient conditions for

A
A

to be

C
n

{C^n}

-scalar. Letting

F
(
s
)

F(s)

be the projection whose range is the

span

{

x
k

|

a
k

is between 

0

 and 

s
}

{\text {span}}\{ {x_k}|{a_k}\;{\text {is between }}0{\text { and }}s\}

, null-space is the closure of

span

{

x
k

|

a
k

is not between 

0

 and 

s
}

{\text {span}}\{ {x_k}|{a_k}\;{\text {is not between }}0{\text { and }}s\}

, we show that

A
A

is

C
1

{C^1}

-scalar if and only if the series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript a Subscript k Baseline greater-than 0 Endscripts phi left-parenthesis upper F left-parenthesis a Subscript k Baseline right-parenthesis x right-parenthesis left-parenthesis StartFraction 1 Over a Subscript k Baseline EndFraction minus StartFraction 1 Over a Subscript k plus 1 Baseline EndFraction right-parenthesis and sigma-summation Underscript a Subscript k Baseline greater-than 0 Endscripts phi left-parenthesis upper F left-parenthesis a Subscript k Baseline right-parenthesis x right-parenthesis left-parenthesis StartFraction 1 Over a Subscript k minus 1 Baseline EndFraction minus StartFraction 1 Over a Subscript k Baseline EndFraction right-parenthesis">

∑

a
k

>
0

φ
(
F
(

a
k

)
x
)

(

1

a
k

−

1

a

k
+
1

)

and

∑

a
k

>
0

φ
(
F
(

a
k

)
x
)

(

1

a

k
−
1

−

1

a
k

)

\sum \limits _{{a_k} > 0} {\varphi (F({a_k})x)\left ( {\frac {1}{{{a_k}}} - \frac {1}{{{a_{k + 1}}}}} \right )} \quad {\text {and}}\quad \sum \limits _{{a_k} > 0} {\varphi (F({a_k})x)\left ( {\frac {1}{{{a_{k - 1}}}} - \frac {1}{{{a_k}}}} \right )}

\]
both converge absolutely, for all

φ
\varphi

in

X
∗

,
x

{X^ * },x

in

X
X

. As a corollary, we get that

i
d

/

d
x

id/dx

, on

{
f

 in 

C
[
0
,
1
]

|

f
(
0
)
=
f
(
1
)
}

\{ f{\text { in }}C[0,1]|f(0) = f(1)\}

, is

C
1

{C^1}

-scalar. Also,

i
d

/

d
x

id/dx

, on

L
1

[
0
,
1
]

{L^1}[0,1]

, is

C
1

{C^1}

-scalar.
Point singularities and conformal metrics on Riemann surfaces
Authors: McOwen Robert C.
Published: July 2010

Given a closed hyperbolic Riemann surface and a finite number of points, we prove the existence and uniqueness of hyperbolic conformal metrics with prescribed singularities or degeneracies at the given points.
Cesàro convergence of martingale difference sequences and the Banach-Saks and Szlenk theorems
Authors: Freniche Francisco J.
Published: July 2010

It is shown that, for

1
≤
p
>
+
∞

1 \leq p > + \infty

, any weakly null martingale difference sequence in

L
p

[

0
,
1

]

{L^p}\left [ {0,1} \right ]

is Cesàro convergent to zero in the

L
p

{L^p}

norm. This result combined with a theorem of Gaposhkin gives an easy proof of two theorems of Banach-Saks and Szlenk at once.
The Faber transform and analytic continuation
Authors: Johnston Elgin
Published: July 2010

Let

Ω
⊆
C

\Omega \subseteq C

be a bounded, simply connected domain, and let

{

Φ
n

(
w
)

}

n
=
0

∞

\left \{ {{\Phi _n}\left ( w \right )} \right \}_{n = 0}^\infty

be the Faber polynomials associated with

Ω
\Omega

. Given

f

(
z
)

=

∑

k
=
0

∞

c
k

z
k

f\left ( z \right ) = \sum \nolimits _{k = 0}^\infty {{c_k}{z^k}}

analytic in

Δ

(

0
,
1

)

\Delta \left ( {0,1} \right )

we consider the function
\[

F

(
w
)

=

∑

k
=
0

∞

c
k

Φ
k

(
w
)

.

F\left ( w \right ) = \sum \limits _{k = 0}^\infty {{c_k}{\Phi _k}\left ( w \right )} .

\]
We show that with proper restrictions on

∂
Ω

\partial \Omega

, the existence of an analytic continuation of

f
f

across a subarc of

C

(

0
,
1

)

C\left ( {0,1} \right )

implies the existence of an analytic continuation of

F
F

across a subarc of

∂
Ω

\partial \Omega

. Some converse results are also established.
An application of Banach limits
Authors: Ahmad Z. U. , Mursaleen
Published: July 2010

Let

l
∞

{l_\infty }

denote the Banach space of bounded sequences,

σ
\sigma

an injection of the set of positive integers into itself having no finite orbits, and

T
T

the operator defined on

l
∞

{l_\infty }

by

T
y

(
n
)

=
y

(

σ
n

)

Ty\left ( n \right ) = y\left ( {\sigma n} \right )

. A positive linear functional

L

\mathcal {L}

with

‖

L

‖

=
1

\left \| \mathcal {L} \right \| = 1

, is called a

σ
\sigma

-mean if

L

(
y
)

=

L

(

T
y

)

\mathcal {L}\left ( y \right ) = \mathcal {L}\left ( {{T_y}} \right )

for all

y
y

in

l
∞

{l_\infty }

. A sequence

y
y

is said to be

σ
\sigma

-convergent, denoted

y
∈

V
σ

y \in {V_\sigma }

, if

L

(
y
)

\mathcal {L}\left ( y \right )

takes the same value, called

σ
−
lim
y

\sigma - \lim y

, for all

σ
\sigma

-means

L

\mathcal {L}

. P. Schaefer [6] gave necessary and sufficient conditions on a matrix

A
A

to ensure that

A

(
c
)

⊂

V
σ

A\left ( c \right ) \subset {V_\sigma }

, where

c
c

is the space of convergent sequences, and additional conditions ensuring that

σ
−
lim
A
y
=
lim
y

\sigma - \lim Ay = \lim y

for all

y
∈
c

y \in c

, denoting the class of matrices satisfying these conditions by

(

c
,

V
σ

)

1

{\left ( {c,{V_\sigma }} \right )_1}

and calling them the

σ
\sigma

-regular matrices. In this paper, we use such matrices to find the sum of a sequence of Walsh functions.
An asymptotic fixed point theorem for a locally convex space
Authors: Burton T. A. , Dwiggins D. P.
Published: July 2010

The asymptotic fixed point theorem of Horn is extended to a locally convex topological vector space. Applications to differential equations with infinite delay are discussed.
Weak (1,1) boundedness of singular integrals with nonsmooth kernel
Authors: Hofmann Steve
Published: July 2010

If

Ω
∈

L
q

(

S
1

)

\Omega \in {L^q}\left ( {{S^1}} \right )

for some

q
>
1
,

∫

S
1

Ω
=
0

q > 1,\int _{{S^1}} {\Omega = 0}

, and

Ω
\Omega

is homogeneous of degree 0, then the operator defined in two dimensions by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript epsilon Baseline f left-parenthesis x right-parenthesis equals integral Underscript StartAbsoluteValue y EndAbsoluteValue greater-than epsilon Endscripts f left-parenthesis x minus y right-parenthesis normal upper Omega left-parenthesis y right-parenthesis StartAbsoluteValue y EndAbsoluteValue Superscript negative 2 Baseline d y">

T
ε

f

(
x
)

=

∫

|
y
|

>
ε

f

(

x
−
y

)

Ω

(
y
)

|
y
|

−
2

d
y

{T_\varepsilon }f\left ( x \right ) = \int _{\left | y \right | > \varepsilon } {f\left ( {x - y} \right )\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}dy}

is of weak-type

(
1
,
1
)

(1,1)

with bound independent of

ε
>
0

\varepsilon > 0

.
The logically simplest form of the infinity axiom
Authors: Parlamento Franco , Policriti Alberto
Published: July 2010

We provide a way of expressing the existence of infinite sets in the first order set theoretic language, which is of the lowest possible logical complexity.
Two theorems on truth table degrees
Authors: Downey R. G.
Published: July 2010

In this article we solve two questions of Odifreddi on the r.e.

tt

{\text {tt}}

-degrees. First we construct an r.e.

tt

{\text {tt}}

-degree with anticupping property. In fact, we construct r.e.

tt

{\text {tt}}

-degrees

a

,

b

{\mathbf {a}},{\mathbf {b}}

with

0

>

a

>

b

{\mathbf {0}} > {\mathbf {a}} > {\mathbf {b}}

and such that for all (not necessarily r.e.)

tt

{\text {tt}}

-degrees

c

{\mathbf {c}}

if

a

∪

c

≥

b

{\mathbf {a}} \cup {\mathbf {c}} \geq {\mathbf {b}}

then

a

≤

c

{\mathbf {a}} \leq {\mathbf {c}}

. This result also has ramifications in, for example, the r.e.

wtt

{\text {wtt}}

-degrees. Finally we solve another question of Odifreddi by constructing an r.e.

tt

{\text {tt}}

-degree with no greatest r.e.

m
m

-degree.
A proportionality principle for partitioning problems
Authors: Hill Theodore P.
Published: July 2010

In a general class of measure-partitioning or fair-division problems, the extremal case occurs when the measures are proportional. Applications are given to classical and recent fair-division problems, and to statistical decision theory, mathematical physics, Banach space theory, and inequalities for continuous random variables.
Spaces every quotient of which is metrizable
Authors: Jayanthan A. J. , Kannan V.
Published: July 2010

We characterise those topological spaces for which every quotient image is metrizable. This supplements the earlier known results in this direction, in a fairly complete manner.
Finite-dimensional complement theorems: examples and results
Authors: Sher R. B. , Venema G. A.
Published: July 2010

Examples are given which show the necessity of various hypotheses in the known finite dimensional complement theorems. In addition, several positive results are presented which improve one direction of such theorems.
More nondevelopable spaces in 𝑀𝑂𝐵𝐼
Authors: Chaber J.
Published: July 2010

We construct two spaces with a point-countable base, the first is not weakly

θ
\theta

-refinable, while the second is not a

σ
#

{\sigma ^\# }

-space. Both spaces can be represented as open and compact images of metacompact developable Hausdorff spaces. This answers questions of H. R. Bennett and R. F. Gittings.
Random trees in random graphs
Authors: Bender E. A. , Wormald N. C.
Published: July 2010

We show that a random labeled

n
n

-vertex graph almost surely contains isomorphic copies of almost all labeled

n
n

-vertex trees, in two senses. In the first sense, the probability of each edge occurring in the graph diminishes as

n
n

increases, and the set of trees referred to as "almost all" depends on the random graph. In the other sense, the probability of an edge occurring is fixed, and the relevant set of trees is predetermined. The same method applies to show that almost all labeled

n
n

-tournaments contain isomorphic copies of almost all labeled oriented

n
n

-trees.
Enumerating 2-cell imbeddings of connected graphs
Authors: Mull Bruce P. , Rieper Robert G. , White Arthur T.
Published: July 2010

A systematic approach is developed for enumerating congruence classes of

2
2

-cell imbeddings of connected graphs on closed orientable

2
2

-manifolds. The method is applied to the wheel graphs and to the complete graphs. Congruence class genus polynomials and congruence class imbedding polynomials are introduced, to summarize important information refining the enumeration.
On Gauss’s first proof of the fundamental theorem of algebra
Authors: Gersten S. M. , Stallings John R.
Published: July 2010

Gauss’s first proof of the Fundamental Theorem of Algebra is shown to be related to basic properties of free groups.
Tame objects for finite commutative Hopf algebras
Authors: Waterhouse William C.
Published: July 2010

Let

S
S

be an

A
A

-module algebra for a commutative Hopf algebra

A
A

, both projective of the same rank over a commutative ring. Let

I

{\mathbf {I}}

be the space of integrals in

A
A

. Then

S
S

is an invertible

A
A

-module iff it is a faithful module which satisfies the "trace surjectivity" condition that 1 is in

I

S

{\mathbf {I}}S

.
Injective dimension of some divisible modules over a valuation domain
Authors: Bazzoni Silvana
Published: July 2010

Let

R
R

be a valuation domain of global dimension

n
+
1

n + 1

. Given an infinite direct product of injective envelopes of (torsion) cyclic modules, let

D

n
−
k

{D_{n - k}}

be the submodule consisting of the elements having support of cardinality less than

ℵ

n
−
k

{\aleph _{n - k}}

. We prove that the injective dimension of

D

n
−
k

{D_{n - k}}

is at most

k
k

and, using

⋄
\diamond

-axiom, we prove that

D

n
−
2

{D_{n - 2}}

has injective dimension exactly 2.
Quasi-atoms and complete distributivity
Authors: Deng Zi Ke
Published: July 2010

We introduce quasi-atoms in completely distributive complete lattices and use them to treat the representation of these lattices and their connection with a binary relation.
Polarity in a completely distributive complete lattice
Authors: Deng Zi Ke
Published: July 2010

We introduce

p
p

-bases in completely distributive complete polarity lattices and give a procedure for generating these lattices by

p
p

-bases.
Subnormal subgroups in 𝑈(𝑍𝐺)
Authors: Gonçalves Jairo , Ritter Jürgen , Sehgal Sudarshan
Published: July 2010

Let

U
U

be the unit group of the integral group ring of a finite group

G
G

. We prove that every subgroup of

U
U

containing

G
G

and almost subnormal in

U
U

contains a noncyclic free group unless

G
G

is abelian or a Hamiltonian

2
2

-group.
A sequence well dispersed in the unit square
Authors: Lambert J. P.
Published: July 2010

Distributional properties of sequences of points in the unit square have been studied extensively and there is considerable interest in sequences which are well distributed according to various criteria. Dispersion is a measure of sequence density and an important concept connected with regularity of distribution. We introduce an infinite, dyadic, easily-generated sequence which is particularly well distributed, in the sense of having dispersion of lowest possible order of magnitude.
Differential basis, 𝑝-basis, and smoothness in characteristic 𝑝>0
Authors: Tyc Andrzej
Published: July 2010

It is shown that every differential basis of a Noetherian ring

R
R

of prime characteristic

p
p

over an arbitrary subring

k
k

is a

p
p

-basis of

R
R

over

k
k

. Moreover, if

k
k

is a field, then

R
R

is smooth over

k
k

, provided

R
R

has a differential basis over

k
k

and the ring

R

⊗
k

k

p

−
1

R{ \otimes _k}{k^{{p^{ - 1}}}}

is reduced.
Spots over analytically normal local domains
Authors: Sally Judith D.
Published: July 2010

A result of Lipman on

2
2

-dimensional analytically normal local domains is extended to dimension

d
>
2

d > 2

, thereby enlarging the class of local domains known to have certain finiteness properties with respect to birational extensions.
A finiteness theorem for linear semigroups
Authors: Zalcstein Yechezkel
Published: July 2010

A classical theorem of Burnside on the finiteness of periodic linear groups is generalized to semigroups.
Nil algebras with nonradical tensor square
Authors: Puczyłowski Edmund R.
Published: July 2010

The paper contains some simple observations on the tensor square of algebras. Applied to the well-known Golod examples, they allow us to produce nil algebras with nonradical tensor square.
An inequality for the integral means of a Hadamard product
Authors: Pavlović Miroslav
Published: July 2010

Motivated by Colzani’s paper [1] we prove that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript q Baseline left-parenthesis r comma f asterisk g right-parenthesis less-than-or-equal-to left-parenthesis 1 minus r right-parenthesis Superscript 1 minus 1 slash p Baseline double-vertical-bar f double-vertical-bar p double-vertical-bar g double-vertical-bar q comma 0 greater-than r greater-than 1 comma">

M
q

(
r
,
f
∗
g
)
≤

(
1
−
r

)

1
−
1

/

p

‖
f
‖

p

‖
g
‖

q
,

0

 > 

r

 > 

1
,

{M_q}(r,f * g) \leq {(1 - r)^{1 - 1/p}}\left \| f \right \|p\left \| g \right \|q,\quad \;0{\text { > }}r{\text { > }}1,

\]
where

0

 > 

p

 > 

1
,
p
≤
q
≤
∞

0{\text { > }}p{\text { > }}1,p \leq q \leq \infty

and

f
∗
g

f * g

is the Hadamard product of

f
∈

H
p

f \in {H^p}

and

g
∈

H
q

g \in {H^q}

.
Browder spectral systems
Authors: Curto Raúl E. , Dash A. T.
Published: July 2010

For two spectral systems

σ
1

{\sigma _1}

and

σ
2

{\sigma _2}

on a Banach space

X

\mathcal {X}

, the associated Browder spectral system is

σ

b
;
1
,
2

:=

σ
1

∪

σ
2
′

{\sigma _{b;1,2}}: = {\sigma _1} \cup {\sigma ’_2}

. We prove that

σ

b
;
1
,
2

{\sigma _{b;1,2}}

possesses the projection and spectral mapping properties whenever

σ
1

{\sigma _1}

and

σ
2

{\sigma _2}

do (and satisfy a few additional mild assumptions). We also calculate

σ

b
;
1
,
2

{\sigma _{b;1,2}}

for tensor products. The results extend several previous works on Browder spectra.
Support points of subordination families
Authors: Hallenbeck D. J.
Published: July 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a function

F
F

analytic in the unit disc

Δ
\Delta

. Let

H
s
(
F
)

Hs(F)

denote the closed convex hull of

s
(
F
)

s(F)

and supp

s
(
F
)

s(F)

the set of support points of

s
(
F
)

s(F)

. We prove the following Theorem. Let

F
F

be analytic in

Δ
\Delta

and satisfy (1)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H s left-parenthesis upper F right-parenthesis equals left-brace integral Underscript partial-differential normal upper Delta Endscripts upper F left-parenthesis x z right-parenthesis d mu left-parenthesis x right-parenthesis colon mu a probablity measure on partial-differential normal upper Delta right-brace">

H
s
(
F
)
=
{

∫

∂
Δ

F
(
x
z
)
d
μ
(
x
)
:
μ

a

probablity

measure

on

∂
Δ

}

Hs(F) = \{ \int _{\partial \Delta } {F(xz)d\mu (x):\mu \;{\text {a}}\;{\text {probablity}}\;{\text {measure}}\;{\text {on}}\;\partial \Delta } \}

and (2)

F
(
z
)
=
G
(
z
)

/

(
z
−

x
0

)
α

F(z) = G(z)/{(z - {x_0})^\alpha }

where

G
G

is analytic in

Δ
\Delta

, continuous in

Δ
¯

\bar \Delta

,

G
(

x
0

)
≠
0

G({x_0}) \ne 0

and

α
>
1

\alpha > 1

. Then supp

s
(
F
)
=

{

F
(
x
z
)
:

|

x

|

=
1

}

s(F) = \left \{ {F(xz):|x| = 1} \right \}

.
On joint hyponormality of operators
Authors: Athavale Ameer
Published: July 2010

A notion of joint hyponormality is introduced for a collection of bounded linear operators on a separable Hilbert space.
An integral criterion for normal functions
Authors: Aulaskari Rauno , Lappan Peter
Published: July 2010

A new characterization for normal functions is given. It is shown that a function

f
f

meromorphic in the unit disk is a normal function if and only if for each

δ
>
0

\delta > 0

and each

p
>
2

p > 2

there exists a constant

K
f

(
δ
,
p
)

{K_f}(\delta ,p)

such that, for each hyperbolic disk

Ω
\Omega

with hyperbolic radius

δ
\delta

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-integral Underscript normal upper Omega Endscripts left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript p minus 2 Baseline left-parenthesis f Superscript not-equals Baseline left-parenthesis z right-parenthesis right-parenthesis Superscript p Baseline d upper A left-parenthesis z right-parenthesis less-than-or-equal-to upper K Subscript f Baseline left-parenthesis delta comma p right-parenthesis comma">

∬
Ω

(
1
−

|
z
|

2

)

p
−
2

(

f
≠

(
z
)
)

p

d
A
(
z
)
≤

K
f

(
δ
,
p
)

,

\iint _\Omega {{{(1 - {{\left | z \right |}^2})}^{p - 2}}{{({f^ \ne }(z))}^p}dA(z) \leq {K_f}(\delta ,p)},

\]
where

f
≠

(
z
)

{f^ \ne }(z)

denotes the spherical derivative of

f
f

and

d
A
(
z
)

dA(z)

is the Euclidean element of area. It is shown by example that this characterization is not valid for

p
=
2

p = 2

.
Local uncertainty inequalities for compact groups
Authors: Price John F. , Sitaram Alladi
Published: July 2010

Conditions are established on

α
,
β
∈

R

\alpha ,\beta \in {\mathbf {R}}

for there to exist a constant

K
=
K
(
α
,
β
)

K = K(\alpha ,\beta )

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript gamma element-of upper E Endscripts d left-parenthesis gamma right-parenthesis trace left-parenthesis ModifyingAbove f With caret left-parenthesis gamma right-parenthesis Superscript asterisk Baseline ModifyingAbove f With caret left-parenthesis gamma right-parenthesis right-parenthesis less-than-or-equal-to upper K left-parenthesis sigma-summation Underscript gamma element-of upper E Endscripts d left-parenthesis gamma right-parenthesis squared right-parenthesis Superscript alpha Baseline double-vertical-bar w Superscript beta Baseline f double-vertical-bar Subscript 2">

∑

γ
∈
E

d
(
γ
)
tr
⁡
(

f
^

(
γ
)

∗

f
^

(
γ
)
)
≤
K

(

∑

γ
∈
E

d

(
γ
)

2

)

α

‖

w
β

f

‖

2

\sum \limits _{\gamma \in E} {d(\gamma )\operatorname {tr} (\hat f{{(\gamma )}^ * }\hat f(\gamma )) \leq K{{\left ( {\sum \limits _{\gamma \in E} {d{{(\gamma )}^2}} } \right )}^\alpha }{{\left \| {{w^\beta }f} \right \|}_2}}

\]
for all

f
∈

L
1

(
G
)

f \in {L^1}(G)

and

E
⊆

G
^

E \subseteq \hat G

where

G
G

is a compact metric group,

G
^

\hat G

is its dual,

f
^

\hat f

is the Fourier transform of

f
f

and

w
:
G
→

R

+

w:G \to {{\mathbf {R}}^ + }

is the function taking

x
∈
G

x \in G

to the area of the ball in

G
G

with centre

e
e

and

x
x

on its boundary. This is followed by a partial analogy for compact riemannian manifolds.
The area of the complement of a conformally rigid domain
Authors: Overholt Marius
Published: July 2010

A hyperbolic subdomain

D
D

of

C
^

{\mathbf {\hat C}}

is said to be (conformally) rigid if any conformal map from

D
D

into

C
^

{\mathbf {\hat C}}

is either a Möbius transformation, or has Schwarzian norm larger than a positive constant depending on

D
D

only. We show that the complement of a conformally rigid domain has zero area.
A comparison theorem for linear difference equations
Authors: Eloe P. W.
Published: July 2010

Let a be real,

I
=
{
a
,
a
+
1
,
⋯
,
b
}

I = \{ a,a + 1, \cdots ,b\}

where

b
−
a

b - a

is a positive integer or

I
=
{
a
,
a
+
1
,
…
}

I = \{ a,a + 1, \ldots \}

. Let

n
n

be a positive integer and let

I
n

=
{
a
,
a
+
1
,
…
,
b
+
n
}

{I^n} = \{ a,a + 1, \ldots ,b + n\}

if

b
>
∞

or

I
n

=
I

b > \infty \;{\text {or}}\;{I^n} = I

otherwise. Consider the

n
n

th order difference equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P u left-parenthesis m right-parenthesis equals sigma-summation Underscript j equals 0 Overscript n Endscripts alpha Subscript j Baseline left-parenthesis m right-parenthesis u left-parenthesis m plus j right-parenthesis equals 0 comma alpha Subscript n Baseline left-parenthesis m right-parenthesis equals 1 comma alpha 0 left-parenthesis m right-parenthesis not-equals 0 comma m element-of upper I">

P
u
(
m
)
=

∑

j
=
0

n

α
j

(
m
)
u
(
m
+
j
)
=
0
,

α
n

(
m
)
=
1
,

α
0

(
m
)
≠
0
,

m
∈
I

Pu(m) = \sum \nolimits _{j = 0}^n {{\alpha _j}(m)u(m + j) = 0,\;{\alpha _n}(m) = 1,\;{\alpha _0}(m) \ne 0,\;m \in I}

. It is shown that if

0
≤
r
(
m
)
≤
q
(
m
)
,

m
∈
I

0 \leq r(m) \leq q(m),\;m \in I

and if the equations

P
u
(
m
)
=
0

Pu(m) = 0

and

P
u
(
m
)
+
q
(
m
)
u
(
m
)
=
0

Pu(m) + q(m)u(m) = 0

are disconjugate on

I
n

{I^n}

, then the equation

P
u
(
m
)
+
r
(
m
)
u
(
m
)
=
0

Pu(m) + r(m)u(m) = 0

is disconjugate on

I
n

{I^n}

.
Polynomials associated to characters
Authors: Turull Alexandre
Published: July 2010

The present paper concerns certain divisibility properties of the (integer) values of some polynomials naturally defined from characters and modules of finite groups. A generalization of a theorem of K. Brown is obtained in this context.
Multiple nontrivial solutions of semilinear elliptic equations
Authors: Hirano Norimichi
Published: July 2010

We give a condition for a semilinear elliptic equation to have two nontrivial solutions. Our condition does not demand any differentiability of the nonlinear term.
𝑝-harmonic functions in the plane
Authors: Manfredi Juan J.
Published: July 2010

Given

p
>
1

p > 1

, let

u
u

be a solution to

div
⁡
(
∇
u

|

p
−
2

∇
u
)
=
0

\operatorname {div}(\nabla u{|^{p - 2}}\nabla u) = 0

, on a domain

Ω
\Omega

of the plane. Using the theory of quasiregular mappings we prove that the zeros of

∇
u

\nabla u

are isolated in

Ω
\Omega

, obtain bounds for the Hölder exponent of

∇
u

\nabla u

and prove a strong form of the comparison principle.
Martin compactifications of the punctured disk with close to rotation free densities
Authors: Tada Toshimasa
Published: July 2010

The purpose of this paper is to prove that Martin compactifications

Ω
P
∗

\Omega _P^*

and

Ω
Q
∗

\Omega _Q^*

of the punctured unit disk

Ω
:
0
>

|

z

|

>
1

\Omega :0 > |z| > 1

with respect to equations

Δ
u
=
P
u

\Delta u = Pu

and

Δ
u
=
Q
u

\Delta u = Qu

, respectively, are homeomorphic to each other if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper P left-parenthesis z right-parenthesis minus upper Q left-parenthesis z right-parenthesis EndAbsoluteValue equals upper O left-parenthesis StartAbsoluteValue z EndAbsoluteValue Superscript negative 2 Baseline right-parenthesis left-parenthesis z right-arrow 0 right-parenthesis">

|

P
(
z
)
−
Q
(
z
)

|

=
O
(

|

z

|

−
2

)
(
z
→
0
)

|P(z) - Q(z)| = O(|z{|^{ - 2}})(z \to 0)

and

P
(
z
)
=
P
(

|

z

|

)
(
z
∈
Ω
)

P(z) = P(|z|)(z \in \Omega )

.
On the maximal Riesz-transforms along surfaces
Authors: Chen Lung-Kee
Published: July 2010

Let

b
(
t
)

b(t)

be an arbitrary bounded radial function. For

x
=
(

x
1

,

x
2

)
,
t
=
(

t
1

,

t
2

)

x = ({x_1},{x_2}),t = ({t_1},{t_2})

in

R
2

,

|
t
|

=

(

t
1

+

t
2

)

1

/

2

{R^2},\left | t \right | = {({t_1} + {t_2})^{1/2}}

, we establish that the following maximal Riesz-transforms along the surfaces

(

t
1

,

t
2

,

|

t

|

a

)
,
a
>
0

({t_1},{t_2},|t{|^a}),a > 0

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Superscript asterisk Baseline f left-parenthesis x right-parenthesis equals sup Underscript epsilon greater-than 0 Endscripts vertical-bar integral Underscript StartAbsoluteValue t EndAbsoluteValue greater-than epsilon Endscripts f left-parenthesis x 1 minus t 1 comma x 2 minus t 2 comma x 3 minus StartAbsoluteValue t EndAbsoluteValue Superscript a Baseline right-parenthesis b left-parenthesis t right-parenthesis StartFraction t 1 Over StartAbsoluteValue t EndAbsoluteValue cubed EndFraction d t vertical-bar">

T
∗

f
(
x
)
=

sup

ε
>
0

|

∫

|

t

|

>
ε

f
(

x
1

−

t
1

,

x
2

−

t
2

,

x
3

−

|

t

|

a

)
b
(
t
)

t
1

|

t

|

3

d
t

|

{T^*}f(x) = \sup \limits _{\varepsilon > 0} \left | {\int _{|t| > \varepsilon } {f({x_1} - {t_1},{x_2} - {t_2},{x_3} - |t{|^a})b(t)\left . {\frac {{{t_1}}}{{|t{|^3}}}dt} \right |} } \right .

\]
are bounded in

L
p

(

R
3

)

{L^p}({R^3})

for all

1
>
p
>
∞

1 > p > \infty

. The

n
n

-dimensional result can be found at the end of this paper.
Continuous functions with everywhere infinite variation with respect to sequences
Authors: Buczolich Z.
Published: July 2010

We prove that if

{

a
n

}

n
=
1

∞

\{ {a_n}\} _{n = 1}^\infty

is such that

a
n

↘
0

{a_n} \searrow 0

and
\[

lim

n
→
∞

(

a
n

+
1

/

a
n

)
=
1
,

\lim \limits _{n \to \infty } ({a_n}_{ + 1}/{a_n}) = 1,

\]
then for the typical continuous function

f
f

we have
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S Subscript n 0 Baseline colon equals sigma-summation Underscript n equals n 0 Overscript normal infinity Endscripts StartAbsoluteValue f left-parenthesis x Subscript n Baseline Subscript plus 1 Baseline right-parenthesis minus f left-parenthesis x Subscript n Baseline right-parenthesis EndAbsoluteValue equals plus normal infinity">

S

n
0

:=

∑

n
=

n
0

∞

|

f
(

x
n

+
1

)
−
f
(

x
n

)

|

=
+
∞

{S_{{n_0}}}: = \sum \limits _{n = {n_0}}^\infty | f({x_n}_{ + 1}) - f({x_n})| = + \infty

\]
whenever

x
∈
[
0
,
1
−

a

n
0

]

x \in [0,1 - {a_{{n_0}}}]

and

x
n

∈
[
x
+

a

n
+
1

,
x
+

a
n

]

{x_n} \in [x + {a_{n + 1}},x + {a_n}]

. Based on our result in a previous paper, we know that the above theorem fails to hold if

a

n
+
1

/

a
n

=
λ
>
1

{a_{n + 1}}/{a_n} = \lambda > 1

. We also prove that if

{

a
n

}

n
=
1

∞

\{ {a_n}\} _{n = 1}^\infty

is such that

a
n

↘
0

{a_n} \searrow 0

, then for the typical continuous function

f
f

we have

S

n
0

=
+
∞

 if 

x
n

=
x
+

a
n

{S_{{n_0}}} = + \infty {\text { if }}{x_n} = x + {a_n}

and

x
∈
[
0
,
1
−

a

n
0

]

x \in [0,1 - {a_{{n_0}}}]

.
Hyperbolic lengths of geodesics surrounding two punctures
Authors: Hempel Joachim A. , Smith Simon J.
Published: July 2010

For the plane regions

Ω
1

=

{

|
z
|

>
R
,
z
≠
0
,
1

}

{\Omega _1} = \left \{ {\left | z \right | > R,z \ne 0,1} \right \}

with

R
>
1

R > 1

, and

Ω
2

=

C

∖

{

0
,
1
,
p

}

{\Omega _2} = {\mathbf {C}} \setminus \left \{ {0,1,p} \right \}

with

|
p
|

=
R
>
1

\left | p \right | = R > 1

, we describe, as

R
→
∞

R \to \infty

, the hyperbolic lengths of the geodesies surrounding 0 and 1. Upper and lower bounds for the lengths are also stated, and these results are used to obtain inequalities, which are precise in a certain sense, for the length of the geodesic surrounding 0 and 1 in an arbitrary plane region

Ω
\Omega

satisfying

Ω
1

⊂
Ω
⊂

Ω
2

{\Omega _1} \subset \Omega \subset {\Omega _2}

.
Functions whose derivative has positive real part
Authors: London R. R.
Published: July 2010

In this paper we find a sharp upper bound for

|

z

f
′

(
z
)

/

f

(
z
)

|

\left | {z\,f’\left ( z \right ) / f\left ( z \right )} \right |

, where

f
f

is a normalised analytic function with Re

f
′

(
z
)

>
0

f’\left ( z \right ) > 0

in the unit disc.
Coefficient bounds for convex functions of bounded type
Authors: Wirths K.-J.
Published: July 2010

A normalized univalent function is called convex of bounded type if the curvature of the curve bounding the image domain of the unit disc lies between two fixed positive numbers. Sharp bounds for the modulus of the second and the third Taylor coefficient of such functions are derived. The proof concerning the second coefficient is based on a maximum-minimum principle for locally univalent functions.
A note on unique continuation for Schrödinger’s operator
Authors: Kenig Carlos E. , Sogge Christopher D.
Published: July 2010

In this paper we shall prove a unique continuation theorem for Schrödinger’s operator,

i
∂

/

∂
t
−
Δ

i\partial /\partial t - \Delta

. This will be a consequence of "uniform Sobolev inequalities" for operators which are the Schrödinger operator plus lower order terms in

x
x

.
A learning theorem for linear operators
Authors: Mycielski Jan
Published: July 2010

We prove a convergence theorem in linear dynamic approximation theory which yields estimates of certain series of errors.
Totally accretive operators
Authors: deLaubenfels Ralph
Published: July 2010

Let

A
A

be a (possibly unbounded) linear operator on a Banach space. We show that, when

A
A

generates a uniformly bounded strongly continuous semigroup

{

e

−
t
A

}

t
≥
0

{\left \{ {{e^{ - tA}}} \right \}_{t \geq 0}}

, then

A
2

{A^2}

generates a bounded holomorphic semigroup (BHS) of angle

θ
\theta

if and only if

A
A

generates a BHS of angle

θ

/

2
+
π

/

4

\theta / 2 + \pi / 4

. We show that each power of

A
A

generates a uniformly bounded strongly continuous semigroup if and only if

A
A

generates a BHS of angle

π

/

2

\pi / 2

if and only if each power of

A
A

generates a BHS of angle

π

/

2

\pi / 2

. If

A
A

is a linear operator on a Hilbert space, then each power of

A
A

generates a strongly continuous contraction semigroup if and only if

A
A

is positive selfadjoint.
Products of completion regular measures
Authors: Gryllakis Constantinos
Published: July 2010

Let

X
=

∏

i
∈
I

X
i

X = {\prod _{i \in I}}{X_i}

and

Y
=

∏

j
∈
J

Y
j

Y = {\prod _{j \in J}}{Y_j}

, where all

X
i

,

Y
j

{X_i},{Y_j}

are separable metric spaces. Let

μ
\mu

and

ν
\nu

be completion regular Radon probability measures on

X
X

and

Y
Y

respectively. Then

μ
×
ν

\mu \times \nu

on

X
×
Y

X \times Y

is completion regular. This solves a problem of J. R. Choksi and D. H. Fremlin.
Small sets of infinite type are benign for the \overline∂-Neumann problem
Authors: Boas Harold P.
Published: July 2010

An explicit construction shows that the

∂
¯

\bar \partial

-Neumann operator and the Bergman and Szegő projections are globally regular in every smooth bounded pseudoconvex domain whose set of boundary points of infinite type has Hausdorff two-dimensional measure equal to zero. On the other hand there are examples of domains with globally regular

∂
¯

\bar \partial

-Neumann operator but whose infinite-type points fill out an open subset of the boundary.
Best approximation in metric spaces
Authors: Khalil Roshdi
Published: July 2010

A metric space

(

X
,
d

)

\left ( {X,d} \right )

is called an

M
M

-space if for every

x
x

and

y
y

in

X
X

and for every

r
∈

[

0
,
λ

]

r \in \left [ {0,\lambda } \right ]

we have

B

[

x
,
r

]

∩
B

[

y
,
λ
−
r

]

=

{
z
}

B\left [ {x,r} \right ] \cap B\left [ {y,\lambda - r} \right ] = \left \{ z \right \}

for some

z
∈
X

z \in X

, where

λ
=
d

(

x
,
y

)

\lambda = d\left ( {x,y} \right )

. It is the object of this paper to study

M
M

-spaces in terms of proximinality properties of certain sets.
A note on Paley-Wiener-Zygmund stochastic integrals
Authors: Park Chull , Skoug David
Published: July 2010

A Fubini theorem for multiparameter Paley-Wiener-Zygmund stochastic integrals is established which unifies various stochastic integration formulas. We also obtain similar results for double stochastic integrals of Paley-Wiener-Zygmund type.
The influence of the initial distribution on a random walk
Authors: Stadje Wolfgang
Published: July 2010

Let

T
1

,

T
2

,
…

{T_1},{T_2}, \ldots

be i.i.d. random variables,

S
n

=

T
1

+
⋯
+

T
n

{S_n} = {T_1} + \cdots + {T_n}

; let

X
X

and

Y
Y

be independent of

(

T
n

)

n
≥
1

{\left ( {{T_n}} \right )_{n \geq 1}}

. We study the total variation distance between the distributions of

X
+

S
n

X + {S_n}

and

Y
+

S
n

Y + {S_n}

, especially its speed of convergence to 0 in the case that some

S
j

{S_j}

is not singular.
Continuity of excessive harmonic functions for certain diffusions
Authors: Pop-Stojanović Z. R.
Published: July 2010

This paper shows that under certain analytic conditions concerning its potential kernel, a strong Markov process, which is transient and with continuous sample paths, has all of its excessive harmonic functions, which are not identically infinite, continuous, Harnack inequality holds, and a family of its excessive functions is equi-continuous on compacts.
Discrete symmetry, toral symmetry and the Euler characteristic of manifolds
Authors: Papadima Ştefan
Published: July 2010

We consider a closed manifold

M
M

and establish various connections between its Euler characteristic and certain numerical invariants defined so as to measure the abundance of free discrete (respectively rationally free toral) group actions supported by

M
M

.
Can the fundamental (homotopy) group of a space be the rationals?
Authors: Shelah Saharon
Published: July 2010

We prove that for any topological space which is metric, compact (hence separable) path connected and locally path connected, its homotopy group is not the additive group of the rational, moreover if it is not finitely generated then it has the cardinality of the continuum.
Sequential conditions and free products of topological groups
Authors: Morris Sidney A. , Thompson H. B.
Published: July 2010

If

A
A

and

B
B

are nontrivial topological groups, not both discrete, such that their free product

A
∐
B

A\coprod B

is a sequential space, then it is sequential of order

ω
1

{\omega _1}

.
Localizations in universal topological categories
Authors: Cagliari F. , Mantovani S.
Published: July 2010

For some familiar topological categories it is shown that the subcategory of indiscrete spaces is the only nontrivial localization.
Tutte polynomials and link polynomials
Authors: Jaeger François
Published: July 2010

We show how the Tutte polynomial of a plane graph can be evaluated as the "homfly" polynomial of an associated oriented link. Then we discuss some consequences for the partition function of the Potts model, the Four Color Problem and the time complexity of the computation of the homfly polynomial.
Local rigidity of symmetric spaces of nonpositive curvature
Authors: Schroeder Viktor , Strake Martin
Published: July 2010

We prove a rigidity theorem for Riemannian manifolds of nonpositive curvature, whose metric is locally symmetric in a neighborhood of the boundary.
Unconditional bases in countable-ℒ₁ spaces
Authors: Díaz J. C.
Published: July 2010

In this paper, we show that a countably-

L

1

{\mathcal {L}_1}

space which has an unconditional basis is isomorphic to some echelon sequence space of order 1. As a consequence, a countably-

L

1

{\mathcal {L}_1}

space with a basis is nuclear if all its bases are unconditional (this gives a partial answer to a conjecture of Wojtynski). We also study those countably-

L

1

{\mathcal {L}_1}

spaces on which a Fréchet lattice structure can be defined.
A new proof and a generalization of a theorem of de Bruijn
Authors: Aziz Abdul
Published: July 2010

Using a recently developed interpolation formula, we present elementary new and simple proofs of De Bruijn’s theorem and Zygmund’s inequality concerning the integral mean estimates for polynomials. We also present a generalization of De Bruijn’s theorem which leads to a refinement of a theorem of Erdös and Lax.
Measure requirements on distributive lattices for Boolean algebras and topological applications
Authors: Huerta Carlos C.
Published: July 2010

In this note we give a measure requirement for a distributive lattice to be complemented, and therefore a boolean algebra. Applying this result and using measure style arguments, we get various topological results.
A remark on pseudojump operators
Authors: Lachlan A. H. , Yi X.
Published: July 2010

Let

{

W
n

}

n
∈
ω

{\left \{ {{W_n}} \right \}_{n \in \omega }}

be an enumeration of the recursively enumerable sets. In answer to a question of Jockusch and Shore we show that there exist

i
i

and

j
j

such that for all

e
e

either

W
e

{W_e}

is recursive or at least one of

(

W
e

⊕

W
i

W
e

)

({W_e} \oplus W_i^{{W_e}})

and

(

W
e

⊕

W
j

W
e

)

({W_e} \oplus W_j^{{W_e}})

is not recursively enumerable.
A characterization of 𝐻² classes on rank one symmetric spaces of noncompact type
Authors: Cifuentes Patricio
Published: July 2010

A characterization of the Hardy class

H
2

{H^2}

on a rank one symmetric space of noncompact type by a Littlewood-Paley type operator defined through the Green potential of the norm square of the invariant gradient.
A proof of the Fefferman-Stein-Strömberg inequality for the sharp maximal functions
Authors: Fujii Nobuhiko
Published: July 2010

We give another proof of a theorem of Strömberg for the Fefferman-Stein sharp maximal functions. Our method is based on a decomposition lemma which is due to the arguments of Carleson, Garnett and Jones for the functions of BMO, and it is valid for a two-weight setting under a condition which is equivalent to the

A
∞

{A_\infty }

condition in the case of equal weights.
A theorem on polynomial-star approximation
Authors: Davie A. M. , Gamelin T. W.
Published: July 2010

We prove that the unit ball of a Banach space is polynomial-star dense in the unit ball of its bidual. This strengthens Goldstine’s theorem on weak-star density.
Spectral properties for operators in a Lie algebra
Authors: Prado Humberto
Published: July 2010

Given an exponentiable Lie algebra

L

\mathcal {L}

of operators on a Hilbert space

H

\mathcal {H}

, we study the spectrum of those self-adjoint, non-adnilpotent operators

−
i
A

- iA

, with

A
A

in

L

\mathcal {L}

, for a certain class of solvable Lie algebras

L

\mathcal {L}

.
Measurable homomorphisms of locally compact groups
Authors: Kleppner Adam
Published: July 2010

Let

G
G

and

H
H

be locally compact groups and

φ
\varphi

a homomorphism from

G
G

into

H
H

. Suppose that

φ

−
1

(
U
)

{\varphi ^{ - 1}}\left ( U \right )

is measurable for every open set

U
⊂
H

U \subset H

. It is known under some conditions, for example, if

H
H

is

σ
\sigma

-compact, that

φ
\varphi

is continuous. Here it is shown that this result is true without any countability restrictions on

G
G

and

H
H

. The proof depends on the observation that the regular representation of

H
H

is a homomorphism.
A note on the variety of plane curves with nodes and cusps
Authors: Kang Pyung-Lyun
Published: July 2010

Let

P

N

{\mathbb {P}^N}

be the projective space of all degree

d
d

plane curves and

V
(
d
,
δ
,
κ
)

V(d,\delta ,\kappa )

the subvariety of

P

N

{\mathbb {P}^N}

consisting of all reduced and irreducible curves with

δ
\delta

notes and

κ
\kappa

cusps. Then

V
(
d
,
δ
,
κ
)

V(d,\delta ,\kappa )

is irreducible if

κ
≤

1
2

(
d
+
1
)
−
g

\kappa \leq \frac {1}{2}(d + 1) - g

.
On tree-like continua which are homogeneous with respect to confluent light mappings
Authors: Krupski Paweł
Published: July 2010

If

X
X

is a tree-like continuum with property

K
K

which is homogeneous with respect to confluent light mappings, then

X
X

contains no two non-degenerate subcontinua with the one-point intersection. This is a generalization of C. L. Hagopian’s result concerning homogeneous

X
X

.
On harmonic Boehmians
Authors: Mikusiński Piotr
Published: July 2010

Existence of generalized functions (called Boehmians) satisfying the Laplace equation which are not

C
∞

{C^\infty }

-functions is proved.
Some dual statements concerning Wiener measure and Baire category
Authors: Simon K.
Published: July 2010

This paper deals with the duality between Wiener measure and Baire category on

C
0
1

C_0^1

, the set of continuous functions

f
:
[
0
,
1
]
→
[
0
,
1
]

f:[0,1] \to [0,1]

endowed with the supremum norm. We prove that some properties shared by a residual set of continuous functions, e.g. the typical level set structure, and the existence of periodic points of order 3 hold a.e. with respect to the Wiener measure.
On the singular rank of a representation
Authors: Li Jian-Shu
Published: July 2010

Consider the reductive dual pair

(

S

p

2
n

,

O

p
,
q

)

({\text {S}}{{\text {p}}_{2n}},{{\text {O}}_{p,q}})

. We prove that if

π
\pi

is a representation of

S

p

2
n

{\text {S}}{{\text {p}}_{2n}}

coming from duality correspondence with some representation of

O

p
,
q

{{\text {O}}_{p,q}}

then the wave front set of

π
\pi

has rank

≤
p
+
q

\leq p + q

. For

p
+
q
>
n

p + q > n

this implies a result stated (but not proved) by Howe.
A functional equation for the Joukowski transformation
Authors: Baran Mirosław
Published: July 2010

The purpose of this paper is to solve a functional equation which arises from the Joukowski transformation. The functional equation has a simple geometric interpretation which is given.
Factoring Fourier transforms with zeros in a strip
Authors: Dickson D. G.
Published: July 2010

f
f

is the Fourier transform of an infinitely differentiable function of compact support on

R

{\mathbf {R}}

if, and only if,

f
f

is entire and of exponential type with

|

f

(
x
)

|

=
O

(

|

x

|

−
N

)

\left | {f\left ( x \right )} \right | = O\left ( {|x{|^{ - N}}} \right )

for each

N
>
0

N > 0

as

|

x

|

→
∞

|x| \to \infty

for real

x
x

. In some sense, such an

f
f

has its zeros close to the real axis and has positive density of zeros

F
F

with

n

(
r
)

=
D
r
+
o

(
r
)

n\left ( r \right ) = Dr + o\left ( r \right )

. It is shown here that if the zeros of

f
f

are in a strip parallel to the real axis and if

n

(
r
)

=
D
r
+
O

(
1
)

n\left ( r \right ) = Dr + O\left ( 1 \right )

, then

f
f

is the product of two such transforms with zero densities

D

/

2

D/2

and indicators one-half of the indicator of

f
f

. There is a factorable

f
f

in

D

^

(

R

)

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

with zeros on a line and not satisfying the stricter density condition. Analogous results hold for transforms of distributions of compact support on

R

{\mathbf {R}}

. The study was motivated by the outstanding problem of Ehrenpreis that asks if

D

(

R

)

∗

D

(

R

)

=

D

(

R

)

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

.
Finite-dimensional subalgebras in matrix rings over transcendental division algebras
Authors: Lichtman A. I. , Wehrfritz B. A. F.
Published: July 2010

We prove in this article that for some classes of division algebras

D
D

over a field

F
F

every finite dimensional semisimple subalgebra of

D

n
×
n

{D^{n \times n}}

must be conjugate to a subalgebra of

F

n
×
n

{F^{n \times n}}

.
Small solutions of cubic congruences
Authors: Cochrane Todd
Published: July 2010

Let

C
(

x

)

C({\mathbf {x}})

be a cubic form in

n
n

variables over

Z

{\mathbf {Z}}

and

p
p

be a prime. Then for

0
>
σ
>

2
3

0 > \sigma > \frac {2}{3}

the congruence

C
(

x

)
≡
0
(
mod
p
)

C({\mathbf {x}}) \equiv 0(\bmod p)

has a nonzero solution

x
x

with

max

|

x
i

|

≪

p

1

/

3
+
σ

\max \left | {{x_i}} \right | \ll {p^{1/3 + \sigma }}

, provided that

n
>
8

/

σ

n > 8/\sigma

, (where the constant in the

≪
\ll

depends on

n
n

and

σ
\sigma

).
Probability measure functors preserving the ANR-property of metric spaces
Authors: Nhu Nguyen To , Cu Ta Khac
Published: July 2010

Let

P
k

(
X
)

{P_k}\left ( X \right )

denote the set of all probability measures on a metric space

X
X

whose supports consist of no more than

k
k

points, equipped with the Fedorchuk topology. We prove that if

X
∈

ANR

X \in {\text {ANR}}

then

P
k

(
X
)

∈

ANR

{P_k}\left ( X \right ) \in {\text {ANR}}

for every

k
∈

N

k \in {\mathbf {N}}

. This implies that for each

k
∈

N

k \in {\mathbf {N}}

the functor

P
k

{P_k}

preserves the topology of separable Hilbert space.
Affine conormal of convex hypersurfaces
Authors: Yau Chi-Ming
Published: July 2010

The geometry of convex hypersurfaces in real affine space is analyzed using the affine conormal. A weak version of Chern’s conjecture, characterizing paraboloids among convex graphs, is proved. In addition, it is shown that a closed convex affine hypersurface with constant affine total curvature is an ellipsoid.
Automorphic-differential identities in rings
Authors: Bergen Jeffrey
Published: July 2010

Let

R
R

be a ring and

f
f

an endomorphism obtained from sums and compositions of left multiplications, right multiplications, automorphisms, and derivations. We prove several results relating the behavior of

f
f

on certain subsets of

R
R

to its behavior on all of

R
R

. For example, we prove (1) if

R
R

is prime with ideal

I
≠
0

I \ne 0

such that

f
(
I
)
=
0

f(I) = 0

, then

f
(
R
)
=
0

f(R) = 0

, (2) if

R
R

is a domain with right ideal

λ
≠
0

\lambda \ne 0

such that

f
(
λ
)
=
0

f(\lambda ) = 0

, then

f
(
R
)
=
0

f(R) = 0

, and (3) if

R
R

is prime and

f
(

λ
n

)
=
0

f({\lambda ^n}) = 0

, for

λ
\lambda

a right ideal and

n
≥
1

n \geq 1

, then

f
(
λ
)
=
0

f(\lambda ) = 0

. We also prove some generalizations of these results for semiprime rings and rings with no non-zero nilpotent elements.
“Place functions”: alias continuous functions on the Stone space
Authors: Fleischer I.
Published: July 2010

Caratheodory’s system of place functions on a Boolean

σ
\sigma

-ring is shown to be canonically isomorphic—order and algebraically—to the system of continuous extended real-valued functions over its Stone space. This enables one to dispense with the imperspicuous formulae and calculations of [CL] in favor of the more straightforward and familiar ones of function spaces; and to subsume the proposed generalization of integration under standard treatment.
Automorphism groups of locally compact reductive groups
Authors: Wu T. S.
Published: July 2010

A topological group

G
G

is reductive if every continuous finite dimensional

G
G

-module is semi-simple. We study the structure of those locally compact reductive groups which are the extension of their identity components by compact groups. We then study the automorphism groups of such groups in connection with the groups of inner automorphisms. Proposition. Let

G
G

be a locally compact reductive group such that

G

/

G
0

G/{G_0}

is compact. Then

I

(

G
0

)

I\left ( {{G_0}} \right )

is dense in

A
0

(
G
)

{A_0}\left ( G \right )

.