bluuebunny/crossref_metadata_2025 huggingface dataset:
Characterization of Schwartz spaces by their holomorphic duals
Authors: Bjon Sten , Lindström Mikael
Published: June 2010

Let

U
U

be an open subset of a locally convex space

E
E

, and let

H
c

(
U
,
F
)

{H_c}(U,F)

denote the vector space of holomorphic functions into a locally convex space

F
F

, endowed with continuous convergence. It is shown that if

F
F

is a semi-Montel space, then the bounded subsets of

H
c

(
U
,
F
)

{H_c}(U,F)

are relatively compact. Further it is shown that

E
E

is a Schwartz space iff the continuous convergence structure on the algebra

H
(
U
)

H(U)

of scalar-valued holomorphic functions on

U
U

, coincides with local uniform convergence. Using this, an example of a nuclear Fréchet space

E
E

is given, such that the locally convex topology associated with continuous convergence on

H
(
E
)

H(E)

is strictly finer than the compact open topology. Thus, the behavior of the space

H
c

(
E
)

{H_c}(E)

differs in this respect from that of its subspace

L
c

E

{L_c}E

of linear forms and that of its superspace

C
c

(
E
)

{C_c}(E)

of continuous functions.
The Bloch space and BMO analytic functions in the tube over the spherical cone
Authors: Békollé David
Published: June 2010

We prove that the Bloch space coincides with the space BMOA in the tube over the spherical cone of

R

3

{{\mathbf {R}}^3}

; this extends a well-known one-dimensional result.
Systems of functional-differential equations with asymptotically constant solutions
Authors: Trench William F. , Kusano Takaŝi
Published: June 2010

Sufficient conditions are given for a nonlinear system of differential equations with deviating arguments to have solutions which approach finite limits as

t
→
∞

t \to \infty

. No specific assumptions other than continuity are imposed on the deviating arguments. The nonlinearities may be superlinear, sublinear, or singular in form, or a mixture of these. Some of the results are global.
Four topologically equivalent measures in the Cantor space
Authors: Navarro-Bermúdez Francisco J. , Oxtoby John C.
Published: June 2010

We show that one of the binomial numbers discovered by K. J. Huang provides an example of topologically equivalent measures in

2

N

{2^{\mathbf {N}}}

that are not trivially homeomorphic.
Comparison theorems for eigenvalue problems for 𝑛th order differential equations
Authors: Hankerson Darrel , Peterson Allan
Published: June 2010

We give a comparison theorem for eigenvalues for a

(
k
,
n
−
k
)

(k,n - k)

-conjugate boundary value problem for the systems

(
−
1

)

n
−
k

L
y
=
λ
P
(
t
)
y

{( - 1)^{n - k}}Ly = \lambda P(t)y

and

(
−
1

)

n
−
k

L
z
=
Λ
Q
(
t
)
z

{( - 1)^{n - k}}Lz = \Lambda Q(t)z

, where

P
(
t
)

P(t)

and

Q
(
t
)

Q(t)

are continuous

m
×
m

m \times m

matrix functions. We assume that the corresponding scalar equation

L
x
=
0

Lx = 0

is

(
j
,
n
−
j
)

(j,n - j)

-disconjugate for

k
−
1
≤
j
≤
n
−
1

k - 1 \leq j \leq n - 1

. A special case of this is when

L
x
=
0

Lx = 0

is disconjugate; our results are new even in this case.
Bounded harmonic functions on coverings
Authors: Toledo Domingo
Published: June 2010

We prove that the space of bounded harmonic functions on a Galois cover of a compact Riemannian manifold is either one dimensional or infinite dimensional.
Existence of periodic solutions of second order differential equations with delay
Authors: Metzen Gerhard
Published: June 2010

We prove the existence of

2
π

2\pi

-periodic solutions of second order ordinary differential equations with delay. In particular we study the effect the delay has with respect to the solvability of such problems. Resonance and nonresonance problems are considered.
Estimates for Green’s functions
Authors: Ramm A. G. , Li Lige
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l Subscript q Baseline equals minus nabla squared plus q left-parenthesis x right-parenthesis comma x element-of upper R cubed comma 0 less-than-or-equal-to q less-than-or-equal-to c left-parenthesis 1 plus StartAbsoluteValue x EndAbsoluteValue right-parenthesis Superscript negative a Baseline comma a greater-than 2 comma l Subscript q Baseline upper G Subscript q Baseline left-parenthesis x comma y right-parenthesis equals delta left-parenthesis x minus y right-parenthesis">

l
q

=
−

∇
2

+
q
(
x
)
,
x
∈

R
3

,
0
≤
q
≤
c

(
1
+

|
x
|

)

−
a

,
a

 > 

2
,

l
q

G
q

(
x
,
y
)
=
δ
(
x
−
y
)

{l_q} = - {\nabla ^2} + q(x),x \in {R^3},0 \leq q \leq c{(1 + \left | x \right |)^{ - a}},a{\text { > }}2,{l_q}{G_q}(x,y) = \delta (x - y)

. If

q
≥
p
≥
0
,
q
≢
p

q \geq p \geq 0,q \not \equiv p

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row c vertical-bar x minus y vertical-bar Superscript negative 1 Baseline greater-than upper G Subscript q Baseline left-parenthesis x comma y right-parenthesis greater-than upper G Subscript p Baseline left-parenthesis x comma y right-parenthesis less-than-or-equal-to left-parenthesis 4 pi StartAbsoluteValue x minus y EndAbsoluteValue right-parenthesis Superscript negative 1 Baseline comma x not-equals y EndLayout">

c

|

x
−
y

|

−
1

 > 

G

q

(
x
,
y
)

 > 

G
p

(
x
,
y
)
≤

(
4
π

|

x
−
y

|

)

−
1

,
x
≠
y

\begin {gathered} c\left | {x - y\left | {^{ - 1}} \right .{\text { > }}{{\text {G}}_q}(x,y){\text { > }}} \right .{G_p}(x,y) \leq {(4\pi \left | {x - y} \right |)^{ - 1}},x \ne y \end {gathered}

, for some positive

c
=
c
(
q
)

c = c(q)

. If

p
≢
0

p \not \equiv 0

then

G
p

 > 

(
4
π

|

x
−
y

|

)

−
1

,
x
≠
y

{G_p}{\text { > }}{(4\pi \left | {x - y} \right |)^{ - 1}},x \ne y

.
Best uniform approximation by bounded analytic functions
Authors: Papadimitrakis M.
Published: June 2010

This paper gives a counterexample to the conjecture that the continuity of the conjugate

f
~

\tilde f

of an

f
∈
C

(
T
)

f \in C\left ( T \right )

implies the continuity of the best uniform approximation

g
∈

H
∞

(
T
)

g \in {H^\infty }\left ( T \right )

of

f
f

. It also states two conditions which imply the continuity of

g
g

.
Subnormal elements of 𝐶*-algebras
Authors: Williams L. R.
Published: June 2010

An algebraic characterization of a subnormal operator on Hilbert space is given. The result also characterizes subnormal elements of certain abstract

C
∗

{C^ * }

-algebras
Locally finite families, completely separated sets and remote points
Authors: Henriksen M. , Peters T. J.
Published: June 2010

It is shown that if

X
X

is a nonpseudocompact space with a

σ
\sigma

-locally finite

π
\pi

-base, then

X
X

has remote points. Within the class of spaces possessing a

σ
\sigma

-locally finite

π
\pi

-base, this result extends the work of Chae and Smith, because their work utilized normality to achieve complete separation. It provides spaces which have remote points, where the spaces do not satisfy the conditions required in the previous works by Dow, by van Douwen, by van Mill, or by Peters. The lemma: "Let

X
X

be a space and let

{

C
ξ

:
ξ
>
α
}

\{ {C_\xi }:\xi > \alpha \}

be a locally finite family of cozero sets of

X
X

. Let

{

Z
ξ

:
ξ
>
α
}

\{ {Z_\xi }:\xi > \alpha \}

be a family of zero sets of

X
X

such that for each

ξ
>
α
,

Z
ξ

⊂

C
ξ

\xi > \alpha ,{Z_\xi } \subset {C_\xi }

. Then

∪

ξ
>
α

Z
ξ

{ \cup _{\xi > \alpha }}{Z_\xi }

is completely separated from

X

/

∪

ξ
>
α

C
ξ

X/{ \cup _{\xi > \alpha }}{C_\xi }

", is a fundamental tool in this work. An example is given which demonstrates the value of this tool. The example also refutes an appealing conjecture—a conjecture for which the authors found that there existed significant confusion within the topological community as to its truth or falsity.
Nonhomogeneity of certain Baire sets in 𝛽𝑋
Authors: García-Ferreira Salvador
Published: June 2010

We prove that every nonempty Baire set of

β
X

\beta X

contained in

β
X
−
X

\beta X - X

is not homogeneous.
Nontrivial uses of trivial rings
Authors: Richman Fred
Published: June 2010

Four theorems about commutative rings are proved with the aid of the notion of a trivial ring.
Modular group algebras of 𝑁-groups
Authors: Ullery William
Published: June 2010

For a prime

p
p

, let

F
F

be the field with

p
p

elements and let

G
G

be a reduced

p
p

-primary abelian

N
N

-group. If

F
G
≅
F
H

FG \cong FH

for some group

H
H

, then it is shown that

G
≅
H

G \cong H

.
A projection formula for the Askey-Wilson polynomials and an application
Authors: Rahman Mizan
Published: June 2010

A projection formula for

p
n

(
x
;
a
,
b
,
c
,
d

|

q
)

{p_n}(x;a,b,c,d|q)

, the Askey-Wilson polynomials, is obtained by using a generalization of Askey and Wilson’s

q
q

-beta integral. The result is used to find a

q
q

-analogue of the Feldheim-Vilenkin formula for ultraspherical polynomials. A

q
q

-analogue of the ultraspherical polynomials, other than the one due to Rogers, is also introduced.
Compactness of the \overline∂-Neumann operator
Authors: Krantz Steven G.
Published: June 2010

This paper gives an elementary construction of smooth, bounded pseudoconvex domains which satisfy Condition

R
R

but which do not have compact

∂
¯

\bar \partial

-Neumann operator.
The differentiability of the hairs of 𝑒𝑥𝑝(𝑍)
Authors: da Silva M. Viana
Published: June 2010

We prove that the hairs of the exponential-like maps

f
(
z
)
=
λ

e
z

f(z) = \lambda {e^z}

are smooth curves. This answers affirmatively a question of Devaney and Krych. The proof is constructive in the sense that a dynamically defined

C
∞

{C^\infty }

parametrization is presented.
Neighborhoods of points in codimension-one submanifolds lie in codimension-one spheres
Authors: Ancel Fredric D.
Published: June 2010

For

n
≥
4

n \geq 4

, let

M
M

be an

(
n
−
1
)

(n - 1)

-manifold embedded in an

n
n

-manifold

N
N

. For each point

p
p

of

M
M

, there is an

(
n
−
1
)

(n - 1)

-sphere

Σ
\Sigma

in

N
N

such that

Σ
∩
M

\Sigma \cap M

is a neighborhood of

p
p

in

M
M

.
Non-𝐺-equivalent Moore 𝐺-spaces of the same type
Authors: Doman Ryszard
Published: June 2010

Let

G
G

be a finite group. By a Moore

G
G

-space we mean a

G
G

-space

X
X

such that for each subgroup

H
H

of

G
G

the fixed point space

X
H

{X^H}

is a Moore space of type

(

M
H

,
n
)

({M_H},n)

, where

n
>
1

n > 1

is a fixed integer and

M
H

{M_H}

are abelian groups. In this paper it is shown that there exist infinitely many non-

G
G

-homotopy equivalent Moore

G
G

-spaces of certain given type.
Generalized open mapping theorems for bilinear maps, with an application to operator algebras
Authors: Dixon P. G.
Published: June 2010

Cohen [4] gave an example of a surjective bilinear mapping between Banach spaces which was not open, and Horowitz [8] gave a much simpler example. We build on Horowitz’ example to produce a similar result for bilinear mappings such that every element of the target space is a linear combination of

n
n

elements of the range. An immediate application is that Bercovici’s construction [1] of an operator algebra with property

(

A

1

)

({\mathbb {A}_1})

but not

(

A

1

(
r
)
)

({\mathbb {A}_1}(r))

can be extended to achieve property

(

A

1

/

n

)

({\mathbb {A}_{1/n}})

without

(

A

1

/

n

(
r
)
)

({\mathbb {A}_{1/n}}(r))

.
A factorization theorem for unfoldings of analytic functions
Authors: Suwa Tatsuo
Published: June 2010

Let

f
~

\tilde f

and

g
g

be holomorphic function germs at 0 in

C

n

×

C

n

=

{

(

x
,
s

)

}

{{\mathbf {C}}^n} \times {{\mathbf {C}}^n} = \left \{ {\left ( {x,s} \right )} \right \}

. If

d
x

g
Λ

d
x

f
~

=
0

{d_x}g\Lambda {d_x}\tilde f = 0

and if

f

(
x
)

=

f
~

(

x
,
0

)

f\left ( x \right ) = \tilde f\left ( {x,0} \right )

is not a power or a unit, then there exists a germ

λ
\lambda

at 0 in

C

n

×

C

n

{{\mathbf {C}}^n} \times {{\mathbf {C}}^n}

such that

g

(

x
,
s

)

=
λ

(

f
~

(

x
,
s

)

,
s

)

g\left ( {x,s} \right ) = \lambda \left ( {\tilde f\left ( {x,s} \right ),s} \right )

. The result has the implication that the notion of an RL-morphism in the unfolding theory of foliation germs generalizes that of a right-left morphism in the function germ case.
Complex retractions and envelopes of holomorphy
Authors: Trapani S.
Published: June 2010

In this paper we show that if a domain

Ω
\Omega

of a Stein manifold

X
X

is a "holomorphic deformation retract" of a domain of holomorphy

D
⊆
X

D \subseteq X

, then

Ω
\Omega

has a univalent envelope of holomorphy

Ω
∗

⊆
X

{\Omega ^*} \subseteq X

.
Mazur’s intersection property and a Kreĭn-Mil′man type theorem for almost all closed, convex and bounded subsets of a Banach space
Authors: Georgiev Pando Grigorov
Published: June 2010

Let

V

\mathcal {V}

(resp.

V

∗

{\mathcal {V}^*}

) be the set of all closed, convex and bounded (resp.

w
∗

{w^*}

-compact and convex) subsets of a Banach space

E
E

(resp. of its dual

E
∗

{E^*}

) furnished with the Hausdorff metric. It is shown that if there exists an equivalent norm

|

|

⋅

|

|

|| \cdot ||

in

E
E

with dual

|

|

⋅

|

|

∗

|| \cdot |{|^*}

such that

(

E
,

|

|

⋅

|

|

)

\left ( {E,|| \cdot ||} \right )

has Mazur’s intersection property and

(

E
∗

,

|

|

⋅

|

|

∗

)

\left ( {{E^*},||\cdot |{|^*}} \right )

has

w
∗

{w^*}

-Mazur’s intersection property, then (1) there exists a dense

G
δ

{G_\delta }

subset

V

0

{\mathcal {V}_0}

of

V

\mathcal {V}

such that for every

X
∈

V

0

X \in {\mathcal {V}_0}

the strongly exposing functionals form a dense

G
δ

{G_\delta }

subset of

E
∗

{E^*}

; (2) there exists a dense

G
δ

{G_\delta }

subset

V

0
∗

\mathcal {V}_0^*

of

V

∗

{\mathcal {V}^*}

such that for every

X
∗

∈

V

0
∗

{X^*} \in \mathcal {V}_0^*

the

w
∗

{w^*}

-strongly exposing functionals form a dense

G
δ

{G_\delta }

subset of

E
E

. In particular every

X
∈

V

0

X \in {\mathcal {V}_0}

is the closed convex hull of its strongly exposed points and every

X
∗

∈

V

0
∗

{X^*} \in \mathcal {V}_0^*

is the

w
∗

{w^*}

-closed convex hull of its

w
∗

{w^*}

-strongly exposed points.
Radial symmetry of the first eigenfunction for the 𝑝-Laplacian in the ball
Authors: Bhattacharya Tilak
Published: June 2010

We prove the radial symmetry of the eigenfunction corresponding to the first eigenvalue of the equation:

div
⁡

(

|

∇
u

|

p
−
2

∇
u

)

+
λ

|

u

|

p
−
2

u
=
0

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

, when

Ω
\Omega

is a ball in

R
n

{R^n}

and

1
>
p
>
∞

1 > p > \infty

.
A de Rham type theorem for orbit spaces
Authors: Verona Andrei
Published: June 2010

Let

G
G

be a compact Lie group and

M
M

be a smooth

G
G

-space. We prove that the real cohomology algebra of the orbit space

M

/

G

M/G

is isomorphic to the homology algebra of the de Rham complex of

G
G

-basic differential forms on

M
M

.
Two new topological cardinal inequalities
Authors: Sun Shu Hao
Published: June 2010

In this paper, two new cardinal inequalities are obtained. The first one is a common generalization of inequalities of Hajnal-Juhasz and Sapirovskii; the second one generalizes the inequality of Arhangelskiĭ.
Some explicit real analytic trivializations of the Teichmüller curves
Authors: Earle Clifford J.
Published: June 2010

The barycentric extension of homeomorphisms from the unit circle to the closed unit disk produces real analytic trivializations of the Teichmüller curves.
Almost fixed point theorems
Authors: Idzik Adam
Published: June 2010

A closed cover version of Hazewinkel and van de Vel’s theorem on the existence of

γ
\gamma

-almost fixed points for set-valued functions and for real Hausdorff topological vector spaces is proved. Generalizations of well-known fixed point theorems of the Schauder-Tychonoff type are presented.
Analyticity of homology classes
Authors: Tognoli Alberto
Published: June 2010

Let

W
W

be a real analytic manifold and

{
α
}
∈

H
p

(
W
,

Z
2

)

\{ \alpha \} \in {H_p}(W,{Z_2})

. We shall say that

{
α
}

\{ \alpha \}

is analytic if there exists a compact analytic subset

S
S

of

W
W

, such that:

{
α
}
=

\{ \alpha \}= {}

fundamental class of

S
}

S\}

. The purpose of this short paper is to prove Theorem 1. Let

W
W

be a paracompact real analytic manifold; then any homology class

{
α
}
∈

H
p

(
W
,

Z
2

)

\{ \alpha \} \in {H_p}(W,{Z_2})

is analytic. We remember that a similar result does not hold in the real algebraic case (see [1]).
The existence of 𝐾(𝜋,1)4-manifolds which are rational homology 4-spheres
Authors: Luo Feng
Published: June 2010

We construct rational homology

4
4

-sphere manifolds with contractible universal cover.
On 𝑞-derangement numbers
Authors: Wachs Michelle L.
Published: June 2010

We derive a

q
q

-analogue of the classical formula for the number of derangements of an

n
n

element set. Our derivation is entirely analogous to the classical derivation, but relies on a descent set preserving bijection between the set of permutations with a given derangement part and the set of shuffles of two permutations.
On the almost everywhere existence of the ergodic Hilbert transform
Authors: Gallardo Diego , Martín-Reyes F. J.
Published: June 2010

Let

(
X
,

M

,
μ
)

(X,\mathfrak {M},\mu )

be a finite measure space,

T
T

an invertible measure-preserving transformation and

υ
\upsilon

a positive measurable function. For

p
=
1

p = 1

, we prove that the ergodic Hubert transform
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H f left-parenthesis upper X right-parenthesis equals li m Subscript n right-arrow normal infinity Baseline sigma-summation Underscript i equals negative n Overscript n Endscripts prime f left-parenthesis upper T Superscript i Baseline x right-parenthesis slash i">

H
f
(
X
)
=

li

m

n
→
∞

∑

i
=
−
n

n

′

f
(

T
i

x
)

/

i

Hf(X) = {\text {li}}{{\text {m}}_{n \to \infty }}\sum \nolimits _{i = - n}^n {’f({T^i}x)/i}

exists a.e. for every

f
f

in

L
1

(
υ
d
μ
)

{L^1}(\upsilon d\mu )

if and only if

in

f

i
≥
0

υ
(

T
i

x
)
>
0

{\text {in}}{{\text {f}}_{i \geq 0}}\upsilon ({T^i}x) > 0

a.e. We also solve the problem for

1
>
p
≤
2

1 > p \leq 2

. In this case the condition is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="su p Subscript k greater-than-or-equal-to 1 Baseline k Superscript negative 1 Baseline sigma-summation Underscript i minus 0 Overscript k minus 1 Endscripts upsilon Superscript negative 1 slash left-parenthesis p minus 1 right-parenthesis Baseline left-parenthesis upper T Superscript i Baseline x right-parenthesis greater-than normal infinity">

su

p

k
≥
1

k

−
1

∑

i
−
0

k
−
1

υ

−
1

/

(
p
−
1
)

(

T
i

x
)
>
∞

{\text {su}}{{\text {p}}_{k \geq 1}}{k^{ - 1}}\sum \nolimits _{i - 0}^{k - 1} {{\upsilon ^{ - 1/(p - 1)}}} ({T^i}x) > \infty

a.e. If the transformation

T
T

is ergodic, the characterizing conditions become that

1

/

υ
∈

L
∞

1/\upsilon \in {L^\infty }

and

υ

−
1

/

(
p
−
1
)

∈

L
1

(
μ
)

{\upsilon ^{ - 1/(p - 1)}} \in {L^1}(\mu )

, respectively. These characterizations, together with some recent results, give, for

1
≤
p
≤
2

1 \leq p \leq 2

, that

H
f
(
x
)

Hf(x)

exists a.e. for every

f
f

in

L
p

(
υ
d
μ
)

{L^p}(\upsilon d\mu )

if and only if the sequence of the Césàro-averages
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Superscript negative 1 Baseline left-parenthesis f left-parenthesis x right-parenthesis plus f left-parenthesis upper T x right-parenthesis plus ellipsis f left-parenthesis upper T Superscript k minus 1 Baseline x right-parenthesis right-parenthesis">

k

−
1

(
f
(
x
)
+
f
(
T
x
)
+
…
f
(

T

k
−
1

x
)
)

{k^{ - 1}}(f(x) + f(Tx) + \ldots f({T^{k - 1}}x))

converge a.e. for every

f
f

in

L
p

(
υ
d
μ
)

{L^p}(\upsilon d\mu )

. This equivalence has recently been obtained by Jajte for a unitary operator, not necessarily positive, acting on

L
2

{L^2}

.
Remark on Witten’s modular forms
Authors: Brylinski Jean-Luc
Published: June 2010

We give a simple proof of the modular invariance of a power series which Witten [4] attaches to an even-dimensional closed manifold whose first Pontryagin class is torsion. The proof uses only the functional equation satisfied by classical theta functions.
On the submetacompactness of products
Authors: Yajima Yukinobu
Published: June 2010

In this paper, we raise the question of whether the product

X
×
Y

X \times Y

of a regular space

X
X

with a

σ
\sigma

-closure-preserving cover by compact sets and a submetacompact space

Y
Y

is submetacompact, and we show that the product

X
×
Y

X \times Y

is weakly submetacompact.
A characterization of complex hypersurfaces in 𝐶^{𝑚}
Authors: Dajczer Marcos
Published: June 2010

We show that an isometric immersion of a connected Kaehler manifold

M

2
n

{M^{2n}}

into the euclidean space with (real) codimension two is holomorphic with respect to some complex structure of

R

2
n
+
2

{{\mathbf {R}}^{2n + 2}}

provided that the index of nullity

μ
\mu

of the curvature tensor satisfies

μ
>
2
n
−
4

\mu > 2n - 4

everywhere.
On the regularity properties for solutions of the Cauchy problem for the porous media equation
Authors: Hayasida Kazuya
Published: June 2010

We consider the Cauchy problem for the equation

∂
t

u
=
Δ

u
m

{\partial _t}u = \Delta {u^m}

in

R
N

×
(
0
,
T
)

{R^N} \times (0,T)

. We assume that

1
>
m
>
3
N

/

(
3
N
−
2
)

1 > m > 3N/(3N - 2)

and the initial data

u
0

{u_0}

is in

C
0
1

(

R
N

)

C_0^1({R^N})

and

u
0

≥
0

{u_0} \geq 0

in

R
N

{R^N}

. Then we prove that the second derivatives of

u
m

{u^m}

with respect to the space-variable are in

L
2

(

R
N

×
(
0
,
T
)
)

{L^2}({R^N} \times (0,T))

.
Nonnegative solutions for a class of radially symmetric nonpositone problems
Authors: Castro Alfonso , Shivaji R.
Published: June 2010

We consider the existence of radially symmetric non-negative solutions for the boundary value problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row minus normal upper Delta u left-parenthesis x right-parenthesis equals lamda f left-parenthesis u left-parenthesis x right-parenthesis right-parenthesis double-vertical-bar x double-vertical-bar less-than-or-equal-to 1 comma x element-of upper R Superscript upper N Baseline left-parenthesis upper N greater-than-or-equal-to 2 right-parenthesis 2nd Row u left-parenthesis x right-parenthesis equals 0 double-vertical-bar x double-vertical-bar equals 1 EndLayout">

−
Δ
u
(
x
)
=
λ
f
(
u
(
x
)
)

‖
x
‖

≤
1
,
x
∈

R
N

(
N
≥
2
)

u
(
x
)
=
0

‖
x
‖

=
1

\begin {array}{*{20}{c}} { - \Delta u(x) = \lambda f(u(x))\quad \left \| x \right \| \leq 1,x \in {R^N}(N \geq 2)} \\ {u(x) = 0\quad \left \| x \right \| = 1} \\ \end {array}

\]
where

λ
>
0
,
f
(
0
)
>
0

\lambda > 0,f(0) > 0

(non-positone),

f
′

≥
0

f’ \geq 0

and

f
f

is superlinear. We establish existence of non-negative solutions for

λ
\lambda

small which extends some work of our previous paper on non-positone problems, where we considered the case

N
=
1

N = 1

. Our work also proves a recent conjecture by Joel Smoller and Arthur Wasserman.
Continuity properties of optimal stopping value
Authors: Elton John
Published: June 2010

The optimal stopping value of a sequence (finite or infinite) of integrable random variables is lower semicontinuous for the topology of convergence in distribution, when restricted to a collection with uniformly integrable negative parts. It is continuous for finite sequences which are adapted by a continuous invertible "triangular" function to independent sequences, such as partial averages; this is our main result. The proof depends on conditional weak convergence, uniform on compact sets, for such processes. A topological result on the inverses of triangular functions on iteratively connected domains may be of independent interest (§3).
Existence of multiple periodic solutions for a semilinear evolution equation
Authors: Hirano Norimichi
Published: June 2010

In this paper, we consider the existence of multiple periodic solutions for the problem
\[

d
u

d
t

+
L
u
=
g
(
u
)
+
h
,
t
>
0
,
u
(
0
)
=
u
(
T
)
,

\frac {{du}}{{dt}} + Lu = g(u) + h,t > 0,u(0) = u(T),

\]
where

L
L

is a uniformly strongly elliptic operator with domain

D
(
L
)
=

H
0
m

(
Ω
)
,
g
:
R
→
R

D(L) = H_0^m(\Omega ),g:R \to R

is a continuous mapping,

T
>
0

T > 0

and

h
:
(
0
,
T
)
→

H
0
m

(
Ω
)

h:(0,T) \to H_0^m(\Omega )

is a measurable function.
Lack of uniform stabilization for noncontractive semigroups under compact perturbation
Authors: Triggiani R.
Published: June 2010

Let

G
(
t
)
,
t
≥
0

G(t),t \geq 0

, be a strongly continuous semigroup on a Hilbert space

X
X

(or, more generally, on a reflexive Banach space with the approximating property), with infinitesimal generator

A
A

. Let: (i) either

G
(
t
)

G(t)

or

G
∗

(
t
)

{G^ * }(t)

be strongly stable, yet (ii) not uniformly stable as

t
→
+
∞

t \to + \infty

. Then, for any compact operator

B
B

on

X
X

, the semigroup

S
B

(
t
)

{S_B}(t)

generated by

A
+
B

A + B

cannot be uniformly stable as

t
→
+
∞

t \to + \infty

. This result is ’optimal’ within the class of compact perturbations

B
B

. It improves upon a prior result in [G.1] by removing the assumption that

G
(
t
)

G(t)

be a contraction for positive times. Moreover, it complements a result in [R.1] where

G
(
t
)

G(t)

was assumed to be a group, contractive for negative times. Our proof is different from both [R.1 and G.1]. Application include physically significant dynamical systems of hyperbolic type in feedback form, where the results of either [R.1 or G.1] are not applicable, as the free dyamics is not a contraction.
Composition operators induced by functions with supremum strictly smaller than 1
Authors: Zorboska Nina
Published: June 2010

We give some partial solutions to the following problem. Does a function analytic in the unit disc

D
D

with supremum strictly smaller than 1 , induce a bounded composition operator on all weighted Hardy spaces

H
2

(
β
)

{H^2}(\beta )

.
Note on arc-length and harmonic measure
Authors: Spraker John S.
Published: June 2010

In this note it is shown that the only Smirnov domain for which arc-length measure is harmonic measure is the disk. The proof depends on some facts about inner and outer functions and arc-length preserving maps.
A cusp closing theorem
Authors: Schroeder Viktor
Published: June 2010

Using a modification of a cusp closing result of Thurston, we construct compact Riemannian manifolds of nonpositive sectional curvature which have rank one (in the sense of Brin, Ballmann, and Eberlein) but which contain embedded flat tori of codimension 2. The metric can even be made analytic.
Scattered spaces and the class 𝑀𝑂𝐵𝐼
Authors: Bennett H. R. , Chaber J.
Published: June 2010

We show that every regular scattered space with a point-countable base is an open and compact image of a scattered metacompact Moore space and, consequently, is an element of the minimal class of regular spaces containing all the scattered metric spaces and invariant under open and compact mappings. This gives a characterization of a subclass of the class MOBI.
On the coherence and weak dimension of the rings 𝑅⟨𝑥⟩ and 𝑅(𝑥)
Authors: Glaz Sarah
Published: June 2010

Let

R
R

be a commutative ring. We first derive necessary and sufficient conditions for the rings

R

⟨
x
⟩

R\left \langle x \right \rangle

and

R
(
x
)

R(x)

to be coherent. Next, for stably coherent rings of finite weak dimension exact relations are found between the weak dimension of

R
R

and that of

R

⟨
x
⟩

R\left \langle x \right \rangle

and

R
(
x
)

R(x)

. These relations are used to determine necessary and sufficient conditions for

R

⟨
x
⟩

R\left \langle x \right \rangle

and

R
(
x
)

R(x)

to be Von Neumann regular or semihereditary.
On a theorem of Feit and Tits
Authors: Kleidman Peter B. , Liebeck Martin W.
Published: June 2010

Feit and Tits [3] lay the groundwork for determining the smallest degree of a projective representation of a finite extension of a finite simple group

G
G

. Provided

G
G

is not of Lie type in characteristic 2, they determine precisely when this degree is smaller than the degree of a projective representation of

G
G

itself. We complete this project by extending their results to the groups of Lie type in characteristic 2.
The conjugacy problem for finite graph products
Authors: Horadam K. J.
Published: June 2010

A finite graph product is the fundamental group of a finite graph of groups. Finite graph products with finite cyclic edge groups are shown to inherit a solvable conjugacy problem from their vertex groups under certain conditions, of which the most important is that all the edge group generators in each vertex group are powers of a common central element.
Simple near-rings associated with meromorphic products
Authors: Maxson C. J. , Smith K. C.
Published: June 2010

Let

H
H

be a subgroup of

G
2

{G^2}

and let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 0 left-parenthesis upper G comma 2 comma upper H right-parenthesis equals StartSet f element-of upper M 0 left-parenthesis upper G right-parenthesis vertical-bar f left-parenthesis upper H right-parenthesis subset-of-or-equal-to upper H EndSet period">

M
0

(

G
,
2
,
H

)

=

{

f
∈

M
0

(
G
)

|

f
(
H
)
⊆
H

}

.

{M_0}\left ( {G,2,H} \right ) = \left \{ {f \in {M_0}\left ( G \right )|f(H) \subseteq H} \right \}.

\]
In this paper we characterize in terms of properties of

H
H

when

M
0

(

G
,
2
,
H

)

{M_0}\left ( {G,2,H} \right )

is a simple near-ring.
On strong uniqueness in one-sided 𝐿¹-approximation of differentiable functions
Authors: Kroó András , Sommer Manfred , Strauss Hans
Published: June 2010

We consider the problem of one-sided

L
1

{L^1}

-approximation in

C
1

[

0
,
1

]

{C^1}\left [ {0,1} \right ]

. A sharp estimate for the rate of strong uniqueness for arbitrary unicity subspaces of

C
1

[

0
,
1

]

{C^1}\left [ {0,1} \right ]

is given.
Hochschild and cyclic homology are far from being homotopy functors
Authors: Geller S. , Weibel C.
Published: June 2010

Given a homology theory

H
∗

(
A
)

{H_*}(A)

on rings, based on a natural chain complex, one can form a new theory

H
∗
h

(
A
)

H_*^h(A)

which is universal with respect to the homotopy property

H
∗

(
A
)
≃

H
∗

(
A
[
t
]
)

{H_*}(A) \simeq {H_*}(A[t])

. We show that the homotopy theories

H

H
∗
h

HH_*^h

and

H

C
∗
h

HC_*^h

associated to Hochschild and cyclic homology are both zero. On the other hand, if

H

C
∗
−

HC_*^ -

denotes Goodwillie’s variant of cyclic homology, and

A
A

contains a field of characteristic 0, we show that

(
H

C
−

)
∗
h

A

(H{C^ - })_*^hA

is Connes’ periodic cyclic homology

H

P
∗

(
A
)

H{P_*}(A)

.
Marczewski sets, measure and the Baire property. II
Authors: Walsh John T.
Published: June 2010

An example is given (in ZFC) of a hereditarily Marczewski set which does not have the Baire property in the wise sense and is not Lebesgue measurable. It is then shown that Marczewski sets, which do not satisfy the c.c.c., possess a hull property.
Thurston norm minimizing surfaces and skein trees for links in 𝑆³
Authors: Thompson Abigail
Published: June 2010

This paper gives a method for constructing all links in

S
3

{S^3}

, beginning with the unknot and adding at most one to the norm of the link at each stage. This has two corollaries. The first is that links with ’minimal’ skein trees are fibered. The second is a complete list of all links with skein trees of height two.
On the imbedding of a direct product into a zero-dimensional commutative ring
Authors: Gilmer Robert , Heinzer William
Published: June 2010

This paper addresses questions related to results of M. Arapovic concerning imbeddability of a commutative unitary ring

R
R

in a zero-dimensional ring. The case where

R
R

is a product of zero-dimensional rings is of special interest. We show (1) if the zero ideal of

R
R

admits a unique representation as an irredundant intersection of (strongly primary) ideals, then

R
R

need not be imbeddable in a zero-dimensional ring, and (2) for a family

{

R
α

}

\left \{ {{R_\alpha }} \right \}

of zero-dimensional rings,

R
=
∏

R
α

R = \prod {R_\alpha }

is imbeddable in a zero-dimensional ring if and only if

R
R

itself is zero-dimensional.
Global preservation of nodal structure in coupled systems of nonlinear Sturm-Liouville boundary value problems
Authors: Cantrell Robert Stephen
Published: June 2010

In this paper, we examine the solution set to the coupled system (

∗
*

)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace Subscript minus left-parenthesis p 2 left-parenthesis x right-parenthesis upsilon prime left-parenthesis x right-parenthesis right-parenthesis Sub Superscript prime Subscript plus q 2 left-parenthesis x right-parenthesis upsilon left-parenthesis x right-parenthesis equals mu upsilon left-parenthesis x right-parenthesis plus upsilon left-parenthesis x right-parenthesis dot g left-parenthesis u left-parenthesis x right-parenthesis comma upsilon left-parenthesis x right-parenthesis right-parenthesis comma Superscript minus left-parenthesis p 1 left-parenthesis x right-parenthesis u prime left-parenthesis x right-parenthesis right-parenthesis Super Superscript prime Superscript plus q 1 left-parenthesis x right-parenthesis u left-parenthesis x right-parenthesis equals lamda u left-parenthesis x right-parenthesis plus u left-parenthesis x right-parenthesis dot f left-parenthesis u left-parenthesis x right-parenthesis comma upsilon left-parenthesis x right-parenthesis right-parenthesis">

{

−
(

p
2

(
x
)

υ
′

(
x
)

)
′

+

q
2

(
x
)
υ
(
x
)
=
μ
υ
(
x
)
+
υ
(
x
)
⋅
g
(
u
(
x
)
,
υ
(
x
)
)
,

−
(

p
1

(
x
)

u
′

(
x
)

)
′

+

q
1

(
x
)
u
(
x
)
=
λ
u
(
x
)
+
u
(
x
)
⋅
f
(
u
(
x
)
,
υ
(
x
)
)

\left \{ {_{ - ({p_2}(x)\upsilon ’(x))’ + {q_2}(x)\upsilon (x) = \mu \upsilon (x) + \upsilon (x) \cdot g(u(x),\upsilon (x)),}^{ - ({p_1}(x)u’(x))’ + {q_1}(x)u(x) = \lambda u(x) + u(x) \cdot f(u(x),\upsilon (x))}} \right .

\]
where

λ
,
μ
∈
R
,
x
∈
[
a
,
b
]

\lambda ,\mu \in R,x \in [a,b]

, and the system (

∗
*

) is subject to zero Dirichlet boundary data on

u
u

and

υ
\upsilon

. We determine conditions on

f
f

and

g
g

which permit us to assert the existence of continua of solutions to (

∗
*

) characterized by

u
u

having

n
−
1

n - 1

simple zeros in

(
a
,
b
)
,
υ

(a,b),\upsilon

having

m
−
1

m - 1

simple zeros in

(
a
,
b
)

(a,b)

, where

n
n

and

m
m

are positive but not necessarily equal integers. Moreover, we also determine conditions under which these continua link solutions to (

∗
*

) of the form

(
λ
,
μ
,
u
,
0
)

(\lambda ,\mu ,u,0)

with

u
u

having

n
−
1

n - 1

simple zeros in

(
a
,
b
)

(a,b)

to solutions of (

∗
*

) of the form

(
λ
,
μ
,
0
,
υ
)

(\lambda ,\mu ,0,\upsilon )

with

υ
\upsilon

having

m
−
1

m - 1

simple zeros in

(
a
,
b
)

(a,b)

.
Algebraic structure in complex function spaces
Authors: Ellis A. J.
Published: June 2010

Let

M
M

be a complex function space containing constants, and let

Z
Z

be the complex state space of

M
M

. If

M
M

is linearly isometric to a uniform algebra and if

Z
Z

is affinely homeomorphic to the complex state space of a uniform algebra then we prove that

M
M

is a uniform algebra. Neither of the two conditions taken separately imply this conclusion.
The ghost of an index theorem
Authors: Harte Robin
Published: June 2010

The "index theorem" whose spirit we invoke says that the index of a product of Fredholm operators is the sum of their indexes; our extension is to operators with "generalized inverses".
A short proof of the Grigorchuk-Cohen cogrowth theorem
Authors: Szwarc Ryszard
Published: June 2010

Let

G
G

be a group generated by

g
1

,
…
,

g
r

{g_1}, \ldots ,{g_r}

. There are exactly

2
r

(
2
r
−
1

)

n
−
1

2r{(2r - 1)^{n - 1}}

reduced words in

g
1

,
…
,

g
r

{g_1}, \ldots ,{g_r}

of length

n
n

. Part of them, say

γ
n

{\gamma _n}

represents identity element of

G
G

. Let

γ
=
lim
sup

γ
n

1

/

n

\gamma = \lim \sup \gamma _n^{1/n}

. We give a short proof of the theorem of Grigorchuk and Cohen which states that

G
G

is amenable if and only if

γ
=
2
r
−
12

\gamma = 2r - 12

. Moreover we derive some new properties of the generating function

∑

γ
n

z
n

\sum {{\gamma _n}{z^n}}

.
Nowhere dense 𝑃-subsets of 𝜔
Authors: Just Winfried
Published: June 2010

It is relatively consistent with ZFC that no nowhere dense

P
P

-subset of

ω
∗

\omega *

is homeomorphic to the space

ω
∗

{\omega ^ * }

itself.
Approximation of analytic multifunctions
Authors: Slodkowski Zbigniew
Published: June 2010

Set-valued generalizations of analytic functions are defined by a form of local maximum principle. It is shown that they are identical with limits of decreasing sequences of multifunctions whose graphs are locally covered by graphs of single-valued analytic mappings.
An entropy inequality for the bi-multivariate hypergeometric distribution
Authors: Kochman Fred , Murray Alan , West Douglas B.
Published: June 2010

Given parameters

r
¯

=

r
1

,
…
,

r
m

\bar r = {r_1}, \ldots ,{r_m}

and

c
¯

=

c
1

,
…
,

c
n

\bar c = {c_1}, \ldots ,{c_n}

with

∑

r
i

=
∑

c
j

=
N

\sum {{r_i}} = \sum {{c_j}} = N

, the bi-multivariate hypergeometric distribution is the distribution on nonnegative integer

m
×
n

m \times n

matrices with row sums

r
¯

\bar r

and column sums

c
¯

\bar c

defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Prob left-parenthesis upper A right-parenthesis equals product r Subscript i Baseline factorial product c Subscript j Baseline factorial slash left-parenthesis upper N factorial product a Subscript i j Baseline factorial right-parenthesis">

Prob

(
A
)

=
∏

r
i

!
∏

c
j

!

/

(

N
!
∏

a

i
j

!

)

{\text {Prob}}\left ( A \right ) = \prod {{r_i}} !\prod {{c_j}} ! / \left ( {N!\prod {{a_{ij}}!} } \right )

. It is shown that the entropy of this distribution is a Schur-concave function of the block-size parameters.
On the existence of idempotent liftings
Authors: Grekas S.
Published: June 2010

An existence theorem for idempotent liftings is proved. This implies that every compact measure space with full support and separable measure algebra admits an idempotent lifting.
On nilpotent derivations of prime rings
Authors: Chuang Chen-Lian
Published: June 2010

Let

R
R

be a prime ring with center

Z
Z

and let

U
U

be a noncentral Lie ideal of

R
R

. Suppose that

d
d

is a derivation of

R
R

such that

d
n

(
u
)
∈
Z

{d^n}(u) \in Z

for all

u
∈
U

u \in U

, where

n
n

is a fixed integer. It is shown that either

d
n

(
R
)
=
0

{d^n}(R) = 0

or

R
R

is an order of a

4
4

-dimensional simple algebra over a field of characteristic 2.
On abelian quotients of primitive groups
Authors: Aschbacher Michael , Guralnick Robert M.
Published: June 2010

It is shown that if

G
G

is a primitive permutation group on a set of size

n
n

, then any abelian quotient of

G
G

has order at most

n
n

. This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if

M

/

K

M/K

is a minimal extension and

L

/

K

L/K

is an abelian extension contained in the normal closure of

M
M

, then the degree of

L

/

K

L/K

is at most the degree of

M

/

K

M/K

.
On the two definitions of the Conley index
Authors: Kurland Henry L.
Published: June 2010

The two definitions of the homotopy equivalences between Conley index spaces of an isolated invariant set, the original one of Conley [C] as completed by the author in [K1] and the more recent definition of Salamon [S], are shown to define the same homotopy classes without reference to the difficult proof of [K1] showing the Conley index to be a connected simple system. The equivalences of the original definition are useful in describing certain geometric situations in terms of the index; examples are given.
Complemented copies of 𝑐₀ in vector valued Hardy spaces
Authors: Dowling Patrick N.
Published: June 2010

Let

X
X

be a complex Banach space containing a copy of

c
0

{c_0}

, let

T

\mathbb {T}

be the unit circle and let

D
D

be the open unit disk in the complex plane. Then

H
p

(

T

,
X
)

{H^p}(\mathbb {T},X)

contains a complemented copy of

c
0

{c_0}

for

1
≤
p
>
∞

1 \leq p > \infty

. The corresponding result for

H
p

(
D
,
X
)

{H^p}(D,X)

fails for

1
≤
p
≤
∞

1 \leq p \leq \infty

.
Almost complete tilting modules
Authors: Happel Dieter , Unger Luise
Published: June 2010

Let

A
A

be a finite-dimensional hereditary algebra over an algebraically closed field. All modules will be finite-dimensional left

A
A

-modules. We are concerned with partial tilting modules which can be completed to a tilting module by one indecomposable module which will be called a complement. As a main result we show that such a partial tilting module allows (up to isomorphism) at most two complements and there are two such complements if and only if the partial tilting module is sincere.
Extension of operators from subspaces of 𝑐₀(Γ) into 𝐶(𝐾) spaces
Authors: Johnson W. B. , Zippin M.
Published: June 2010

It is shown that for every

ε
>
0

\varepsilon > 0

, every bounded linear operator

T
T

from a subspace

X
X

of

c
0

(
Γ
)

{c_0}\left ( \Gamma \right )

into a

C

(
K
)

C\left ( K \right )

space has an extension

T

{\mathbf {T}}

from

c
0

(
Γ
)

{c_0}\left ( \Gamma \right )

into the

C

(
K
)

C\left ( K \right )

space such that

‖

T

‖

≤

(

1
+
ε

)

‖
T
‖

\left \| {\mathbf {T}} \right \| \leq \left ( {1 + \varepsilon } \right )\left \| T \right \|

. Even when

Γ
\Gamma

is countable,

T
T

is compact, and

X
X

has codimension 1 in

c
0

{c_0}

, the "

ε
\varepsilon

" cannot be replaced by 0. These results answer questions raised by J. Lindenstrauss and A. Pełczynski in 1971.
Ideal theoretic complete intersections in 𝑃³_{𝐾}
Authors: Thoma Apostolos
Published: June 2010

We describe the monomial curves in

P
K
3

P_K^3

(

K
K

algebraically closed field of characteristic zero) that are set theoretical complete intersections on two binomial surfaces. We prove that they are exactly those which are ideal theoretic complete intersections. Using that, we get explicitly all monomial curves that are ideal theoretic complete intersections and a minimal generating basis for their ideals.
A diameter pinching sphere theorem for positive Ricci curvature
Authors: Wu Jyh Yang
Published: June 2010

In this note we generalize Shiohama’s volume pinching sphere theorem to a diameter pinching sphere theorem for positive Ricci curvature.
Isometries on conservative subalgebras of bounded sequences
Authors: Connor J. , Loomis I.
Published: June 2010

Using extreme point techniques, we show that if

A
A

is a closed subalgebra of the bounded sequences which contain

c
c

, then any linear isometry of

A
A

onto itself is a permutation up to a modulus one multiplication. If the subalgebra

A
A

is generated by an ideal, then a permutation

P
P

maps

A
A

onto itself if and only if

P
P

maps

μ
\mu

-null sets to

μ
\mu

-null sets where

μ
\mu

is a 0,1-valued finitely additive measure associated with the ideal. In particular, if

T
T

is a nonnegative regular summability method, we characterize the isometries which map the bounded strongly

T
T

-summable sequences onto themselves and give a concrete sufficient condition for a permutation to map the bounded strongly Cesaro summable sequences onto themselves.
On a theorem of Deskins
Authors: Zimmermann Irene
Published: June 2010

In this note we present a solvability criterion for finite groups. We show that a finite group

G
G

is solvable if in every maximal subgroup chain of length 3 of

G
G

at least one term is a submodular subgroup of

G
G

. This generalizes an earlier result of Deskins.
Optimal matching and empirical measures
Authors: Yukich J. E.
Published: June 2010

Using results from optimal matching, we find the exact order of convergence for

β
(

P
n

,
P
)

\beta ({P_n},P)

and

ρ
(

P
n

,
P
)

\rho ({P_n},P)

, where

β
\beta

denotes the dual bounded Lipschitz metric,

ρ
\rho

the Prokhorov metric and

P
n

{P_n}

the

n
n

th empirical measure associated to

P
P

, the uniform measure on the unit square. The results solve a long-open problem in empirical measures.
Nonlinear two-point boundary value problems at resonance without Landesman-Lazer condition
Authors: Iannacci R. , Nkashama M. N.
Published: June 2010

The purpose of this paper is to study the solvability of a semilinear two-point boundary value problem of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded.
On the road coloring problem
Authors: Friedman Joel
Published: June 2010

Let

G
=
(
V
,
E
)

G = (V,E)

be a strongly connected, aperiodic, directed graph having outdegree 2 at each vertex. A red-blue coloring of

G
G

is a coloring of the edges with the colors red and blue such that each vertex has one red edge and one blue edge leaving it. Given such a coloring, we define

R
:
V
→
V

R:V \to V

by

R
(
v
)
=
w

R(v) = w

iff there is a red edge from

v
v

to

w
w

. Similarly we define

B
:
V
→
V

B:V \to V

.

G
G

is said to be collapsible if some composition of

R
R

’s and

B
B

’s maps

V
V

to a single vertex. The road coloring problem is to determine whether

G
G

has a collapsible coloring. It has been conjectured that all such

G
G

have a collapsible coloring. Since

G
G

has outdegree 2 everywhere and is strongly connected, the adjacency matrix,

A
A

, of

G
G

has a positive left eigenvector

w
=
(
w
(

v
1

)
,
…
,
w
(

v
n

)
)

w = (w({v_1}), \ldots ,w({v_n}))

with eigenvalue 2 , i.e.

w
A
=
2
w

wA = 2w

. Furthermore, we can assume that

w
w

’s components are integers with no common factor. We call

w
(
v
)

w(v)

the weight of

v
v

. Let

W
≡

Σ

v
∈
V

w
(
v
)

W \equiv {\Sigma _{v \in V}}w(v)

, defined to be the weight of the graph. We will prove that if

G
G

has a simple cycle of length relatively prime to

W
W

, then

G
G

is collapsibly colorable.
The extension of the theorems of Č. V. Stanojević and V. B. Stanojević
Authors: Sheng Shu Yun
Published: June 2010

The new necessary-sufficient conditions for

L
1

{L^1}

convergence of Fourier series are obtained; Č. V. Stanojevic̀ and V. B. Stanojevic̀’s theorem [5] and Singh and Sharma’s theorem [2] are modified; the convergence theorem for a function sequence in

L
1

{L^1}

space is obtained; and the extensions are made for the Sheng and Yang theorem [6] and Singh and Sharma’s results.
Representation of tree permutations by words
Authors: Maroli John A.
Published: June 2010

The problem of solving equations in groups can be stated as follows: given a group

G
G

and a free group

F
=
F

(

x
1

,

x
2

,
…

)

F = F\left ( {{x_1},{x_2}, \ldots } \right )

, for which pairs

(

w
,
g

)

\left ( {w,g} \right )

with

w
=
w

(

x
1

,

x
2

,
…

)

∈
F
,
g
∈
G

w = w\left ( {{x_1},{x_2}, \ldots } \right ) \in F,g \in G

, is it possible to find elements

g
i

∈
G

{g_i} \in G

such that

w

(

g
1

,

g
2

,
…

)

=
g

w\left ( {{g_1},{g_2}, \ldots } \right ) = g

? We investigate the corresponding question of solving equations in the group

A

(
Ω
)

A\left ( \Omega \right )

of all automorphisms of a transitive tree

Ω
\Omega

. If the tree has isomorphic cones at a branch point, then certain equations of the form

x
n

=
g

{x^n} = g

cannot be solved (Theorem 2.3). If the tree is sufficiently transitive, we find large classes of equations

w
=
g

w = g

which can be solved (Theorems 2.13, 2.16).
Centralizing mappings on von Neumann algebras
Authors: Brešar Matej
Published: June 2010

Let

R
R

be a ring with center

Z
(
R
)

Z(R)

. A mapping

F
F

of

R
R

into itself is called centralizing if

F
(
x
)
x
−
x
F
(
x
)
∈
Z
(
R
)

F(x)x - xF(x) \in Z(R)

for all

x
∈
R

x \in R

. The main result of this paper states that every additive centralizing mapping

F
F

on a von Neumann algebra

R
R

is of the form

F
(
x
)
=
c
x
+
ζ
(
x
)
,
x
∈
R

F(x) = cx + \zeta (x),x \in R

, where

c
∈
Z
(
R
)

c \in Z(R)

and

ζ
\zeta

is an additive mapping from

R
R

into

Z
(
R
)

Z(R)

. We also consider

α
\alpha

-derivations and some related mappings, which are centralizing on rings and Banach algebras