bluuebunny/crossref_metadata_2025 huggingface dataset:
Dimensions and measures of quasi self-similar sets
Authors: Falconer K. J.
Published: July 2010

We show that sets with certain quasi self-similar properties have equal Hausdorff and box-packing dimensions and also have positive and finite Hausdorff measure at the dimensional value. A number of applications of these results to particular examples are given.
Distributions and the Lie algebras their bases can generate
Authors: Hermes Henry
Published: July 2010

The problem is to determine when a smooth,

k
k

-dimensional distribution

D
k

{D^k}

defined on an

n
n

-manifold

M
n

{M^n}

, locally admits a vector field basis which generates a nilpotent, solvable or even finite-dimensional Lie algebra. We show that for every

2
≤
k
≤
n
−
1

2 \leq k \leq n - 1

there exists a (nonregular at

p
∈

M
n

p \in {M^n}

) distribution

D
k

{D^k}

on

M
n

{M^n}

which does not locally (near

p
p

) admit a vector field basis generating a solvable Lie algebra. From classical results on the equivalence problem, it is shown that for

1
≤
k
≤
4

1 \leq k \leq 4

and

D
k

{D^k}

regular at

p
∈

M
4

p \in {M^4}

,

D
k

{D^k}

admits a local vector field basis generating a nilpotent Lie algebra.
Continuity properties of the spectrum of operators on Lebesgue spaces
Authors: Barnes Bruce A.
Published: July 2010

Fix

1
≤
p
≤
s
≤
∞

1 \leq p \leq s \leq \infty

. Let

T
x

,
x
∈

[

p
,
s

]

{T_x},x \in \left [ {p,s} \right ]

, be the collection of bounded linear operators on the Lebesgue spaces

L
x

{L^x}

determined by some fixed operator

T
T

. This paper concerns continuity properties of the map

x
→
σ

(

T
x

)

x \to \sigma \left ( {{T_x}} \right )

.
Functions not constant on fractal quasi-arcs of critical points
Authors: Norton Alec
Published: July 2010

This paper provides geometric sufficient conditions for an arc to be a critical set for some function not constant along that arc—an example of which was first discovered by Whitney in 1935. In particular, any fractal subarc of a quasi-circle has this property. The maximum degree of differentiability of the function is closely connected to the arc’s geometry.
A useful proposition for division algebras of small degree
Authors: Haile Darrell
Published: July 2010

Let

F
F

be a field and let

D
D

be an

F
F

-central division algebra of degree

n
n

. We present a short, elementary proof of the following statement: There is an

n
−
1

n - 1

-dimensional

F
F

-subspace

V
V

of

D
D

such that for every nonzero element

ν
\nu

of

V
V

,

Tr

(
ν
)
=

Tr(

ν

−
1

)
=
0

{\text {Tr}}(\nu ) = {\text {Tr(}}{\nu ^{ - 1}}) = 0

. We then indicate how one can use this result to obtain the basic structural results on division algebras of degree three and four (results of Wedderburn and Albert, respectively).
On the intersection of a class of maximal subgroups of a finite group
Authors: Guo Xiu Yun
Published: July 2010

Let

G
G

be a finite group and

π
\pi

a set of primes. We consider the family of subgroups of

G
:

F

=
{
M
:
M
>
⋅
G
,

[
G
:
M

]
π

=
1
,
[
G
:
M
]

G:\mathcal {F} = \{ M:M > \cdot G,{[G:M]_\pi } = 1,[G:M]

is composite} and denote

S
π

(
G
)
=
⋂

{
M
:
M
∈

F

}

{S_\pi }(G) = \bigcap \left \{ M: M \in \mathcal {F} \right \}

if

F

\mathcal {F}

is non-empty, otherwise

S
π

(
G
)
=
G

{S_\pi }(G) = G

. The purpose of this note is to prove Theorem. Let

G
G

be a

π
\pi

-solvable group. Then

S
π

(
G
)

{S_\pi }(G)

has the following properties: (1)

S
π

(
G
)

/

O
π

(
G
)

{S_\pi }(G)/{O_\pi }(G)

is supersolvable. (2)

S
π

(

S
π

(
G
)
)
=

S
π

(
G
)

{S_\pi }({S_\pi }(G)) = {S_\pi }(G)

. (3)

G

/

O
π

(
G
)

G/{O_\pi }(G)

is supersolvable if and only if

S
π

(
G
)
=
G

{S_\pi }(G) = G

.
Souslin subsets of 𝑃(𝜔)-spaces
Authors: Wu Li Sheng
Published: July 2010

A space

X
X

is a

P

(
ω
)

P\left ( \omega \right )

-space if and only if

X
X

is countably

θ
\theta

-refinable and every Souslin subset of

X
X

is a generalized

F
σ

{F_\sigma }

-set in

X
X

.
When is 𝐶(𝑋) a coherent ring?
Authors: Neville Charles W.
Published: July 2010

We prove:

C
(
X
)

C(X)

is coherent if and only if

X
X

is basically disconnected.
The virtual 𝑍-representability of 3-manifolds which admit orientation reversing involutions
Authors: Wang Shi Cheng
Published: July 2010

We prove a result which supports the Waldhausen Conjecture, i.e., suppose

M
M

is an irreducible orientable

3
3

-manifold with

|

π
1

(
M
)

|

=
∞

|{\pi _1}(M)| = \infty

; if

M
M

admits an orientation reversing involution

τ
\tau

, and

M
M

has a nontrivial finite cover, then some finite cover

M
~

\tilde M

of

M
M

has positive first Betti number.
On 2-step solvable groups of finite Morley rank
Authors: Enochs Kathryn , Nesin Ali
Published: July 2010

We prove the following results: Theorem 1. Let

G
G

be a connected, centerless, solvable group of class 2 and of finite Morley rank. Then we can interpret in

G
G

finitely many connected, solvable of class 2 and centerless algebraic groups

G
~

1

,
…
,

G
~

n

{\tilde G_1}, \ldots ,{\tilde G_n}

over algebraically closed fields

K
i

{K_i}

in such a way that

G
G

interpretably imbeds in

G
~

=

G
~

1

⊕
⋯
⊕

G
~

n

\tilde G = {\tilde G_1} \oplus \cdots \oplus {\tilde G_n}

. Furthermore,

G
′

=
(

G
~

)
′

G’ = (\tilde G)’

. Let

F
(
G
)

F(G)

denote the Fitting subgroup of

G
G

. Theorem 2. Let

G
,

G
~

,

G
~

i

G,\tilde G,{\tilde G_i}

be as in Theorem 1. Then (i)

F
(
G
)
=
F
(

G
~

)
∩
G

F(G) = F(\tilde G) \cap G

. (ii)

F
(
G
)

F(G)

has a complement

V
V

in

G
:
G
=
F
⋊
V

G:G = F \rtimes V

. (iii) Elements of

F
(
G
)

F(G)

are unipotent elements of

G
G

in

G
~

\tilde G

. (iv) If the characteristic of each base field

K
i

{K_i}

of

G
~

i

{\tilde G_i}

is different from 0, then

V
V

is definable and its elements are semi-simple in

G
~

\tilde G

.
Functional characterizations of Baire spaces
Authors: Holá Ľubica
Published: July 2010

We give functional characterizations of Baire spaces in the class of topological spaces with a

σ
\sigma

-locally finite almost-base. This answers positively a question asked in [5] for topological spaces with a

σ
\sigma

-locally finite almost-base.
The general form of Green’s theorem
Authors: Jurkat W. B. , Nonnenmacher D. J. F.
Published: July 2010

Using a recently developed Perron-type integration theory, we prove a new form of Green’s theorem in the plane, which holds for any rectifiable, closed, continuous curve under very general assumptions on the vector field. In particular, Cauchy’s integral theorem can be deduced in its presently most general form.
A property of polynomial curves over a field of positive characteristic
Authors: Daigle D.
Published: July 2010

Let

k

{\mathbf {k}}

be an algebraically closed field of characteristic

p
>
0

p > 0

. We show that if

F
∈

k

[

X
,
Y

]

F \in {\mathbf {k}}\left [ {X,Y} \right ]

is a rational curve with one place at infinity and with nonprincipal bidegree, such that

θ

(
F
)

∈

k

[

X
p

,
Y

]

\theta \left ( F \right ) \in {\mathbf {k}}\left [ {{X^p},Y} \right ]

for some automorphism

θ
\theta

of

k

[

X
,
Y

]

{\mathbf {k}}\left [ {X,Y} \right ]

, then

θ
\theta

can be chosen to be either linear or of "de Jonquière" type. We also give consequences of that fact for the problem of classifying the embeddings of the line in the plane.
A local bifurcation theorem for 𝐶¹-Fredholm maps
Authors: Fitzpatrick P. M. , Pejsachowicz Jacobo
Published: July 2010

The Krasnosel’skii Bifurcation Theorem is generalized to

C
1

{C^1}

-Fredholm maps. Let

X
X

and

Y
Y

be Banach spaces,

F
:
R
×
X
→
Y

F:R \times X \to Y

be

C
1

{C^1}

-Fredholm of index 1 and

F
(
λ
,
0
)
≡
0

F(\lambda ,0) \equiv 0

. If

I
⊆
R

I \subseteq R

is a closed, bounded interval at whose endpoints

∂
F

∂
x

∂
F

∂
x

(
λ
,
0
)

\frac {{\partial F}}{{\partial x}}\frac {{\partial F}}{{\partial x}}(\lambda ,0)

is invertible, and the parity of

∂
F

∂
x

(
λ
,
0
)

\frac {{\partial F}}{{\partial x}}(\lambda ,0)

on

I
I

is -1, then

I
I

contains a bifurcation point of the equation

F
(
λ
,
x
)
=
0

F(\lambda ,x) = 0

. At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.
Inequalities involving multivariate convex functions. II
Authors: Neuman Edward
Published: July 2010

In this paper we offer some inequalities involving multivariate convex functions. Among other things a refinement of classical Jensen’s inequality as well as an extension of Fejér’s inequality to the case of

s
s

-variate

(
s
≥
1
)

(s \geq 1)

functions are included. These results are obtained with the aid of the generalized simplex splines.
On the type of Wiener-Hopf 𝐶*-algebras
Authors: Sheu Albert Jeu-Liang
Published: July 2010

In this paper, we show that if the positive cone

P
P

of a locally compact group

G
G

does not satisfy a regularity condition then the corresponding Wiener-Hopf

C
∗

{C^ * }

-algebra

W

(
P
)

\mathcal {W}(P)

is not of type I while the converse does not hold, and that if

C
∗

(
G
)

{C^ * }(G)

is not of type I then neither is

W

(
P
)

\mathcal {W}(P)

. Thus a conjecture and a question, both proposed by P. Muhly and J. Renault in their important systematic treatment of general Wiener-Hopf

C
∗

{C^ * }

-algebras using groupoid

C
∗

{C^*}

-algebras, are settled.
Convex functions with restricted curvature
Authors: Styer D. , Wright D. J.
Published: July 2010

Given

0
≤

R
1

≤

R
2

≤
∞

0 \leq {R_1} \leq {R_2} \leq \infty

, we consider a class of normalized convex functions

f
f

in the unit disk

D

{\mathbf {D}}

, for which

∂
f
(

D

)

\partial f({\mathbf {D}})

satisfies a Blaschke Rolling Theorem condition with radii

R
1

{R_1}

and

R
2

{R_2}

. This class contains the convex functions of bounded type. We study the geometry of the image region

f
(

D

)

f({\mathbf {D}})

and various covering and distortion properties.
The masses in a symmetric centered solution of the 𝑛-body problem
Authors: Diacu Florin Nicolae
Published: July 2010

We prove that if a planar solution of the

n
n

-body problem has a symmetry axis, fixed with respect to the considered frame, and the center of mass of the particle system lies on this axis during the motion, then the symmetric masses must be equal. We also show that the set of initial conditions leading to symmetric solutions has measure zero and is nowhere dense relative to the set of all initial conditions that define solutions in the space.
Open relation theorem without closedness assumptions
Authors: Dolecki Szymon
Published: July 2010

It is shown that a convex relation such that the preimage of a bounded set has nonempty interior is lower semicontinuous throughout the algebraic interior of its domain. If, besides, the relation is closed-valued, then it is closed throughout the algebraic interior of its domain.
Persistence definitions and their connections
Authors: Freedman H. I. , Moson P.
Published: July 2010

We give various definitions of types of persistence of a dynamical system and establish a hierarchy among them by proving implications and demonstrating counterexamples. Under appropriate conditions, we show that several of the definitions are equivalent.
Teichmüller-type annihilators of 𝐿¹-analytic functions
Authors: Reich Edgar , Shapiro Harold S.
Published: July 2010

Characterizations are obtained for some classes of Teichmüller-type

L
∞

{L^\infty }

functions that are orthogonal to all polynomials in the unit disk.
Product formula for resolvents of normal operators and the modified Feynman integral
Authors: de Bivar-Weinholtz António , Lapidus Michel L.
Published: July 2010

We extend the theory of the "modified Feynman integal" developed by the second author by extending his product formula for the imaginary resolvents of selfadjoint (unbounded) operators to those of normal operators. This enables us to establish the convergence of the "modified Feynman integral" for Hamiltonians with highly singular complex (instead of real) potentials. Such Hamiltonians arise naturally in the study of the Schrödinger equation associated with dissipative quantum mechanical systems. By slightly altering the proof of our results, we also give a very general (operator-theoretic) interpretation of Nelson’s "Feynman integral by analytic continuation in the mass parameter" that is valid for singular potentials with an arbitrary sign. An interesting aspect of our "product formula for the imaginary resolvents of normal operators" is that it extends, and in some sense unifies, the above two approaches to the Feynman integral.
An analogue to Glauberman’s 𝑍𝐽-theorem
Authors: Stellmacher Bernd
Published: July 2010

Let

P
P

be a finite

p
p

-group,

p
p

an odd prime. Using certain versions of

p
p

-stability it is shown that there exists a nontrivial characteristic subgroup

W
W

in

P
P

that is normal in every finite

p
p

-stable group

G
G

satisfying

C
G

(

O
p

(
G
)
)
≤

O
p

(
G
)

{C_G}({O_p}(G)) \leq {O_p}(G)

and

P
∈

Sy

l

p

(
G
)

P \in {\text {Sy}}{{\text {l}}_p}(G)

. Moreover,

W
W

contains every abelian subgroup of

P
P

normalized by

W
W

.
On (0,1,2) interpolation in uniform metric
Authors: Szabados J. , Varma A. K.
Published: July 2010

From the well known theorem of G. Faber it follows that for any given matrix of nodes there exists a continuous function for which the Lagrange interpolation polynomial

L
n

[
f
,
x
]

{L_n}[f,x]

, generated by the

n
n

th row of the matrix, does not tend uniformly to

f
(
x
)

f(x)

. In this paper we shall provide analogous results for the related operator

H

n
,
3

[
f
,
x
]

{H_{n,3}}[f,x]

as defined below.
Stability results for first order projection bodies
Authors: Goodey P. R. , Groemer H.
Published: July 2010

The motivation for this work comes from a result of Minkowski. He showed that if a three-dimensional convex body has the property that all its projections have the same perimeter, then the original body has constant width. Our objective was to extend this to a stability result and not to restrict ourselves to dimension three. The result we obtained shows that if two centrally symmetric bodies have projections which all have approximately the same mean width, then the two bodies are approximately the same up to translation. This is, in effect, a continuity result for the inverse of the spherical Radon transform. It is closely related to recent three-dimensional results of Campi and to work of Bourgain and Lindenstrauss, who consider the volumes of projections rather than their mean widths. The techniques we employ are drawn from the theory of spherical harmonics and from the theory of mixed volumes.
Growth of two-step solvable Lie algebras
Authors: Kobayashi Shigeru , Sanami Manabu
Published: July 2010

It is proved that every finitely generated infinite-dimensional twostep solvable Lie algebra has polynomially bounded growth. As a corollary, it is shown that the universal enveloping algebras of such Lie algebras are not Noetherian but of Krull domain.
The Banach-Tarski theorem on polygons, and the cancellation law
Authors: Gardner R. J. , Laczkovich M.
Published: July 2010

The Banach-Tarski theorem on polygons in

R

2

{{\mathbf {R}}^2}

implies that two polygons are equidecomposable if and only if they are equidissectable. The possibility of strengthening this result in various ways is investigated. We show that if two polytopes in

R

d

{{\mathbf {R}}^d}

are equidecomposable under a finite set of isometries which generates a discrete group, then they are equidissectable using the same isometries. We then give a simple example in

R

{\mathbf {R}}

showing that this is not true for arbitrary finite sets of isometries. A modification of this example is used to answer a question of S. Wagon concerning the cancellation law.
Univalence of Gaussian and confluent hypergeometric functions
Authors: Miller Sanford S. , Mocanu Petru T.
Published: July 2010

Conditions are determined for the univalence convexity and starlikeness of Gaussian and confluent hypergeometric functions. In addition, subordination results are obtained for these classes of functions.
Nonexistence of positively expansive maps on compact connected manifolds with boundary
Authors: Hiraide Koichi
Published: July 2010

In this note we prove that no compact connected manifold with boundary admits a positively expansive map.
Defining the boundary of a homology manifold
Authors: Mitchell W. J. R.
Published: July 2010

We show that the natural homological analogue of the definition of topological manifold with boundary does indeed properly capture the properties required of a homology manifold with boundary. Accordingly existing definitions requiring that the boundary is already a (homology) manifold can be simplified.
On Hankel transform
Authors: Duran Antonio J.
Published: July 2010

In this paper, using some results of the author on Hankel transform in the Schwartz and Gel’fand-Shilov spaces, we characterize the integral operators of Hankel type which are isomorphisms between the spaces

H
μ

{H_\mu }

of Zemanian. As a particular case, we obtain the classical Zemanian results on Hankel transform, some results of Mendez, and improve some results of Lee. Finally, we use these results to characterize the functions

f
f

of the Schwartz space which satisfy

∫
0
∞

t

α
+
n

f
(
t
)
d
t
=
0

\smallint _0^\infty {t^{\alpha + n}}f(t)dt = 0

for all

n
≥
0

n \geq 0

and

α
>
−
1

\alpha > - 1

.
Nongeometric convergence of best 𝐿_{𝑝}(𝑝≠2) polynomial approximants
Authors: Ivanov K. G. , Saff E. B.
Published: July 2010

For an arbitrary function

f
f

analytic in the disk

D
:

|
z
|

>
1

D:\left | z \right | > 1

and continuous in

D
¯

\bar D

, we show that geometric convergence in

D
D

of best

L
p

(
1
≤
p
≤
∞
)

{L_p}(1 \leq p \leq \infty )

polynomial approximants to

f
f

on

C
:

|
z
|

=
1

C:\left | z \right | = 1

is assured only when

p
=
2

p = 2

.
The irrationals are not recursively enumerable
Authors: Mansfield Richard
Published: July 2010

The set of rationals is not recursive over any field of infinite transcendence degree.
Partitions of vector spaces
Authors: Bešlagić Amer
Published: July 2010

The following question is answered: If the real line is partitioned into countable sets, is there a Hamel basis that picks at most one element from each member of the partition?
On 𝐿²-plurigenera of not-log-canonical Gorenstein isolated singularities
Authors: Tomari Masataka , Watanabe Kimio
Published: July 2010

We will give an application of S. Izumi’s theorems of order functions of analytic spaces to classification problems of isolated singularities by means of plurigenus

δ
m

{\delta _m}

.
On the weighted integrability of maximal ergodic ratios
Authors: Andersen Kenneth F.
Published: July 2010

Necessary and sufficient conditions on pairs of nonnegative weight functions are given which ensure that the maximal ergodic ratios defined by means of a measure preserving conservative ergodic transformation satisfy a reverse weak type inequality. From this, weighted integrability results may be deduced.
Random times and time projections
Authors: Barnett Chris , Wilde Ivan F.
Published: July 2010

Random times and their associated time projections are discussed within the context of quantum probability theory. A stochastic integral representation for time projections is obtained, and their order structure is investigated. A quantum analogue of the classical result relating the range and bounded stoppings of the stochastic integral is proved.
On approximately inner automorphisms of certain crossed product 𝐶*-algebras
Authors: Dădărlat Marius , Pasnicu Cornel
Published: July 2010

Let

G
G

be a compact connected topological group having a dense subgroup isomorphic to

Z

\mathbf {Z}

. Let

C
(
G
)

∝

⋊

Z

C(G) \stackrel {\rtimes }{\propto } \mathbf {Z}

be the crossed product

C
∗

C^*

-algebra of

C
(
G
)

C(G)

with

Z

\mathbf {Z}

, where

Z

\mathbf {Z}

acts on

G
G

by rotations. Automorphisms of

C
(
G
)

∝

⋊

Z

C(G) \stackrel {\rtimes }{\propto } \mathbf {Z}

leaving invariant the canonical copy of

C
(
G
)

C(G)

are shown to be approximately inner iff they act trivially on

K
1

(
C
(
G
)

∝

⋊

Z

)

K_1(C(G) \stackrel {\rtimes }{\propto } \mathbf {Z})

.
Singular integrals with power weights
Authors: Hofmann Steve
Published: July 2010

This note contains proofs of weighted weak-type

(
1
,
1
)

(1,1)

and weighted

L
p

{L^p}

inequalities (with power weights

|
x
|

α

{\left | x \right |^\alpha }

) for singular integrals whose kernels satisfy Hörmander’s condition, and also various size conditions. Some counter-examples are also given, yielding sharp results.
Integrability of double lacunary sine series
Authors: Móricz Ferenc
Published: July 2010

We consider mainly the series

∑

∑

a

j
k

sin
⁡

2
j

x
sin
⁡

2
k

y

\sum {\sum {{a_{jk}}\sin {2^j}x\sin {2^k}y} }

, which converges to a finite function

f
(
x
,
y
)

f(x,y)

a.e. if

∑

∑

a

j
k

2

>
∞

\sum {\sum {a_{jk}^2 > \infty } }

. We show that the (Lebesgue) integrability of

x

−
1

y

−
1

f
(
x
,
y
)

{x^{ - 1}}{y^{ - 1}}f(x,y)

over the two-dimensional torus is essentially controlled by the quantity
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript m equals 1 Overscript normal infinity Endscripts sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts left-parenthesis sigma-summation Underscript j equals m Overscript normal infinity Endscripts sigma-summation Underscript k equals n Overscript normal infinity Endscripts a Subscript j k Superscript 2 Baseline right-parenthesis Superscript 1 slash 2">

∑

m
=
1

∞

∑

n
=
1

∞

(

∑

j
=
m

∞

∑

k
=
n

∞

a

j
k

2

)

1

/

2

\sum \nolimits _{m = 1}^\infty {\sum \nolimits _{n = 1}^\infty {{{\left ( {\sum \nolimits _{j = m}^\infty {\sum \nolimits _{k = n}^\infty {a_{jk}^2} } } \right )}^{1/2}}} }

. Our result is an extension of the corresponding one by M. C. Weiss [4] from one-dimensional to two-dimensional lacunary sine series.
The Cesàro operator is bounded on 𝐻¹
Authors: Siskakis Aristomenis G.
Published: July 2010

The purpose of this note is to give a direct proof that the Cesàro operator is bounded on the Hardy space

H
1

{H^1}

.
Some random fixed point theorems for condensing and nonexpansive operators
Authors: Xu Hong Kun
Published: July 2010

Some random versions of deterministic fixed point theorems for condensing and nonexpansive operators are obtained.
The construction of global attractors
Authors: Barge Marcy , Martin Joe
Published: July 2010

The purpose of this note is to show that every inverse limit space of an interval mapping can be realized as a global attractor for a homeomorphism of the plane.
A characterization of sums of 2𝑛th powers of global meromorphic functions
Authors: Ruiz Jesús M.
Published: July 2010

Let

M
M

be a real analytic manifold. In this note we prove Theorem. Let

X
X

be a compact analytic set of

M
M

and

∑
\sum

its singular locus. Then, a meromorphic function

h
h

on

X
X

is a sum of

2
n

2n

-th powers of meromorphic functions if and only if, for every analytic curve

σ
:

(

−
ε
,
ε

)

→
X

\sigma :\left ( { - \varepsilon ,\varepsilon } \right ) \to X

not contained in

∑
\sum

, it holds

h
∘
σ
=
a

t
m

+
⋯

h \circ \sigma = a{t^m} + \cdots

, with

a
>
0

a > 0

and

2
n

2n

dividing

m
m

.
A square shape of the graph of iterates of multifunctions: a complete controllability result
Authors: Leizarowitz Arie
Published: July 2010

We consider a set valued function

T
:
K
→

2
K

T:K \to {2^K}

from a domain

K
K

into itself. We look for conditions under which the graph of

T

(
n
)

{T^{(n)}}

(the

n
n

th iterate of

T
T

) will be equal to

K
×
K

K \times K

for some integer

n
n

. When the graph of

T
T

is convex a sufficient (though not necessary) condition is that neither

T
(
x
)

T(x)

nor

T

−
1

(
x
)

{T^{ - 1}}(x)

are contained in the boundary of

K
K

whenever

x
x

is there. We show that a necessary and sufficient condition is that there are neither forward nor backward trajectories which remain in the boundary for all times. In the introduction, we remark on the significance of this problem for the study of infinite-horizon control systems.
Projectivity of the Whitehead product in spheres
Authors: Lin Wên Hsiung
Published: July 2010

The Whitehead square
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket l Subscript 2 Sub Superscript i Subscript minus 1 Baseline comma l Subscript 2 Sub Superscript i Subscript minus 1 Baseline right-bracket element-of pi Subscript 2 Sub Superscript i plus 1 Subscript minus 3 Baseline left-parenthesis upper S Superscript 2 Super Superscript i Superscript minus 1 Baseline right-parenthesis">

[

l

2
i

−
1

,

l

2
i

−
1

]

∈

π

2

i
+
1

−
3

(

S

2
i

−
1

)

\left [ {{l_{{2^i} - 1}},{l_{{2^i} - 1}}} \right ] \in {\pi _{{2^{i + 1}} - 3}}({S^{{2^i} - 1}})

is shown to be projective for

i
≤
10

i \leq 10

.
Finite projectivity and contravariant finiteness
Authors: Igusa K. , Smalø S. O. , Todorov G.
Published: July 2010

Let

Λ
\Lambda

be an artin algebra, let

mod
Λ

\bmod \Lambda

be the category of finitely generated

Λ
−

modules

\Lambda - {\text {modules}}

, and let

p

d
1

mod
Λ

p{d_1} \bmod \Lambda

be the subcategory of

mod
Λ

\bmod \Lambda

consisting of the modules of projective dimension less than or equal to one. In this paper it is proved that if the projective dimension of the injective envelope of

Λ
\Lambda

as a

Λ
−

module

\Lambda - {\text {module}}

is less than or equal to one, then

p

d
1

mod
Λ

p{d_1}\bmod \Lambda

is contravariantly finite in

mod
Λ

\bmod \Lambda

. It also contains an example of finitistic projective dimension one where

p

d
1

mod
Λ

p{d_1}\bmod \Lambda

is not contravariantly finite in

mod
Λ

\bmod \Lambda

.
Algebras with large homological dimensions
Authors: Kirkman Ellen , Kuzmanovich James
Published: July 2010

An example is given of a semiprimary ring with infinite finitistic dimension. The construction shows that the global dimensions of finite dimensional algebras of finite global dimension cannot be bounded by a function of only Loewy length and the number of nonisomorphic simple modules.
A finiteness property of affine algebras
Authors: Bass Hyman
Published: July 2010

Let

A
A

be a finitely presented reduced commutative

k
k

-algebra and

J
J

an ideal in

A
A

. If

A
A

embeds in

A

/

J

A/J

then

J
=
0

J = 0

.
The degree of smooth non-arithmetically Cohen-Macaulay threefolds in 𝑃⁵
Authors: Miró-Roig Rosa M.
Published: July 2010

In [B], Banica considers the problem of determining the integers

d
d

such that there are smooth threefolds which are not arithmetically Cohen-Macaulay. Moreover, he gives a partial answer to this question. In this note, using liaison, we will complete his answer.
An isomorphism theorem for commutative modular group algebras
Authors: Ullery William
Published: July 2010

For each positive integer

n
n

and limit ordinal

μ
\mu

, a new class of abelian

p
p

-groups, called

A
n

(
μ
)

{A_n}(\mu )

-groups, are introduced. These groups are shown to be uniquely determined up to isomorphism by numerical invariants which include, but are not restricted to, their Ulm-Kaplansky invariants. As an application of this uniqueness theorem, we prove an isomorphism result for group algebras: Let

H
H

be an

A
n

(
μ
)

{A_n}(\mu )

-group and

F
F

a field of characteristic

p
p

. It is shown that if

K
K

is a group such that the group algebras FH and FK are

F
F

-isomorphic, then

H
H

and

K
K

are isomorphic.
Permutation polynomials and resolution of singularities over finite fields
Authors: Wan Da Qing
Published: July 2010

A geometric approach is introduced to study permutation polynomials over a finite field. As an application, we prove that there are no permutation polynomials of degree

2
l

2l

over a large finite field, where

l
l

is an odd prime. This proves that the Carlitz conjecture is true for

n
=
2
l

n = 2l

. Previously, the conjecture was known to be true only for

n
≤
16

n \leq 16

.
A note on graded algebras
Authors: Puczyłowski Edmund R.
Published: July 2010

It is proved that if an algebra over a field of characteristic

p
>
0

p > 0

is graded by a finite

p
p

-group, then its ideals having nilpotent intersection with the identity component are nilpotent themselves.
The index of normal Fredholm elements of 𝐶*-algebras
Authors: Mingo J. A. , Spielberg J. S.
Published: July 2010

Examples are given of normal elements of

C
∗

{C^*}

-algebras that are invertible modulo an ideal and have nonzero index, in contrast to the case of Fredholm operators on Hilbert space. It is shown that this phenomenon occurs only along the lines of these examples.
A characterization of spectral operators
Authors: Radjabalipour Mehdi
Published: July 2010

A characterization of spectral operators due to N. Dunford is simplified. Especially, his complicated Condition (D) is replaced by a very simple one.
Factorization of quasi-differential operators
Authors: Everitt W. N. , Muldowney James S. , Thandi Neeza
Published: July 2010

A quasi-differential generalization of operators of the form

l
n

u
=

u

(
n
)

+

p
1

u

(
n
−
1
)

+
⋯
+

p
n

u

{l_n}u = {u^{(n)}} + {p_1}{u^{(n - 1)}} + \cdots + {p_n}u

is considered. This type of generalization was first formulated by M. Bôcher (1913). A result of A. Zettl (1971) giving a necessary and sufficient condition that a differential operator

l
n

{l_n}

be factorable into a product of lower order differential operators is extended to quasi-differential expressions.
Oscillations in neutral equations with periodic coefficients
Authors: Ladas G. , Philos Ch. G. , Sficas Y. G.
Published: July 2010

We obtain a necessary and sufficient condition for the oscillation of all solutions of the neutral delay differential equation: (1)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d Over d t EndFraction left-bracket x left-parenthesis t right-parenthesis plus p x left-parenthesis t minus tau right-parenthesis right-bracket plus upper Q left-parenthesis t right-parenthesis x left-parenthesis t minus sigma right-parenthesis equals 0 comma">

d

d
t

[
x
(
t
)
+
p
x
(
t
−
τ
)
]
+
Q
(
t
)
x
(
t
−
σ
)
=
0
,

\tfrac {d}{{dt}}[x(t) + px(t - \tau )] + Q(t)x(t - \sigma ) = 0,

\]
where

p
∈

R

,
Q
∈
C
[
[
0
,
∞
)
,

R

+

]
,
Q

p \in {\mathbf {R}},Q \in C[[0,\infty ),{{\mathbf {R}}^ + }],Q

is

ω
\omega

-periodic with

ω
>
0
,
Q
(
t
)
[
u
n
k
]
0

\omega > 0,Q(t)[unk]0

for

t
≧
0

t \geqq 0

, and there exist positive integers

n
1

{n_1}

and

n
2

{n_2}

such that

τ
=

n
1

ω

\tau = {n_1}\omega

and

σ
=

n
2

ω

\sigma = {n_2}\omega

. More precisely we show that every solution of (1) oscillates if and only if every solution of an associated neutral equation with constant coefficients oscillates.
An end-faithful spanning tree counterexample
Authors: Seymour Paul , Thomas Robin
Published: July 2010

We find an infinitely-connected graph in which every spanning tree has a

2
2

-way infinite path. This disproves Halin’s well-known "end-faithful spanning tree" conjecture and also disproves a recent conjecture of Širáň.
Properties of locally H-closed spaces
Authors: Girou Mike
Published: July 2010

This paper investigates the properties of locally

H
H

-closed spaces with regard to extensions, subspaces, and functions. We solve the

H
H

-closed extension remainder problem by showing that a space is locally

H
H

-closed if and only if it has a

θ
\theta

-closed remainder in some

H
H

-closed extension. In fact, an

H
H

-closed space is Urysohn iff every

H
H

-closed subspace is

θ
\theta

-closed. We solve the locally

H
H

-closed subspace problem by giving a necessary and sufficient condition for a subspace of a locally

H
H

-closed space to be locally

H
H

-closed. In particular, an open subspace of a locally

H
H

-closed space is locally

H
H

-closed if and only if its boundary is a

θ
\theta

-closed subspace of its closure. An

H
H

-closed space is shown to be compact if and only if every open subset is locally

H
H

-closed. A retract of a locally

H
H

-closed space is locally

H
H

-closed.
A Fréchet-Schwartz space with basis having a complemented subspace without basis
Authors: Taskinen Jari
Published: July 2010

Using a method introduced by Pelczyński we show that for a nuclear Fréchet space

E
E

without basis we can find a Fréchet-Schwartz space

F
F

with basis containing a complemented, isomorphic copy of

E
E

.
Some remarks about Rosen’s functions
Authors: Grande Zbigniew
Published: July 2010

The main result is: Each Baire 2 function

f
:
I
→
R

f:I \to R

whose set of continuity points is dense is the pointwise limit of a sequence of Darboux Baire

1
2

\tfrac {1}{2}

functions.
Nonintegrability of superharmonic functions
Authors: Suzuki Noriaki
Published: July 2010

In this article we prove the following: If

u
u

is a nonzero superharmonic function on a proper subdomain

D
D

in

R
n

{R^n}

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper D Endscripts StartAbsoluteValue u left-parenthesis x right-parenthesis EndAbsoluteValue Superscript p Baseline delta Subscript upper D Baseline left-parenthesis x right-parenthesis Superscript n p minus n minus 2 p Baseline d x equals normal infinity comma">

∫
D

|

u
(
x
)

|

p

δ
D

(
x

)

n
p
−
n
−
2
p

d
x
=
∞
,

{\int _D {\left | {u(x)} \right |} ^p}{\delta _D}{(x)^{np - n - 2p}}dx = \infty ,

\]
where

0
>
p
≤
1

0 > p \leq 1

and

δ
D

(
x
)

{\delta _D}(x)

denotes the distance between

x
x

and the boundary of

D
D

.
Pseudomoments for generalized domains of attraction
Authors: Meerschaert Mark M.
Published: July 2010

In this paper we resolve a conjecture of Weiner (1987) concerning the existence of the so-called pseudomoments for a random vector which belongs to the generalized domain of attraction of an operator-stable law. The proof is a direct consequence of the spectral decomposition theorem for generalized domains of attraction in Meerschaert (1991).
When are touchpoints limits for generalized Pólya urns?
Authors: Pemantle Robin
Published: July 2010

Hill, Lane, and Sudderth (1980) consider a Pólya-like urn scheme in which

X
0

,

X
1

,
…

{X_0},{X_1}, \ldots

, are the successive proportions of red balls in an urn to which at the

n
n

th stage a red ball is added with probability

f
(

X
n

)

f({X_n})

and a black ball is added with probability

1
−
f
(

X
n

)

1 - f({X_n})

. For continuous

f
f

they show that

X
n

{X_n}

converges almost surely to a random limit

X
X

which is a fixed point for

f
f

and ask whether the point

p
p

can be a limit if

p
p

is a touchpoint, i.e.

p
=
f
(
p
)

p = f(p)

but

f
(
x
)
>
x

f(x) > x

for

x
≠
p

x \ne p

in a neighborhood of

p
p

. The answer is that it depends on whether the limit of

(
f
(
x
)
−
x
)

/

(
p
−
x
)

(f(x) - x)/(p - x)

is greater or less than 1/2 as

x
x

approaches

p
p

from the side where

(
f
(
x
)
−
x
)

/

(
p
−
x
)

(f(x) - x)/(p - x)

is positive.
Length functions on integral domains
Authors: Anderson David F. , Pruis Paula
Published: July 2010

Let

R
R

be an integral domain and

x
∈
R

x \in R

which is a product of irreducible elements. Let

l
(
x
)

l(x)

and

L
(
x
)

L(x)

denote respectively the inf and sup of the lengths of factorizations of

x
x

into a product of irreducible elements. We show that the two limits,

l
¯

(
x
)

\bar l(x)

and

L
¯

(
x
)

\bar L(x)

, of

l
(

x
n

)

/

n

l({x^n})/n

and

L
(

x
n

)

/

n

L({x^n})/n

, respectively, as

n
n

goes to infinity always exist. Moreover, for any

0
≤
α
≤
1
≤
β
≤
∞

0 \leq \alpha \leq 1 \leq \beta \leq \infty

, there is an integral domain

R
R

and an irreducible

x
∈
R

x \in R

such that

l
¯

(
x
)
=
α

\bar l(x) = \alpha

and

L
¯

(
x
)
=
β

\overline L (x) = \beta

.
Product shifts on 𝐵(𝐻)
Authors: Stacey P. J.
Published: July 2010

A shift on

B
(
H
)

B(H)

is a

∗
*

-endomorphism

α
\alpha

for which

∩
r

α
r

(
B
(
H
)
)
=

C

P

{ \cap _r}{\alpha ^r}(B(H)) = \mathbb {C}P

for some projection

P
P

. The paper discusses some aspects of the classification of shifts on

B
(
H
)

B(H)

up to conjugacy by

∗
*

-automorphisms, with a focus on the shifts arising from an infinite tensor product decomposition of

H
H

.
A trace formula for two unitary operators with rank one commutator
Authors: Boyadzhiev Khristo N.
Published: July 2010

We give a short and independent proof of the Carey-Pincus trace formula for pairs of unitary operators

U
,
V

U,V

with

Rank
⁡
[
U
,
V
]
=
1

\operatorname {Rank} [U,V] = 1

.
The super GAGA principle and families of super Riemann surfaces
Authors: Topiwala Pankaj , Rabin Jeffrey M.
Published: July 2010

We extend the GAGA principle, the Kodaira embedding theorem, and Chow’s lemma to supergeometry and conclude that families of super Riemann surfaces are locally algebraic.
Complete metrics with nonpositive curvature on the disk
Authors: Duc Duong Minh
Published: July 2010

We consider complete metrics on the unit disk in

R

2

{\mathbb {R}^2}

that are equivalent and conformal to the Poincaré metric.
Group-valued charges: common extensions and the infinite Chinese remainder property
Authors: Rao K. P. S. Bhaskara , Shortt R. M.
Published: July 2010

We exhibit two consistent, integer-valued charges (finitely additive measures) which do not have a common, integer-valued extension. More generally, after introducing the notion of an infinitary Chinese remainder property for Abelian groups, we show that if a group has the common extension property, then the group must have the infinite Chinese remainder property. The class of groups with the common extension property is characterised as coincident with the class of cotorsion groups.
On intersection of compacta in Euclidean space. II
Authors: Dranishnikov A. N.
Published: July 2010

Suppose that

X
X

is a compact subset of

n
n

-dimensional Euclidean space

R

n

{\mathbb {R}^n}

. If every map

f
:
Y
→

R

n

f:Y \to {\mathbb {R}^n}

of a compactum

Y
Y

can be approximated by a map avoiding

X
X

then dim

X
×
Y
>
n

X \times Y > n

.
Algebraic curves in 𝑅𝑃(1)×𝑅𝑃(1)
Authors: Gilmer Patrick
Published: July 2010

We derive the analogs for curves in

R
P
(
1
)
×
R
P
(
1
)

RP(1) \times RP(1)

of Fiedler’s congruence for curves in

R
P
(
2
)

RP(2)

. We also prove a result on the complex orientations of dividing curves in

R
P
(
1
)
×
R
P
(
1
)

RP(1) \times RP(1)

.
Finitely cyclic homogeneous continua
Authors: Krupski Paweł , Rogers James T.
Published: July 2010

A curve is finitely cyclic if and only if it is the inverse limit of graphs of genus

≤
k

\leq k

, where

k
k

is some integer. In this paper it is shown that if

X
X

is a homogeneous finitely cyclic curve that is not tree-like, then

X
X

is a solenoid or

X
X

admits a decomposition into mutually homeomorphic, homogeneous, tree-like continua with quotient space a solenoid. Since the Menger curve is homogeneous, the restriction to finitely cyclic curves is essential.
“Lebesgue measure” on 𝑅^{∞}
Authors: Baker Richard
Published: July 2010

We construct a translation invariant Borel measure

λ
\lambda

on

R

∞

=

∏

i
=
1

∞

R

{{\mathbf {R}}^\infty } = \prod _{i = 1}^\infty {\mathbf {R}}

such that for any infinite-dimensional rectangle
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals product Underscript i equals 1 Overscript normal infinity Endscripts left-parenthesis a Subscript i Baseline comma b Subscript i Baseline right-parenthesis comma negative normal infinity greater-than a Subscript i Baseline less-than-or-equal-to b Subscript i Baseline greater-than plus normal infinity">

R
=

∏

i
=
1

∞

(

a
i

,

b
i

)
,
−
∞
>

a
i

≤

b
i

>
+
∞

R = \prod _{i = 1}^\infty ({a_i},{b_i}), - \infty > {a_i} \leq {b_i} > + \infty

, if

0
≤

∏

i
=
1

∞

(

b
i

−

a
i

)
>
+
∞

0 \leq \prod _{i = 1}^\infty ({b_i} - {a_i}) > + \infty

, then

λ
(
R
)
=

∏

i
=
1

∞

(

b
i

−

a
i

)

\lambda (R) = \prod _{i = 1}^\infty ({b_i} - {a_i})

. Because

R

∞

{{\mathbf {R}}^\infty }

is an infinite-dimensional locally convex topological vector space, the measure

λ
\lambda

can not be

σ
\sigma

-finite.
The complexity of a module and elementary abelian subgroups: a geometric approach
Authors: Symonds Peter
Published: July 2010

We present a proof of the theorem of Alperin and Evens that the complexity of a module is determined by the complexities of its restrictions to elementary abelian subgroups. We use only well-known properties of the spectral sequence.
Finite cyclic subgroups determine the spectrum of the equivariant 𝐾-theory
Authors: Bojanowska Agnieszka
Published: July 2010

Equivariant maps inducing an equivalence of the categories of components of the fixed point sets of topologically cyclic subgroups are considered. It is shown that they are the same as those inducing an equivalence of the categories of components of the fixed point sets of finite cyclic subgroups. It follows that equivariant maps inducing a bijection of maximal ideals of the appropriate equivariant

K
K

-theory rings coincide with those which give bijection on the sets of all prime ideals. As a corollary we obtain that a group homomorphism inducing bijection of maximal ideals of the representation rings is an isomorphism.
The action of the Steenrod squares on the modular invariants of linear groups
Authors: Hung Nguyên H. V.
Published: July 2010

We compute the action of the Steenrod squares on the Dickson invariants of the group

G

L
n

=
G
L
(
n
,

Z

/

2
)

G{L_n} = GL(n,{\mathbf {Z}}/2)

and the Mùi invariants of the subgroup

T
n

{T_n}

consisting of all upper triangular matrices with 1 on the main diagonal. Our method is very elementary. Roughly speaking, we read off the above action from the expansion of the Mùi invariants in terms of Dickson and Mùi invariants of fewer variables.
A characterization of spherical series representations of the free group
Authors: Kuhn Gabriella , Steger Tim
Published: July 2010

The spherical series representations of a free nonabelian group are here characterized as those irreducible unitary representations possessing a certain type of matrix coefficient. We make some conjectures on the problem of finding a natural boundary realization for a general irreducible representation of the free group, supposed to be weakly contained in the regular representation.
On the cohomology ring of hyperplane complements
Authors: Jozsa Richard , Rice John
Published: July 2010

Using only the long exact sequence of local cohomology, we give a brief derivation of Orlik and Solomon’s presentation for the cohomology ring of a complement of hyperplanes in a vector space.
On the relative strength of two absolute summability methods
Authors: Bor Hüseyin
Published: July 2010

In this paper we prove a theorem concerning the relative strength of

|

R
,

p
n

|

k

{\left | {R,{p_n}} \right |_k}

and

|

R
,

q
n

|

k

\left | {R,{q_n}} \right | _k

summability methods,

k
>
1

k > 1

, that generalizes a result of Bosanquet [1].
The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity
Authors: Penot Jean-Paul
Published: July 2010

We give geometric proofs of recent results of G. Beer [13]: the Young-Fenchel correspondence

f
→

f
∗

f \to {f^*}

is bicontinuous on the space of closed proper convex functions on a normed vector space

X
X

endowed with the epidistance topology and the polarity operation is continuous on the space of closed convex subsets of

X
X

with the bounded Hausdorff topology. Our methods are in the spirit of a famous result due to Walkup and Wets [31] about the isometric character of the polarity for closed convex cones. We also prove that the Hausdorff distance associated with the cosmic distance on the space of convex subsets of a normed vector space induces the bounded -Hausdorff topology. This shows a link between Beer’s results and the continuity results of [2].
Homogeneous polynomials and invariant subspaces in the polydisc. II
Authors: Nakazi Takahiko , Takahashi Katsutoshi
Published: July 2010

We determine the invariant subspaces

M
M

of

L
2

(

T
2

)

{L^2}({T^2})

for which there is a subspace

S
S

of

M
M

and a positive integer

r
r

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M equals sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts circled-plus left-bracket sigma-summation Underscript j equals 0 Overscript n Endscripts z Superscript j Baseline w Superscript r left-parenthesis n minus j right-parenthesis Baseline upper S right-bracket comma">

M
=

∑

n
=
0

∞

⊕

[

∑

j
=
0

n

z
j

w

r
(
n
−
j
)

S

]

,

M = \sum \limits _{n = 0}^\infty { \oplus \left [ {\sum \limits _{j = 0}^n {{z^j}{w^{r(n - j)}}S} } \right ]} ,

\]
where, for a subspace

A
A

of

L
2

,

T
2

,
[
A
]

{L^2},{T^2},[A]

denotes the closure of

A
A

.
A note on inner actions of Hopf algebras
Authors: Caenepeel Stefaan
Published: July 2010

Let

H
H

be a commutative, cocommutative, and faithfully projective Hopf algebra over a commutative ring

R
R

. A twisted version of inner action of a Hopf algebra, called

H
H

-inner action is introduced, and it is shown that

H
H

acts

H
H

-innerly on an

H
H

-Azumaya algebra, if

Pic
⁡
(

H
∗

)

\operatorname {Pic} ({H^ * })

is trivial.
On the weighted estimate of the solution associated with the Schrödinger equation
Authors: Wang Si Lei
Published: July 2010

Let

u
(
x
,
t
)

u(x,t)

be the solution of the Schrödinger equation with initial data

f
f

in the Sobolev space

H

−
1
+
a

/

2

(

R

n

)

{H^{ - 1 + a/2}}({\mathbb {R}^n})

with

a
>
1

a > 1

. This paper shows that the weighted inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript double-struck upper R Superscript n Endscripts integral Underscript double-struck upper R Endscripts StartAbsoluteValue u left-parenthesis x comma t right-parenthesis EndAbsoluteValue squared d t left-parenthesis 1 plus StartAbsoluteValue x EndAbsoluteValue right-parenthesis Superscript negative a Baseline d x less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript upper H Sub Superscript negative 1 plus a slash 2 Subscript left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis Baseline">

∫

R

n

∫

R

|

u
(
x
,
t
)

|

2

d
t

(
1
+

|
x
|

)

−
a

d
x
≤
C

‖
f
‖

H

−
1
+
a

/

2

(

R

n

)

\int _{{\mathbb {R}^n}} {\int _\mathbb {R} {{{\left | {u(x,t)} \right |}^2}dt{{(1 + \left | x \right |)}^{ - a}}dx \leq C{{\left \| f \right \|}_{{H^{ - 1 + a/2}}({\mathbb {R}^n})}}} }

is false. Another improved weighted inequality is proved for the general case.
Bounded and zero-convergent solutions of a class of Stieltjes integro-differential equations
Authors: Chen Shao Zhu , Huang Qing Guang , Erbe L. H.
Published: July 2010

We consider asymptotic properties of solutions to a class of nonlinear Stieltjes integro-differential equations. Necessary and sufficient conditions are given which guarantee that there exist solutions which do (or do not) have nonzero limits at

∞
\infty

. These extend earlier results of various authors and apply to linear and nonlinear difference equations as well.
Weight structure theorems and factorization of positive operators
Authors: Bloom Steven , Kerman Ron
Published: July 2010

We characterize the conditions under which weighted norm inequalities for a positive operator

T
T

can be obtained by interpolation with change of measure. The results are applied to the construction of all good weight pairs for

T
T

. This construction is used to show that the study of weighted norm inequalities for operators

T
T

that factor as

T
=
P
Q

T = PQ

reduce to that of the weighted norm inequalities for the factors

P
P

and

Q
Q

.