bluuebunny/crossref_metadata_2025 huggingface dataset:
The structure of the automorphism group of a free group on two generators
Authors: Đoković Dragomir Ž.
Published: June 2010

Let

F
2

=
Z
∗
Z

{F_2} = Z * Z

be a free group of rank two. We show that

Aut
⁡

F
2

{\operatorname {Aut} F_2}

can be built up from cyclic groups by using only the free products and semidirect products. Explicitly we have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u t upper F 2 equals left-parenthesis left-parenthesis upper Z asterisk upper Z right-parenthesis right-normal-factor-semidirect-product left-parenthesis upper Z 3 asterisk upper Z 3 right-parenthesis right-parenthesis right-normal-factor-semidirect-product left-parenthesis upper Z 4 right-normal-factor-semidirect-product upper Z 2 right-parenthesis">

Aut
⁡

F
2

=
(
(
Z
∗
Z
)
⋊
(

Z
3

∗

Z
3

)
)
⋊
(

Z
4

⋊

Z
2

)

\operatorname {Aut} {F_2} = ((Z * Z) \rtimes ({Z_3} * {Z_3})) \rtimes ({Z_4} \rtimes {Z_2})

. As a corollary we obtain a simple presentation of

Aut
⁡

F
2

\operatorname {Aut} {F_2}

.
Coefficients and integral means of some classes of analytic functions
Authors: Sheil-Small T.
Published: June 2010

The sharp coefficient bounds for the classes

V
k

{V_k}

of functions of bounded boundary rotation are obtained by a short and elementary argument. Elementary methods are also applied for the coefficients of related classes characterised by a generalised Kaplan condition. The result
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 plus x z right-parenthesis Superscript alpha Baseline left-parenthesis 1 minus z right-parenthesis Superscript negative beta Baseline much-less-than left-parenthesis 1 plus z right-parenthesis Superscript alpha Baseline left-parenthesis 1 minus z right-parenthesis Superscript negative beta">

(
1
+
x
z

)
α

(
1
−
z

)

−
β

≪

(
1
+
z

)
α

(
1
−
z

)

−
β

{(1 + xz)^\alpha }{(1 - z)^{ - \beta }} \ll {(1 + z)^\alpha }{(1 - z)^{ - \beta }}

(

|
x
|

=
1
,
α
⩾
1
,
β
⩾
1
)

(\left | x \right | = 1,\alpha \geqslant 1,\beta \geqslant 1)

is proved simply. It is further shown that the functions

(
1
+
z

)
α

(
1
−
z

)

−
β

{(1 + z)^\alpha }{(1 - z)^{ - \beta }}

are extremal for the

p
p

th means (

p
p

an arbitrary real) of all Kaplan classes

K
(
α
,
β
)

K(\alpha ,\beta )

.
Minimal topologies of para-𝐻-closed spaces
Authors: Zahid Muhammad I.
Published: June 2010

A Hausdorff space is para-

H
H

-closed if every open cover has a locally-finite open refinement (not necessarily covering the space) whose union is dense in the space. We prove that minimal locally-

H
H

-closed, minimal locally-para-

H
H

-closed and minimal para-

H
H

-closed spaces are all minimal-Hausdorff. We also show that para-

H
H

-closed-closed spaces are

H
H

-closed.
On group d.g. near-rings
Authors: Mahmood S. J. , Meldrum J. D. P.
Published: June 2010

Meldrum has generalized the idea of a group ring and has defined a group d.g. near-ring for a faithful d.g. near-ring

(
R
,
S
)

(R,S)

on a multiplicative group

G
G

. In this paper we generalize this idea even further and define a group d.g. near-ring for an arbitrary d.g. near-ring. We also prove some results about the additive group of this d.g. near-ring similar to those proved by Meldrum for a group d.g. near-ring of a faithful d.g. near-ring.
Noetherian subsets of prime spectra
Authors: Hanna Charles C. , Johnson Jon L.
Published: June 2010

If

X
X

is a noetherian subspace of Spec

R
R

, the set of primes of

R
[
x
]

R[x]

lying over

X
X

is also noetherian. A simple consequence is the theorem of Ohm and Pendleton that a ring module-finite over a

j
j

-noetherian ring is

j
j

-noetherian.
Finite-dimensional representations of JB algebras
Authors: Behncke Horst
Published: June 2010

Let

Λ
\Lambda

be the set of isomorphism classes of finite dimensional JB factors, ordered by inclusion. For any

α
∈
Λ

\alpha \in \Lambda

and JB algebra

A

\mathcal {A}

denote by

α

A

_\alpha \mathcal {A}

the set of equivalence classes of factor representations of

A

\mathcal {A}

into a JB factor of type

α
\alpha

. Then

α

A

_\alpha \mathcal {A}

is closed.
Gradient method for nondensely defined closed unbounded linear operators
Authors: Lee Sung J. , Nashed M. Zuhair
Published: June 2010

The paper establishes the convergence of the steepest descent method for least-squares solutions of operator equations in Hilbert spaces for any (nondensely defined, unbounded) closed linear operator with closed range. This is done by using a graph topology, an explicit graph topology adjoint, and existing theory of steepest descent for bounded linear operators.
Spheres in infinite-dimensional normed spaces are Lipschitz contractible
Authors: Benyamini Y. , Sternfeld Y.
Published: June 2010

Let

X
X

be an infinite-dimensional normed space. We prove the following: (i) The unit sphere

{
x
∈
X
:

‖
x
‖

=
1
}

\{ x \in X:\left \| x \right \| = 1\}

is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of

X
X

onto the unit sphere. (iii) There is a Lipschitz map

T
T

of the unit ball into itself without an approximate fixed point, i.e.

inf
{

‖

x
−
T
x

‖

:

‖
x
‖

⩽
1
}
>
0

\inf \{ \left \| {x - Tx} \right \|:\left \| x \right \| \leqslant 1\} > 0

.
A geometric characterization of 𝑁⁺ domains
Authors: Ahern Patrick , Cohn William
Published: June 2010

A connected open set

O

⊆

C

\mathcal {O} \subseteq {\mathbf {C}}

is called an

N
+

{N^ + }

domain if every holomorphic function defined in the unit disc and taking values in

O

\mathcal {O}

is necessarily in the Smirnov class

N
+

{N^ + }

. We show that

O

\mathcal {O}

is an

N
+

{N^ + }

domain if and only if

∞
\infty

is a regular point for the solution of the Dirichlet problem for

O

\mathcal {O}

. We get a similar characterization when

N
+

{N^ + }

is replaced by the class of outer functions.
On analytic injective matrix functions
Authors: Bshouty D. , Hengartner W.
Published: June 2010

An ingenious generalization of univalent functions of one complex variable into matrix functions was introduced by B. Schwarz, along with two conjectures corresponding to the first coefficient problem and the second coefficient problem for the traditional classes

Σ
\Sigma

and

S
S

, respectively. We exhibit two counterexamples for these conjectures.
A note on polynomial operator approximation
Authors: Smith R. R. , Ward J. D.
Published: June 2010

An example is given of an operator

T
T

contained in a block-diagonal algebra of operators

A

\mathcal {A}

, an ideal

J
⊂

A

J \subset \mathcal {A}

and an infinite set of polynomials

P

\mathcal {P}

for which there is a

K
∈
J

K \in J

satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar p left-parenthesis upper T plus upper K right-parenthesis double-vertical-bar Subscript script upper A Baseline equals double-vertical-bar p left-parenthesis upper T plus upper K right-parenthesis double-vertical-bar Subscript script upper A slash upper J">

‖

p
(
T
+
K
)

‖

A

=

‖

p
(
T
+
K
)

‖

A

/

J

{\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}

for any finite subset of

P

\mathcal {P}

but for which there is no

K
∈
J

K \in J

satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar p left-parenthesis upper T plus upper K right-parenthesis double-vertical-bar Subscript script upper A Baseline equals double-vertical-bar p left-parenthesis upper T plus upper K right-parenthesis double-vertical-bar Subscript script upper A slash upper J">

‖

p
(
T
+
K
)

‖

A

=

‖

p
(
T
+
K
)

‖

A

/

J

{\left \| {p(T + K)} \right \|_{\mathcal {A}}} = {\left \| {p(T + K)} \right \|_{\mathcal {A}/J}}

for all

p
∈

P

p \in \mathcal {P}

. This sheds some light on a well-known question of C. Olsen.
Strong convergence of martingales in von Neumann algebras
Authors: Tsukada Makoto
Published: June 2010

In this paper we prove strong and

L
1

{L^1}

-norm convergence of martingales with respect to a faithful normal semifinite weight on a von Neumann algebra.
No homogeneous tree-like continuum contains an arc
Authors: Hagopian Charles L.
Published: June 2010

In 1960 R. H. Bing [4, p. 210] asked the following question. "Is there a homogeneous tree-like continuum that contains an arc?" We answer this question in the negative. This result generalizes Bing’s theorem [4, p. 229] that every atriodic homogeneous tree-like continuum is arcless.
Simultaneous attainability of central Lyapunov and Bohl exponents for ODE linear systems
Authors: Vinograd Robert E.
Published: June 2010

Millionščikov’s Accessibility Theorem for the central Lyapunov exponent of a linear ODE system is extended to simultaneous attainability of both central Lyapunov and Bohl exponents.
On a singular elliptic equation
Authors: Ni Wei Ming
Published: June 2010

In this paper, we study the singular elliptic equation

L
u
+
K
(
x
)

u
p

=
0

Lu + K(x){u^p} = 0

, where

L
L

is a uniformly elliptic operator of divergence form,

p
>
1

p > 1

and

K
(
x
)

K(x)

has a singularity at the origin. We prove that this equation does not possess any positive (local) solution in any punctured neighborhood of the origin if there exist two constants

C
1

{C_1}

,

C
2

{C_2}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 1 StartAbsoluteValue x EndAbsoluteValue Superscript sigma Baseline greater-than-or-slanted-equals upper K left-parenthesis x right-parenthesis greater-than-or-slanted-equals upper C 2 StartAbsoluteValue x EndAbsoluteValue Superscript sigma">

C
1

|

x

|

σ

⩾
K
(
x
)
⩾

C
2

|

x

|

σ

{C_1}|x{|^\sigma } \geqslant K(x) \geqslant {C_2}|x{|^\sigma }

near the origin for some

σ
⩽
−
2

\sigma \leqslant - 2

(with no other condition on the gradient of

K
K

). In fact, an integral condition is derived.
The secondary characteristic classes of solvable foliations
Authors: Pelletier William T.
Published: June 2010

The secondary characteristic classes of a codimension

q
q

foliation of a manifold

M
M

are certain cohomology classes of

M
M

which are constructed from the curvature matrix of a torsion-free connection on the normal bundle of the foliation. We consider the foliation of a real or complex Lie group

G
G

by the left cosets of a closed connected subgroup

H
H

and any foliation obtained from it through dividing by a discrete subgroup

Γ
⊂
G

\Gamma \subset G

. Such foliations are homogeneous. The Fuks-Pittie conjecture for homogeneous foliations is that the secondary classes are generated by

{

h
1

h

i
2

⋯

h

i
λ

τ
1
q

}

\{ {h_1}{h_{{i_2}}} \cdots {h_{{i_\lambda }}}\tau _1^q\}

. We prove the Fuks-Pittie conjecture if

H
H

is reductive or solvable. We also prove an Addition Theorem for the classes which can be applied to reduce the general problem of calculating the secondary classes to the case in which

G
G

is semisimple.
Rotations and linkage of 2-fold Pfister forms
Authors: Fitzgerald Robert W.
Published: June 2010

We show that a pair of

2
2

-fold Pfister forms admit rotations with the same irreducible, separable characteristic polynomial if and only if they are linked.
Combinatorial examples in universal coalgebra. II
Authors: Davis Robert C.
Published: June 2010

Several new categories of graph-like structures are described. These extend and (at least in a sense) complete the list of categories cotripleable over the category of sets for which the right adjoint at 2 has four elements.
An inequality for derivatives of polynomials whose zeros are in a half-plane
Authors: Abi-Khuzam Faruk F.
Published: June 2010

Let

Q
Q

be a real polynomial of degree

N
N

all of whose zeros lie in the half-plane

Re
⁡
z
⩽
0

\operatorname {Re} z \leqslant 0

. Let

M
(
r
,
Q
)

M(r,Q)

be the maximum of

|

Q
(
z
)

|

on

|
z
|

=
r

\left | {Q(z)} \right |{\text {on}}\left | z \right | = r

and

n
(
r
,
0
)

n(r,0)

the counting function of the zeros of

Q
Q

. It is shown that the inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis r comma upper Q Superscript prime Baseline right-parenthesis less-than-or-slanted-equals left-parenthesis 2 r right-parenthesis Superscript negative 1 Baseline left-brace upper N plus n left-parenthesis r comma 0 right-parenthesis right-brace upper M left-parenthesis r comma upper Q right-parenthesis">

M
(
r
,

Q
′

)
⩽

(
2
r

)

−
1

{

N
+
n
(
r
,
0
)

}

M
(
r
,
Q
)

M(r,Q’) \leqslant {(2r)^{ - 1}}\left \{ {N + n(r,0)} \right \}M(r,Q)

holds for

r
>
0

r > 0

. It is also shown that Bernstein’s inequality characterizes polynomials.
A note on Jensen’s covering lemma
Authors: Komjáth P.
Published: June 2010

We show that Jensen’s covering lemma does not hold for order-types instead of cardinalities.
A localisation principle for quadratic spaces over Laurent extensions
Authors: Parimala Raman , Sinclair Parvin
Published: June 2010

We prove here that the localisation principle holds for anisotropic quadratic spaces over

R
[
T
,

T

−
1

]

R[T,{T^{ - 1}}]

, where

R
R

is an integral domain in which 2 is invertible. We also give an example of an isotropic quadratic space over

R
[
T
,

T

−
1

]

R[T,{T^{ - 1}}]

for which the localisation principle does not hold.
Two Tauberian theorems for nonconvolution Volterra integral operators
Authors: Gripenberg Gustaf
Published: June 2010

Two sets of sufficient conditions on the kernel

k
(
t
,
s
)

k(t,s)

are given so that one can prove that if

x
x

is a bounded function such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript StartLayout 1st Row t right-arrow normal infinity 2nd Row tau right-arrow 0 EndLayout Endscripts StartAbsoluteValue x left-parenthesis t plus tau right-parenthesis minus x left-parenthesis t right-parenthesis EndAbsoluteValue equals 0 and limit Underscript t right-arrow normal infinity Endscripts integral Subscript 0 Superscript t Baseline k left-parenthesis t comma s right-parenthesis x left-parenthesis s right-parenthesis d s">

lim

t
→
∞

τ
→
0

|

x
(
t
+
τ
)
−
x
(
t
)

|

=
0

and

lim

t
→
∞

∫
0
t

k
(
t
,
s
)
x
(
s
)
d
s

\lim \limits _{\begin {array}{*{20}{c}} {t \to \infty } \\ {\tau \to 0} \\ \end {array} } \left | {x(t + \tau ) - x(t)} \right | = 0\quad {\text {and}}\quad \lim \limits _{t \to \infty } \int _0^t {k(t,s)x(s)ds}

\]
exists, then

lim

t
→
∞

x
(
t
)

{\lim _{t \to \infty }}x(t)

exists.
CH and open subspaces of 𝐹-spaces
Authors: Dow Alan
Published: June 2010

N. J. Fine and L. Gillman showed that, if one assumes CH, each open subset of an

F
F

-space of weight

c
c

is an

F
F

-space. In this note it is shown that this fact is equivalent to CH.
On resultants
Authors: Myerson Gerald
Published: June 2010

Let

f
f

and

g
g

be polynomials with coefficients in a commutative ring

A
A

. Let

f
f

be monic. We show that the resultant of

f
f

and

g
g

equals the norm from

A
[
x
]

/(

f

)

A[x]{\text {/(}}f{\text {)}}

to

A
A

of

g
g

. As a corollary we deduce that if

c
c

is in

A
A

and also in the ideal generated by

f
f

and

g
g

, then the resultant divides

c
n

{c^n}

, where

n
n

is the degree of

f
f

.
On defining equations for the Jacobian locus in genus five
Authors: Accola Robert D. M.
Published: June 2010

In the space of principally polarized abelian varieties of dimension 5, eight special theta relations can be chosen to define eight hypersurfaces whose intersection contains the Jacobian locus as a component.
The tail 𝜎-field of a finitely additive Markov chain starting from a recurrent state
Authors: Ramakrishnan S.
Published: June 2010

For a Markov chain with an arbitrary nonempty state space, with stationary finitely additive transition probabilities and with initial distribution concentrated on a recurrent state, it is shown that the probability of every tail set is either zero or one. This generalizes and in particular gives an alternative proof of the result due to Blackwell and Freedman [1] in case the state space is countable and all transition probabilities are countably additive.
The spectral characterization of normal hyperbolicity
Authors: Swanson Richard
Published: June 2010

In many cases hyperbolicity in dynamical systems can be expressed in terms of the spectrum of some canonically associated linear operator; e.g., the linearization at a fixed point. Such a characterization is known for Anosov diffeomorphisms and flows. We construct a vector bundle map, based on the tensor product, whose spectrum is decisive for detecting the normal hyperbolicity of a flow or diffeomorphism at an invariant manifold. This resolves a problem raised by Hirsch, Pugh and Shub. In the case of flows, our operator admits an infinitesimal formulation, which allows us to prove that normally hyperbolic systems are stable under reparameterization in many cases.
A note on a problem of Robinson
Authors: Pearce Kent
Published: June 2010

Let

S

\mathcal {S}

be the usual class of univalent analytic functions on

|
z
|

>
1

\left | z \right | > 1

normalized by

f
(
0
)
=
0

f(0) = 0

and

f
′

(
0
)
=
1

f’(0) = 1

. Let

L

\mathfrak {L}

be the linear operator on

S

\mathcal {S}

given by

L

f
=

1
2

(
z
f

)
′

\mathfrak {L}f = \tfrac {1}{2}(zf)’

and let

r

S

t

{r_{{\mathcal {S}_t}}}

be the minimum radius of starlikeness of

L

f

\mathfrak {L}f

for

f
f

in

S

\mathcal {S}

. In 1947 R. M. Robinson initiated the study of properties of

L

\mathfrak {L}

acting on

S

\mathcal {S}

when he showed that

r

S

t

>
.38

{r_{{\mathcal {S}_t}}} > .38

. Later, in 1975, R. W. Barnard gave an example which showed

r

S

t

>
.445

{r_{{\mathcal {S}_t}}} > .445

. It is shown here, using a distortion theorem and Jenkin’s region of variability for

z

f
′

(
z
)

/

f
(
z
)

zf’(z)/f(z)

,

f
f

in

S

\mathcal {S}

, that

r

S

t

>
.435

{r_{{\mathcal {S}_t}}} > .435

. Also, a simple example, a close-to-convex half-line mapping, is given which again shows

r

S

t

>
.445

{r_{{\mathcal {S}_t}}} > .445

.
A gap Tauberian theorem for generalised absolute Abel summability
Authors: Sarvothaman K.
Published: June 2010

A gap Tauberian theorem for generalised absolute Abel summability

|

A
α

|

\left | {{A_\alpha }} \right |

is proved using Mel’nik’s theorem on convolution transforms.
Countable models of stable theories
Authors: Pillay Anand
Published: June 2010

The notion of a normal theory such a theory

T
,

I
(

ℵ
0

,
T
)
=
1

 or 

⩾

ℵ
0

T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}

. theorem that for superstable

T
,

I
(

ℵ
0

,
T
)
=
1

 or 

⩾

ℵ
0

T,\;I({\aleph _0},T) = 1{\text { or }} \geqslant {\aleph _0}

stronger than stability but incomparable is introduced, and it is proved that for We also include a short proof of Lachlan’s (The property of normality is to superstability.)
Almost sure stability of partial sums of uniformly bounded random variables
Authors: Hill Theodore P.
Published: June 2010

Suppose

a
1

{a_1}

,

a
2

,
…

{a_2}, \ldots

is a sequence of real numbers with

a
n

→
∞

{a_n} \to \infty

. If

(

X
1

+
⋯
+

X
n

)

/

a
n

=
α

({X_1} + \cdots + {X_n})/{a_n} = \alpha

a.s. for every sequence of independent nonnegative uniformly bounded random variables

X
1

{X_1}

,

X
2

,
…

{X_2}, \ldots

satisfying some hypothesis condition A, then for every (arbitrarily-dependent) sequence of nonnegative uniformly bounded random variables

Y
1

{Y_1}

,

Y
2

,
…
,

{Y_2}, \ldots ,

,

lim

 sup

(

Y
1

+
⋯
+

Y
n

)

/

a
n

=
α

\lim {\text { sup}}({Y_1} + \cdots + {Y_n})/{a_n} = \alpha

a.s. on the set where the conditional distributions (given the past) satisfy precisely the same condition A. If, in addition,

Σ
∞

a
n

−
2

>
∞

\Sigma ^\infty a_n^{ - 2} > \infty

, then the assumption of nonnegativity may be dropped.
Selectors for Borel sets with large sections
Authors: Maitra Ashok
Published: June 2010

We prove a result asserting the existence of a Borel selector for a Borel set in the product of two Polish spaces. This subsumes a number of results about Borel selectors for Borel sets having large sections.
On Sylow intersections in finite groups
Authors: Robinson Geoffrey R.
Published: June 2010

In general, for a given prime

p
p

and finite group

G
G

, there need not be Sylow

p
p

-subgroups

P
P

and

Q
Q

of

G
G

with

P
∩
Q
=

O
p

(
G
)

P \cap Q = {O_p}(G)

. In this paper we show that if

G
G

is

p
p

-soluble, and

p
p

is not 2 or a Mersenne prime, then such Sylow

p
p

-subgroups exist (also we give conditions guaranteeing the existence of such Sylow subgroups when

p
p

is 2 or a Mersenne prime). We also show that if

G
G

is not

p
p

-soluble, but

p
p

is odd and the components of

G

/

O
p

(
G
)

G/{O_p}(G)

are in a certain class of quasi-simple groups, then there are Sylow

p
p

-subgroups

P
P

and

Q
Q

of

G
G

with

P
∩
Q
=

O
p

(
G
)

P \cap Q = {O_p}(G)

, unless perhaps

p
p

is a Mersenne prime. When

G
G

is

p
p

-soluble, our work extends results of N. Itô [2].
Quotient groups of finite groups
Authors: Ferguson Pamela A.
Published: June 2010

Assume

H
H

and

H
0

{H_0}

are subgroups of the finite group

G
G

with

H
0

⧋

H

H_0 \triangleubar H

. Three theorems are presented which give criteria for the existence of a relative normal complement in

G
G

of

H
H

over

H
0

{H_0}

.
A test for holomorphy in the unit ball of 𝐶ⁿ
Authors: Berenstein Carlos A.
Published: June 2010

A new test for holomorphy in the unit ball of

C

n

{{\mathbf {C}}^n}

in the spirit of Morera’s theorem is given.
Analyticity preserving properties of resolvents for degenerate diffusion operators in one dimension
Authors: Tsuchiya Masaaki
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals a left-parenthesis x right-parenthesis left-parenthesis d squared slash d x squared right-parenthesis plus b left-parenthesis x right-parenthesis left-parenthesis d slash d x right-parenthesis plus c left-parenthesis x right-parenthesis">

L
=
a
(
x
)
(

d
2

/

d

x
2

)
+
b
(
x
)
(
d

/

d
x
)
+
c
(
x
)

L = a(x)({d^2}/d{x^2}) + b(x)(d/dx) + c(x)

be a diffusion operator on a compact interval

I
=
[

r
0

,

r
1

]

I = [{r_0},{r_1}]

investigated by S. N. Ethier. Here, assume that

a
a

,

b
b

and

c
c

are real analytic functions on

I
I

,

a
(
x
)
>
0

a(x) > 0

for

x
∈
(

r
0

,

r
1

)

x \in ({r_0},{r_1})

,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis r Subscript i Baseline right-parenthesis equals 0 less-than-or-slanted-equals left-parenthesis negative 1 right-parenthesis Superscript i Baseline b left-parenthesis r Subscript i Baseline right-parenthesis left-parenthesis i equals 0 comma 1 right-parenthesis">

a
(

r
i

)
=
0
⩽

(
−
1

)
i

b
(

r
i

)
(
i
=
0
,
1
)

a({r_i}) = 0 \leqslant {( - 1)^i}b({r_i})(i = 0,1)

, and both

r
i

(
i
=
0
,
1
)

{r_i}(i = 0,1)

are simple zeros of

a
(
x
)

a(x)

. It is shown that the resolvent

{

G
λ

}

\left \{ {{G_\lambda }} \right \}

for

L
L

has the analyticity preserving property for sufficiently large

λ
\lambda

, so that the equation

(
L
−
λ
)
u
=
f

(L - \lambda )u = f

is solvable in the space of real analytic functions on

I
I

. Some examples are given to show that the condition on

L
L

is best possible.
Virtual permutations of 𝑍[𝑍ⁿ] complexes
Authors: Maller Michael , Whitehead Jennifer
Published: June 2010

We extend the characterization of virtual permutation endomorphisms in the case where

∏
1

(
M
)
=

Z
n

{\prod _1}(M) = {Z^n}

. We show that for endomorphisms of

Z
[

Z
n

]

Z[{Z^n}]

complexes the appropriate eigenvalue condition is that all eigenvalues be roots of units of the group ring

Z
[

Z
n

]

Z[{Z^n}]

. Among these endomorphisms the virtual permutations are detected by

K
0

{K_0}

. The main application is in identifying Morse-Smale isotopy classes on these manifolds.
On the semisimplicity of skew polynomial rings
Authors: Ram Jai
Published: June 2010

Let

R
R

be a ring satisfying the maximal condition on annihilator left ideals and

σ
\sigma

be an automorphism of

R
R

. We show that the Jacobson radical of the skew polynomial ring

R
σ

[
x
]

{R_\sigma }[x]

is nonzero if and only if the prime radical of

R
σ

[
x
]

{R_\sigma }[x]

is nonzero. Furthermore, it is so if and only if the prime radical of

R
R

is nonzero. In general, an example is given of a commutative semisimple algebra

R
R

and an automorphism

σ
\sigma

such that

R
σ

[
x
]

{R_\sigma }[x]

is prime but the Levitzki radical of

R
σ

[
x
]

{R_\sigma }[x]

is nonzero.
On the problem of modified moments
Authors: Lasser Rupert
Published: June 2010

The problem of modified moments is studied. Let

(

P
n

(
x
)

)

n
∞

=
0

\left ( {{P_n}\left ( x \right )} \right )_n^\infty = 0

be an orthogonal polynomial sequence. Given a sequence

(

d
n

)

n
∞

=
0

\left ( {{d_n}} \right )_n^\infty = 0

of real numbers, does there exist a bounded nondecreasing function with infinitely many points of increase such that for every

n
∈

N

0

n \in {{\mathbf {N}}_0}

,

d
n

=

∫

−
∞

∞

P
n

(
x
)
d
μ
(
x
)

{d_n} = \int _{ - \infty }^\infty {{P_n}} (x)d\mu (x)

? Is there any information about the support of

μ
\mu

? A necessary and sufficient condition for the existence of such a function

μ
\mu

is given in terms of the positivity of certain determinants. For certain

(

P
n

(
x
)

)

n
=
0

∞

\left ( {{P_n}\left ( x \right )} \right )_{n = 0}^\infty

a description of the support of

μ
\mu

is established.
Dilations of 𝑉-bounded stochastic processes indexed by a locally compact group
Authors: Ylinen Kari
Published: June 2010

It is proved that a stochastic process (i.e., a Hilbert space valued function) indexed by a locally compact group is

V
V

-bounded (i.e., weakly harmonizable in an appropriate sense) if, and only if, it can be expressed as an orthogonal projection of a process whose covariance function

R
R

satisfies

R
(
s
,
t
)
=
ρ
(

t

−
1

s
)
+
ρ
(
s

t

−
1

)

R(s,t) = \rho ({t^{ - 1}}s) + \rho (s{t^{ - 1}})

for some continuous positive-definite function

ρ
\rho

. The result generalizes a well-known theorem due to H. Niemi, and depends on the noncommutative Grothendieck type inequality of G. Pisier.
The F. and M. Riesz theorem and singular measures
Authors: Saeki Sadahiro
Published: June 2010

Some generalizations of the F. and M. Riesz theorem are given.
On inductively open real functions
Authors: Ricceri Biagio
Published: June 2010

In this note, given a locally connected topological space

X
X

, we characterize those continuous and locally nonconstant real functions on

X
X

which are inductively open there.
A five-color theorem for graphs on surfaces
Authors: Hutchinson Joan P.
Published: June 2010

We prove that if a graph embeds on a surface with all edges suitably short, then the vertices of the graph can be five-colored. The motivation is that a graph embedded with short edges is locally a planar graph and hence should not require many more than four colors.
Sylvester’s problem and Motzkin’s theorem for countable and compact sets
Authors: Borwein Peter B.
Published: June 2010

The following three variations of Sylvester’s Problem are established. Let

A
A

and

B
B

be compact, countable and disjoint sets of points. (1) If

A
A

spans

E
2

{E^2}

(the Euclidean plane) then there must exist a line through two points of

A
A

that intersects

A
A

in only finitely many points. (2) If

A
A

spans

E
3

{E^3}

(Euclidean three-space) then there must exist a line through exactly two points of

A
A

. (3) If

A
∪
B

A \cup B

spans

E
2

{E^2}

then there must exist a line through at least two points of one of the sets that does not intersect the other set.
Two results concerning cardinal functions on compact spaces
Authors: Juhász I. , Szentmiklóssy Z.
Published: June 2010

We show that for

X
X

compact
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T 2 colon left-parenthesis i right-parenthesis d left-parenthesis upper X right-parenthesis less-than-or-slanted-equals s left-parenthesis upper X right-parenthesis dot ModifyingAbove upper F With caret left-parenthesis upper X right-parenthesis">

T
2

:

(
i
)

d

(
X
)

⩽
s

(
X
)

⋅

F
^

(
X
)

{T_2}:\left ( i \right )d\left ( X \right ) \leqslant s\left ( X \right ) \cdot \hat F\left ( X \right )

; (ii) if the pair

(

κ
,

F
^

(
X
)

)

\left ( {\kappa ,\hat F\left ( X \right )} \right )

is a caliber of

X
X

then

π

(
X
)

>
κ

\pi \left ( X \right ) > \kappa

. These strengthen results of Šapirovskii from [3 and 5], respectively. Moreover, (i) settles a problem raised in [2] implying that there are no compact

T
2

{T_2}

κ
\kappa

-examples for any singular cardinal

κ
\kappa

.
Every mapping of the pseudo-arc onto itself is a near homeomorphism
Authors: Smith Michel
Published: June 2010

The theorem stated in the title is proven. It follows then that the homeomorphism group of the pseudo-arc is dense in the space of continuous functions from the pseudo-arc onto itself.
Solvable linear groups over division rings
Authors: Snider Robert L.
Published: June 2010

A solvable subgroup of

G

L

n

(
D
)

{\text {G}}{{\text {L}}_n}\left ( D \right )

whose linear span is

M
n

(
D
)

{M_n}\left ( D \right )

is abelian-by-locally finite.
Meromorphic maps in the Nevanlinna class
Authors: Molzon R. E. , Patrizio G.
Published: June 2010

A meromorphic mapping from a relatively compact domain

D

\mathcal {D}

in a complex manifold to complex projective space is shown to be in the Nevanlinna class provided the mapping omits a set of hyperplanes of positive capacity. As a consequence, such mappings have admissible limits almost everywhere on

∂

D

\partial \mathcal {D}

.
Primitive obstructions in the cohomology of loopspaces
Authors: Williams Frank
Published: June 2010

Let

X
X

and

X
′

X’

be

H
H

-spaces. If

f
:
Ω
X
→
Ω

X
′

f:\Omega X \to \Omega X’

is an

H
H

-map then the obstruction to

f
f

being a homotopy-commutative map is a subset

{

c
2

(
f
)

}

⊂

[

Ω
X
Λ
Ω
X
;

Ω
2

X
′

]

\left \{ {{c_2}(f)} \right \} \subset \left [ {\Omega X\Lambda \Omega X;{\Omega ^2}X’} \right ]

. In this paper we prove:

I
f
[
f
]

If[f]

is in the image of the composition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper P Subscript k plus m Baseline normal upper Omega upper X semicolon upper X Superscript prime Baseline right-bracket right-arrow left-bracket normal upper Sigma normal upper Omega upper X semicolon upper X Superscript prime Baseline right-bracket ModifyingAbove right-arrow With almost-equals left-bracket normal upper Omega upper X semicolon normal upper Omega upper X Superscript prime Baseline right-bracket comma">

[

P

k
+
m

Ω
X
;

X
′

]

→

[

Σ
Ω
X
;

X
′

]

→

≈

[

Ω
X
;
Ω

X
′

]

,

\left [ P_{k + m} \Omega X;X’ \right ] \rightarrow \left [ {\Sigma \Omega X;X’} \right ] \stackrel {\approx }{\rightarrow } \left [ {\Omega X;\Omega X’} \right ],

\]
then

{

c
2

(
f
)

}

\left \{ {{c_2}(f)} \right \}

is in the image of the composition
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper P Subscript k Baseline normal upper Omega upper X normal upper Lamda upper P Subscript m Baseline normal upper Omega upper X semicolon upper X Superscript prime Baseline right-bracket right-arrow left-bracket normal upper Sigma normal upper Omega upper X normal upper Lamda normal upper Sigma normal upper Omega upper X semicolon upper X Superscript prime Baseline right-bracket ModifyingAbove right-arrow With almost-equals left-bracket normal upper Omega upper X normal upper Lamda normal upper Omega upper X semicolon normal upper Omega squared upper X Superscript prime Baseline right-bracket period">

[

P
k

Ω
X
Λ

P
m

Ω
X
;

X
′

]

→

[
Σ
Ω
X
Λ
Σ
Ω
X
;

X
′

]

→

≈

[
Ω
X
Λ
Ω
X
;

Ω
2

X
′

]

.

\left [ P_k \Omega X\Lambda {P_m}\Omega X;X’ \right ] \rightarrow \left [ \Sigma \Omega X\Lambda \Sigma \Omega X;X’ \right ] \stackrel {\approx }{\rightarrow } \left [ \Omega X\Lambda \Omega X;{\Omega ^2}X’ \right ].

\]
Consequently if

α
∈

H
n

(
Ω
X
;

Z
p

)

\alpha \in {H^n}(\Omega X;{Z_p})

is an

A
3

{A_3}

-class in the sense of Stasheff then each element of

{

c
2

(
f
)

}

\left \{ {{c_2}(f)} \right \}

is of the form

∑

c
′

i

⊗

c
i

\sum {{{c’}_i}} \otimes {c_i}

where the

c
i

{c_i}

are primitive.
A characterization of 𝑀₀-spaces
Authors: Junnila Heikki J. K.
Published: June 2010

The following result is obtained. Theorem. A

T
1

{T_1}

-space has a

σ
\sigma

-closure-preserving base consisting of clopen sets iff the space can be embedded in the product of countably many

σ
\sigma

-discrete stratifiable spaces.
𝐶(𝐾,𝐸) contains a complemented copy of 𝑐₀
Authors: Cembranos Pilar
Published: June 2010

Let

E
E

be a Banach space and let

K
K

be a compact Hausdorff space. We denote by

C
(
K
,
E
)

C(K,E)

the Banach space of all

E
E

-valued continuous functions defined on

K
K

, endowed with the supremum norm. We prove in this paper that if

K
K

is infinite and

E
E

is infinite-dimensional, then

C
(
K
,
E
)

C(K,E)

contains a complemented subspace isomorphic to

c
0

{c_0}

.
On collectionwise normality of product spaces. II
Authors: Chiba Keiko
Published: June 2010

In this paper the following results will be obtained. (1) Let

X
X

be a closed image of a normal

M
M

-space and

Y
Y

a paracompact first countable

P
P

-space. If

X
×
Y

X \times Y

is normal, then

X
×
Y

X \times Y

is collectionwise normal. (2) Let

X
X

be a collectionwise normal space and

Y
Y

a

σ
\sigma

-locally compact paracompact space. If

X
×
Y

X \times Y

is normal, then

X
×
Y

X \times Y

is collectionwise normal.
The nonexistence of maximal invariant measures on abelian groups
Authors: Pelc Andrzej
Published: June 2010

It is shown that every

σ
\sigma

-additive

σ
\sigma

-finite invariant measure on an abelian group has a proper

σ
\sigma

-additive invariant extension.
A short proof of a decomposition theorem of a von Neumann algebra
Authors: Awami M. , Thaheem A. B.
Published: June 2010

Let

M
M

be a von Neumann algebra and

S
S

and

T
T

be commuting

∗

^*

-auto-morphisms on

M
M

satisfying the equation:

S
+

S

−
1

=
T
+

T

−
1

S + {S^{ - 1}} = T + {T^{ - 1}}

. It is proved that

M
M

can be decomposed by a central projection

p
p

in

M
M

such that

S
=
T

S = T

on

M
p

Mp

and

S
=

T

−
1

S = {T^{ - 1}}

on

M
(
1
−
p
)

M(1 - p)

.
The consistency strength of certain stationary subsets of 𝒫_{𝜅}𝜆
Authors: Baldwin Stewart
Published: June 2010

If

κ
⩽
λ

\kappa \leqslant \lambda

are uncountable cardinals with

κ
\kappa

regular, let

S

(

κ
,
λ

)

S\left ( {\kappa ,\lambda } \right )

. We investigate the consistency strength of the statement "

S

(

κ
,
λ

)

S\left ( {\kappa ,\lambda } \right )

is stationary in

P

κ

λ

{\mathcal {P}_\kappa }\lambda

," and prove that it is strictly weaker than "

∃
\exists

a Ramsey cardinal," which combines with the lower bound

(

0
#

)

\left ( {{0^\# }} \right )

proven earlier by J. Baumgartner to give a narrow range of the consistency strength of this statement. In addition, we give an example

(

L

[
U
]

)

\left ( {L\left [ U \right ]} \right )

to show that "

∃
λ
>

\exists \lambda >

" does not necessarily imply "

S

(

κ
,

κ
+

)

S\left ( {\kappa ,{\kappa ^ + }} \right )

is stationary."
Radar ambiguity functions, the Heisenberg group, and holomorphic theta series
Authors: Schempp Walter
Published: June 2010

The concept of linear Schrödinger representation of the real Heisenberg nilpotent group and its various realizations is used to link the theory of radar ambiguity functions with nilpotent harmonic analysis. This group-representation theoretic approach allows us to analyze the radar ambiguity functions simultaneously in time and frequency. Moreover, it allows us to determine the group of all transformations that leave the radar ambiguity surfaces invariant and to specify all admissible envelope functions that belong to radar signals of the same finite energy. In particular, an investigation of the radial, i.e., S0(2, R)-invariant radar ambiguity surfaces, gives rise to an identity for Laguerre-Weber functions of different orders, which implies on its part an identity for holomorphic theta series.
Asymptotics of differential systems with deviating arguments
Authors: Trench William F.
Published: June 2010

Conditions are given which imply that a system

X
′

(
t
)
=
F
[
t
,
X
(
g
(
t
)
)
]

X’(t) = F[t,X(g(t))]

has a solution with some components which approach given limits with specified orders of convergence as

t
→
∞

t \to \infty

, while the other components have specified orders of magnitude. The integral smallness conditions on

F
F

permit conditional convergence of some of the improper integrals that occur, and it is not required that

lim

t
→
∞

g
(
t
)
=
∞

{\lim _{t \to \infty }}g(t) = \infty

.
A chaotic function whose nonwandering set is the Cantor ternary set
Authors: Du Bau Sen
Published: June 2010

We introduce a continuous function from [0, 1] onto itself whose nonwandering set is the Cantor ternary set

C
C

. This function is also chaotic in the sense of Li and Yorke, but with the property that almost all points of [0, 1] are eventually fixed. Those points of

C
C

which are periodic, eventually periodic, asymptotically periodic or recurrent are also explicitly identified.
Adding a 2-handle to a 3-manifold: an application to property 𝑅
Authors: Jaco William
Published: June 2010

A sufficient condition is given for adding a

2
2

-handle to a

3
3

-manifold and obtaining a

3
3

-manifold with incompressible boundary. The main result is then used to show that a knot in the

3
3

-sphere admitting a partially unknotted spanning surface has Property

R
R

.
A Wallman-type compactification for convergence spaces
Authors: Gazik R. J. , Park B. H. , Richardson G. D.
Published: June 2010

A compactification for any given convergence space is constructed. The compactification has the unique continuous extension property subject to reasonable assumptions on the range space. Properties of the compactification are studied.
On the nilpotency index of the radical of a group algebra. VIII
Authors: Motose Kaoru
Published: June 2010

A characterization of a finite

p
p

-solvable group

G
G

with

t
(
G
)
>
(
r
+
2
)
(
p
−
1
)
+
1

t(G) > (r + 2)(p - 1) + 1

is given under certain conditions, where

t
(
G
)

t(G)

is the nilpotency index of the radical of a group algebra of a finite

p
p

-solvable group

G
G

with a

p
p

-Sylow subgroup of order

p
r

{p^r}

over a field of characteristic

p
p

.
Double commutants of operators quasisimilar to normal operators
Authors: Takahashi Katsutoshi
Published: June 2010

It is shown that double commutants of operators quasi-similar to normal operators are reflexive.
A simple evaluation of Askey and Wilson’s 𝑞-beta integral
Authors: Rahman Mizan
Published: June 2010

By using the well-known sum of

2

ϕ
1

(

a
,
b
;
c
;
c

/

a
b

)

_2{\phi _1}\left ( {a,b;c;c/ab} \right )

and Sears’ identity for the sum of two nonterminating balanced

3

ϕ
2

_3{\phi _2}

series, a simple evaluation is given for Askey and Wilson’s

q
q

-beta type integral
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript negative 1 Superscript 1 Baseline StartFraction h left-parenthesis x semicolon 1 right-parenthesis h left-parenthesis x semicolon negative 1 right-parenthesis h left-parenthesis x semicolon StartRoot q EndRoot right-parenthesis h left-parenthesis x colon minus StartRoot q right-parenthesis EndRoot Over h left-parenthesis x semicolon a right-parenthesis h left-parenthesis x semicolon b right-parenthesis h left-parenthesis x semicolon c right-parenthesis h left-parenthesis x semicolon d right-parenthesis EndFraction StartFraction d x Over StartRoot 1 minus x squared EndRoot EndFraction comma">

∫

−
1

1

h
(
x
;
1
)
h
(
x
;
−
1
)
h
(
x
;

q

)
h
(
x
:
−

q
)

h
(
x
;
a
)
h
(
x
;
b
)
h
(
x
;
c
)
h
(
x
;
d
)

d
x

1
−

x
2

,

\int _{ - 1}^1 {\frac {{h(x;1)h(x; - 1)h(x;\sqrt q )h(x: - \sqrt {q)} }}{{h(x;a)h(x;b)h(x;c)h(x;d)}}} \frac {{dx}}{{\sqrt {1 - {x^2}} }},

\]
where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max left-parenthesis StartAbsoluteValue q EndAbsoluteValue comma StartAbsoluteValue a EndAbsoluteValue comma StartAbsoluteValue b EndAbsoluteValue comma StartAbsoluteValue c EndAbsoluteValue comma StartAbsoluteValue d EndAbsoluteValue greater-than 1 right-parenthesis">

max

(

|
q
|

,

|
a
|

,

|
b
|

,

|
c
|

,

|
d
|

>
1

)

\max \left ( {\left | q \right |,\left | a \right |,\left | b \right |,\left | c \right |,\left | d \right | > 1} \right )

.
A vanishing theorem for open orbits on complex flag manifolds
Authors: Schmid Wilfried , Wolf Joseph A.
Published: June 2010

A real reductive Lie group

G
G

acts on complex flag manifolds

G

C

{G_{\mathbf {C}}}

/(parabolic subgroup). The open orbits

D
=
G
(
x
)

D = G(x)

are precisely the homogeneous complex manifolds

G

/

H

G/H

, where

H
H

is the centralizer of a torus. We prove that

D
D

is (

(
s
+
1
)

(s + 1)

)-complete in the sense of Andreotti and Grauert, with

s
s

= complex dimension of a maximal compact subvariety of

D
D

. Thus

H
q

(
D
,

F

)
=
0

{H^q}(D,\mathcal {F}) = 0

for

q
>
s

q > s

and any coherent sheaf

F

→
D

\mathcal {F} \to D

. This vanishing theorem is needed for the realization of certain unitary representations on Dolbeault cohomology groups of homogeneous vector bundles.
On 𝑝-adic congruence of some class functions on a finite group
Authors: Blau Harvey I.
Published: June 2010

Certain

p
p

-adic integer-valued class functions on a finite group are shown to be congruent modulo a suitable power of

p
p

. This is applied to prove and extend a result of Plesken on central characters of a

p
p

-block with cyclic defect group.
The relationship between the zeros of best approximations and differentiability
Authors: Borwein Peter B.
Published: June 2010

We examine the relationship between the analytic properties of continuous functions on

[
−
1
,
1
]

[ - 1,1]

and the location of the roots of the sequence of best polynomial approximations. We show that if the approximants have no zeros in a certain ellipse then the function being approximated must be analytic in this ellipse. We also show that the rate at which the zeros of the

n
n

th approximant tend to the interval

[
−
1
,
1
]

[ - 1,1]

determines the global differentiability of the function under consideration.
Covering homotopy properties of maps between C.W. complexes or ANRs
Authors: Steinberger Mark , West James
Published: June 2010

Serre fibrations between C.W. complexes are fibrations in the category of compactly generated weak Hausdorff spaces, but not, by example, in the category of all spaces. Under suitable hypotheses the notions of Dold fibration in the two categories coincide. Examples are given of maps with ANR base, total space and fibres which are not Dold fibrations but whose pullbacks over maps out of locally finite-dimensional paracompact spaces are bundles.
The chain recurrent set for maps of the circle
Authors: Block Louis , Franke John E.
Published: June 2010

For a continuous map of the circle to itself we give necessary and sufficient conditions for the chain recurrent set to be precisely the set of periodic points. We also examine the possible types of examples which can occur, where the set of periodic points is closed and nonempty, but there are nonperiodic, chain recurrent points.
Two easy examples of zero-dimensional spaces
Authors: Mysior A.
Published: June 2010

We present simple examples of a realcompact zerodimensional space which is not

N
N

-compact and an

N
N

-compact space which is not strongly zerodimensional.
Comparison of two methods of multiplying distributions
Authors: Tysk Johan
Published: June 2010

There is no canonical way of defining a product of distributions. In the present paper we compare two different methods of defining a product of distributions. These methods are based on the sequential and functional-analytic approaches to distributions.
Completely bounded maps on 𝐶*-algebras
Authors: Suen Ching Yun
Published: June 2010

In this paper we give a simpler proof of an extension theorem for completely bounded maps defined on subspaces of

C
∗

{C^ * }

-algebras, a new proof of a theorem of Wittstock [8, Satz 4.5], and a series of propositions by extending the techniques of [7] to work in the case of a

C
C

-bimodule action on the

C
∗

{C^ * }

-algebras involved.
The two multiplications on 𝐵𝑈
Authors: Hubbuck J. R. , Mahmud Z.
Published: June 2010

We prove that

B

U
p

B{U_p}

has only two standard

H
H

-structures.
An arithmetic property of the Taylor coefficients of analytic functions with an application to transcendental numbers
Authors: Laohakosol Vichian
Published: June 2010

We extend a result of Popken concerning the numerators of the Taylor coefficients of algebraic functions and combine it with a result of Mahler on lacunary power series to prove an extension of a special case of a result of Cohn on the transcendence of functional values of lacunary power series evaluated at rational points.
Smooth 𝑇ⁿ-valued cocycles for ergodic diffeomorphisms
Authors: Hawkins Jane M.
Published: June 2010

We prove that if

f
f

is any conservative, ergodic diffeomorphism of a smooth, connected, paracompact manifold, then in the set of smooth

T
n

{T^n}

-valued cocycles on

X
X

for the

Z

{\mathbf {Z}}

-action determined by

f
f

, there is a residual set which give ergodic skew product extensions.
A rigidity result for holomorphic immersions of surfaces in 𝐶𝑃ⁿ
Authors: Rigoli Marco
Published: June 2010

A pinching condition for the Gaussian curvature implies rigidity.
A weighted weak type inequality for the maximal function
Authors: Sawyer E.
Published: June 2010

We show that the operator

S
=

υ

−
1

M
υ

S = {\upsilon ^{ - 1}}M\upsilon

, where

M
M

denotes the HardyLittlewood maximal operator, is of weak type (1,1) with respect to the measure

υ
(
x
)
w
(
x
)
d
x

\upsilon (x)w(x)dx

whenever

υ
\upsilon

and

w
w

are

A
1

{A_1}

weights. B. Muckenhoupt’s weighted norm inequality for the maximal function can then be obtained directly from the P. Jones factorization of

A
p

{A_p}

weights using interpolation with change of measure.
Some remarks about measurable parametrizations
Authors: Pol Roman
Published: June 2010

A result about measurable selections is derived from the classical Yankov-von Neumann selection theorem which yields two theorems about parametrizations of analytic and Borel sets in the plane due to Cenzer and Mauldin [CM] and Srivatsa [S], respectively.
A canonical trace class approximant
Authors: Legg D. A. , Ward J. D.
Published: June 2010

Let

H
H

be a finite-dimensional Hilbert space,

B

(
H
)

B\left ( H \right )

the space of bounded linear operators on

H
H

, and

C
C

a convex subset of

B

(
H
)

B\left ( H \right )

. If

A
A

is a fixed operator in

B

(
H
)

B\left ( H \right )

, then

A
A

has a unique best approximant from

C
C

in the

C
P

{C_P}

norm for

1
>
p
>
∞

1 > p > \infty

. However, in the

C
1

{C_1}

(trace) norm,

A
A

may have many best approximants from

C
C

. In this paper, it is shown that the best

C
p

{C_p}

approximants to

A
A

converge to a select trace class approximant

A
1

{A_1}

as

p
→
1

p \to 1

. Furthermore,

A
1

{A_1}

is the unique trace class approximant minimizing
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript i equals 1 Overscript n Endscripts upper S Subscript i Baseline left-parenthesis upper A minus upper B right-parenthesis ln upper S Subscript i Baseline left-parenthesis upper A minus upper B right-parenthesis">

∑

i
=
1

n

S
i

(

A
−
B

)

ln
⁡

S
i

(

A
−
B

)

\sum \nolimits _{i = 1}^n {{S_i}\left ( {A - B} \right )\operatorname {ln }{S_i}\left ( {A - B} \right )}

over all trace class approximants

B
B

. The numbers

S
i

(
T
)

{S_i}\left ( T \right )

are the eigenvalues of the positive part

|
T
|

\left | T \right |

of

T
T

.