bluuebunny/crossref_metadata_2025 huggingface dataset:
Harmonic maps and a pinching theorem for positively curved hypersurfaces
Authors: Hu H. S. , Pan Y. L. , Shen Y. B.
Published: June 2010

In this paper, we establish a theorem of Liouville type for stable harmonic maps in sufficiently pinched, positively curved hypersurfaces of a space form with nonnegative constant curvature. Similar results for the Euclidean sphere

S
n

{S^n}

have been proved by Y. L. Xin and P. F. Leung, respectively.
The Stone-Čech compactification, the Stone-Čech remainder, and the regular Wallman property
Authors: Kimura Takashi
Published: June 2010

In this paper, the following are proved: (1) the Stone-Čech compactification of a metrizable space is regular Wallman, (2) if the Stone-Čech compactification of a locally compact space whose pseudocompact closed subsets are compact is regular Wallman, then the Stone-Čech remainder is also regular Wallman. Consequently, the Stone-Čech remainder of a locally compact metrizable space is regular Wallman.
Unbounded composition operators on 𝐻²(𝐵₂)
Authors: Cima J. A. , Wogen W. R.
Published: June 2010

Examples are given of holomorphic self-maps of the unit ball on

C

2

{{\text {C}}^2}

which induce unbounded composition operators on the Hardy space

H
2

{H^2}

. In particular, an example is given which is one-to-one on the closed ball. Also, a valence condition on the boundary of this ball is given which is sufficient for unboundedness of the induced composition operator.
Some analogues of Markov and Descartes systems for right disfocality
Authors: Eloe P. W. , Henderson Johnny
Published: June 2010

A necessary and sufficient condition for the disconjagacy of the

n
n

th order linear differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Superscript left-parenthesis n right-parenthesis Baseline plus a 1 left-parenthesis x right-parenthesis y Superscript left-parenthesis n minus 1 right-parenthesis Baseline plus midline-horizontal-ellipsis plus a Subscript n Baseline left-parenthesis x right-parenthesis y equals 0">

y

(
n
)

+

a
1

(
x
)

y

(
n
−
1
)

+
⋯
+

a
n

(
x
)
y
=
0

{y^{(n)}} + {a_1}(x){y^{(n - 1)}} + \cdots + {a_n}(x)y = 0

on a compact interval

I
I

is that there exists a system of solutions

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

such that any one of the following is satisfied: (i)

W
(

y
1

,
…
,

y
k

)
>
0
,
1
≤
k
≤
n

W({y_1}, \ldots ,{y_k}) > 0,1 \leq k \leq n

, on

I
I

; (ii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W left-parenthesis y Subscript i Subscript Sub Subscript 1 Subscript Baseline comma ellipsis comma y Subscript i Subscript Sub Subscript k Subscript Baseline right-parenthesis greater-than 0 comma 1 less-than-or-equal-to i 1 greater-than midline-horizontal-ellipsis greater-than i Subscript k Baseline less-than-or-equal-to n comma 1 less-than-or-equal-to k less-than-or-equal-to n">

W
(

y
i

1

,
…
,

y
i

k

)
>
0
,
1
≤

i
1

>
⋯
>

i
k

≤
n
,
1
≤
k
≤
n

W({y_i}_{_1}, \ldots ,{y_i}_{_k}) > 0,1 \leq {i_1} > \cdots > {i_k} \leq n,1 \leq k \leq n

, on

I
I

; or (iii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W left-parenthesis y Subscript i Baseline comma y Subscript i plus 1 Baseline comma ellipsis comma y Subscript i Baseline Subscript plus k minus 1 Baseline right-parenthesis greater-than 0 comma 1 less-than-or-equal-to i less-than-or-equal-to n minus k plus 1 comma 1 less-than-or-equal-to k less-than-or-equal-to n">

W
(

y
i

,

y

i
+
1

,
…
,

y
i

+
k
−
1

)
>
0
,
1
≤
i
≤
n
−
k
+
1
,
1
≤
k
≤
n

W({y_i},{y_{i + 1}}, \ldots ,{y_i}_{ + k - 1}) > 0,1 \leq i \leq n - k + 1,1 \leq k \leq n

, on

I
I

. Necessary and sufficient criteria for the right disfocality of the linear differential equation on the compact interval

I
I

are established in terms of systems of solutions satisfying conditions which are analogous to those given in (i), (ii), (iii).
A wild Cantor set as the limit set of a conformal group action on 𝑆³
Authors: Bestvina M. , Cooper D.
Published: June 2010

We construct a conformal action of a free group of finite rank on

S
3

{S^3}

whose set of discontinuity

Λ
\Lambda

is a wild Cantor set.
On the final set of a real entire function of exponential type
Authors: Rao Murali , Shen Li-Chien
Published: June 2010

In this paper we have constructed a real entire function of exponential type which is bounded on the real axis such that the entire real axis belongs to its final set.
Representations of infinite permutations by words
Authors: Mycielski Jan
Published: June 2010

We prove several cases of the following theorem: Every free group word which is not a proper power can represent every permutation of an infinite set. The remaining cases will be proved in a forthcoming paper of R. C. Lyndon.
An inequality for selfadjoint operators on a Hilbert space
Authors: Bernstein Herbert J.
Published: June 2010

An elementary inequality of use in testing convergence of eigenvector calculations is proven. If

e
λ

{e_\lambda }

is a unit eigenvector corresponding to an eigenvalue

λ
\lambda

of a selfadjoint operator

A
A

on a Hilbert space

H
H

, then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue left-parenthesis g comma e Subscript lamda Baseline right-parenthesis EndAbsoluteValue squared less-than-or-equal-to StartFraction double-vertical-bar g double-vertical-bar squared double-vertical-bar upper A g double-vertical-bar squared minus left-parenthesis g comma upper A g right-parenthesis squared Over double-vertical-bar left-parenthesis upper A minus lamda upper I right-parenthesis g double-vertical-bar squared EndFraction">

|

(
g
,

e
λ

)

|

2

≤

‖
g
‖

2

‖

A
g

‖

2

−

(
g
,
A
g
)

2

‖

(
A
−
λ
I
)
g

‖

2

{\left | {(g,{e_\lambda })} \right |^2} \leq \frac {{{{\left \| g \right \|}^2}{{\left \| {Ag} \right \|}^2} - {{(g,Ag)}^2}}}{{{{\left \| {(A - \lambda I)g} \right \|}^2}}}

\]
for all

g
g

in

H
H

for which

A
g
≠
λ
g

Ag \ne \lambda g

. Equality holds only when the component of

g
g

orthogonal to

e
λ

{e_\lambda }

is also an eigenvector of

A
A

.
Wu classes
Authors: Stong Robert , Yoshida Toshio
Published: June 2010

This paper shows that a nonbounding closed manifold with

Wu
\operatorname {Wu}

classes

v
i

=
0

{v_i} = 0

for

i
>
k

i > k

has dimension bounded by some function of

k
k

.
A note on Gul′ko compact spaces
Authors: Gruenhage Gary
Published: June 2010

It is known that every Eberlein compact space has a dense

G
δ

{G_\delta }

metrizable subset. We answer a question of M. Talagrand by showing that the same is true for the wider class of Gul’ko compact spaces. We also show that Gul’ko compact spaces satisfy a certain covering property hereditarily.
A weak GAGA statement for arbitrary morphisms
Authors: Neeman Amnon
Published: June 2010

Let

f
:
X
→
Y

f:X \to Y

be an arbitrary morphism of schemes of finite type over

C

{\mathbf {C}}

, and let

f

an

{f^{{\text {an}}}}

be the associated map of analytic spaces. Let

S

\mathcal {S}

be a coherent sheaf on

X
X

. Then

(

f
∗

S

)

an

→

f
∗

an

(

S

an

)

{({f_*}\mathcal {S})^{{\text {an}}}} \to f_*^{{\text {an}}}({\mathcal {S}^{{\text {an}}}})

is injective.
Arens product and the algebra of double multipliers. II
Authors: Wong Pak Ken
Published: June 2010

We show that the algebra of double multipliers of a certain Banach algebra

A
A

can be embedded in the second conjugate space

A

∗
∗

{A^{**}}

of

A
A

. This generalizes the previous work by the author for

B
∗

{B^*}

-algebras.
Picard dimensions of close to rotationally invariant densities
Authors: Tada Toshimasa
Published: June 2010

The purpose of this paper is to show that the Picard dimensions of a rotation-free density

P
P

and a general density

Q
Q

on the punctured unit disk

0
>

|
z
|

>
1

0 > \left | z \right | > 1

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

|

P

(
z
)

−
Q

(
z
)

|

=
O

(

|
z
|

−
2

)

\left | {P\left ( z \right ) - Q\left ( z \right )} \right | = O\left ( {{{\left | z \right |}^{ - 2}}} \right )

as

z
→
0

z \to 0

.
An inequality suggested by Littlewood
Authors: Bennett Grahame
Published: June 2010

It is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n Endscripts a Subscript n Superscript 3 Baseline sigma-summation Underscript m equals 1 Overscript n Endscripts a Subscript m Superscript 2 Baseline sigma-summation Underscript k equals 1 Overscript m Endscripts a Subscript k Baseline less-than-or-equal-to three halves sigma-summation Underscript n Endscripts a Subscript n Superscript 4 Baseline left-bracket sigma-summation Underscript k equals 1 Overscript n Endscripts a Subscript k Baseline right-bracket squared">

∑
n

a
n
3

∑

m
=
1

n

a
m
2

∑

k
=
1

m

a
k

≤

3
2

∑
n

a
n
4

[

∑

k
=
1

n

a
k

]

2

\sum \limits _n {a_n^3} \sum \limits _{m = 1}^n {a_m^2} \sum \limits _{k = 1}^m {{a_k}} \leq \frac {3}{2}\sum \limits _n {a_n^4} {\left [ {\sum \limits _{k = 1}^n {{a_k}} } \right ]^2}

\]
for arbitrary nonnegative numbers

a
1

,

a
2

,
…

{a_1},{a_2}, \ldots

.
On the restricted mean value property
Authors: Fenton P. C.
Published: June 2010

Suppose that

u
u

is continuous in the open unit disc and has the restricted mean value property. It is shown that if

u
u

has finite boundary limits almost everywhere, and if

u
u

possesses a harmonic majorant and minorant, the difference between which has finite radial upper limits everywhere, then

u
u

is harmonic.
Minimal periodic orbits and topological entropy of interval maps
Authors: Du Bau-Sen
Published: June 2010

For any two integers

m
≥
0

m \geq 0

and

n
≥
1

n \geq 1

, we construct continuous functions from

[

0
,
1

]

\left [ {0,1} \right ]

into itself which have exactly one minimal periodic orbit of least period

2
m

(

2
n
+
1

)

{2^m}\left ( {2n + 1} \right )

, but with topological entropy equal to

∞
\infty

.
Constructing metrics with the Heine-Borel property
Authors: Williamson Robert , Janos Ludvik
Published: June 2010

A metric space

(
X
,
d
)

(X,d)

is said to be Heine-Borel if any closed and bounded subset of it is compact. We show that any locally compact and

σ
\sigma

-compact metric space can be made Heine-Borel by a suitable remetrization. Furthermore we prove that if the original metric

d
d

is complete, then this can be done so that the new Heine-Borel metric

d
′

d’

is locally identical to

d
d

, i.e., for every

x
∈
X

x \in X

there exists a neighborhood of

x
x

on which the two metrics coincide.
A simple proof of an extension of the Fuglede-Putnam theorem
Authors: Gong Wei Bang
Published: June 2010

A simple proof is given of the Theorem. If

A
A

and

B
∗

{B^ * }

are hyponormal, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper A upper X minus upper X upper B double-vertical-bar Subscript 2 Baseline greater-than-or-equal-to double-vertical-bar upper A Superscript asterisk Baseline upper X minus upper X upper B Superscript asterisk Baseline double-vertical-bar Subscript 2">

‖

A
X
−
X
B

‖

2

≥

‖

A
∗

X
−
X

B
∗

‖

2

{\left \| {AX - XB} \right \|_2} \geq {\left \| {{A^ * }X - X{B^ * }} \right \|_2}

for every

X
X

in the Hilbert-Schmidt class.
Ergodic group actions with nonunique invariant means
Authors: Chou Ching
Published: June 2010

Let

M
(
X
,
G
)

M(X,G)

be the set of

G
G

-invariant means on

L
∞

(
X
,

B

,
P
)

{L^\infty }(X,\mathcal {B},P)

, where

G
G

is a countable group acting ergodically as measure preserving transformations on a nonatomic probability space

(
X
,

B

,
P
)

(X,\mathcal {B},P)

. We show that if there exists

μ
∈
M
(
X
,
G
)
,
μ
≠
P

\mu \in M(X,G),\mu \ne P

, then

M
(
X
,
G
)

M(X,G)

contains an isometric copy of

β
N
∖
N

\beta N\backslash N

, where

β
N
∖
N

\beta N\backslash N

is considered as a subset of

(

l
∞

)
∗

{({l^\infty })^*}

. This provides an answer to a question raised by J. Rosenblatt in 1981.
Asymptotic behavior of 𝑝-predictions for vector valued random variables
Authors: Cuesta Juan A. , Matrán Carlos
Published: June 2010

Let

(
Ω
,
σ
,
μ
)

(\Omega ,\sigma ,\mu )

be a probability space and let

X
X

be a

B
B

-valued

μ
\mu

-essentially bounded random variable, where

(
B
,

‖

‖

)

(B,\left \| {} \right \|)

is a uniformly convex Banach space. Given

α
\alpha

, a sub-

σ
\sigma

-algebra of

σ
\sigma

, the

p
p

-prediction

1
>
p
>
∞

1 > p > \infty

of

X
X

is defined as the best

L
p

{L_p}

-approximation to

X
X

by

α
\alpha

-measurable random variables. The paper proves that the Pólya algorithm is successful, i.e. the

p
p

-prediction converges to an "

∞
\infty

-prediction" as

p
→
∞

p \to \infty

. First the proof is given for

p
p

-means (

p
p

-predictions given the trivial

σ
\sigma

-algebra), and the general case follows from the characterization of the

p
p

-prediction in terms of the

p
p

-mean of the identity in

B
B

with respect to a regular conditional probability. Notice that the problem was treated in [7], but the proof is not satisfactory (as pointed out in [4]).
Recurrent-proximal linear differential systems with almost periodic coefficients
Authors: Nerurkar Mahesh G.
Published: June 2010

We consider a system of linear differential equations,

x
˙

=
A
(
ω
⋅
t
)
x

\dot x = A(\omega \cdot t)x

, parametrized by a point

ω
∈

T

2

\omega \in {{\mathbf {T}}^2}

, the

2
2

-torus, where

(
ω
,
t
)
→
ω
⋅
t

(\omega ,t) \to \omega \cdot t

denotes an irrational rotation flow on

T

2

{{\mathbf {T}}^2}

. We show that if the rotation number of this flow is well approximable by rationals, then residually many equations (with respect to the

C
k

{C^k}

-topology on a certain class of matrix valued maps

A
(
ω
)

A(\omega )

on

T

2

{{\mathbf {T}}^2}

) exhibit recurrent-proximal behavior. Also the order of differentiability

k
k

of the class in which this generic result holds is related to the "speed" of approximation by rationals.
A transformation for an 𝑛-balanced ₃Φ₂
Authors: Srivastava H. M.
Published: June 2010

An interesting generalization of the familiar

q
q

-extension of the Pfaff-Saalschütz theorem is proved and is applied, for example, to derive a reduction formula for a certain double

q
q

-series. The main theorem (asserting the symmetry in

n
n

and

N
N

of a function

f
(
n
,
N
)

f(n,N)

defined in terms of an

n
n

-balanced basic (or

q

 - 

q{\text { - }}

-) hypergeometric

3

Φ
2

_3{\Phi _2}

series by equation (8)) is essentially a

q
q

-extension of Sheppard’s transformation.
Lower bounds for class numbers of real quadratic and biquadratic fields
Authors: Mollin R. A.
Published: June 2010

In this paper we provide lower bounds for class numbers of real quadratic fields of Richaud-Degert type, as well as for class numbers of related real biquadratic fields.
Normal filters generated by a family of sets
Authors: Di Prisco Carlos Augusto , Uzcategui Carlos Enrique
Published: June 2010

We study normal filters on the set spaces

λ
,

P

κ

(
λ
)

,

[
λ
]

κ

\lambda ,{\mathcal {P}_\kappa }\left ( \lambda \right ),{\left [ \lambda \right ]^\kappa }

, and

(
λ
)

κ

{\left ( \lambda \right )^\kappa }

. We characterize the least normal

γ
\gamma

-complete filter containing a given

γ
\gamma

-complete filter for

γ
≥

ω
1

\gamma \geq {\omega _1}

. If

F

\mathcal {F}

is a

ω
1

{\omega _1}

-complete filter on any of the set spaces mentioned, the least

ω
1

{\omega _1}

-complete normal filter containing it is the filter generated by the sets
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace x element-of upper E vertical-bar alpha 1 comma ellipsis comma alpha Subscript n Baseline element-of x right-arrow x element-of f left-parenthesis alpha 1 comma ellipsis comma alpha Subscript n Baseline right-parenthesis right-brace">

{

x
∈
E

|

α
1

,
…
,

α
n

∈
x
→
x
∈
f

(

α
1

,
…
,

α
n

)

}

\left \{ {x \in E\left | {{\alpha _1}, \ldots ,{\alpha _n} \in x \to x \in f\left ( {{\alpha _1}, \ldots ,{\alpha _n}} \right )} \right .} \right \}

where

f
:

λ

>
ω

→

F

f:{\lambda ^{ > \omega }} \to \mathcal {F}

.
Lindelöf property in function spaces and a related selection theorem
Authors: Marciszewski Witold
Published: June 2010

Let

X
X

be a separable metrizable space. If

K
K

is a compact space whose function space

C
(
K
)

C(K)

is weakly

K

\mathcal {K}

-analytic, then the space

C
p

(
X
,
K
)

{C_p}(X,K)

of continuous maps from

X
X

to

K
K

with the pointwise topology has the Lindelöf property. If

E
E

is a Banach space whose weak topology is

K

\mathcal {K}

-analytic, then each lower semicontinuous map from

X
X

to the family of nonempty closed convex subsets of the unit ball of the dual

E
E

with the weak*-topology admits a continuous selection. This extends some results of Corson and Lindenstrauss.
The preparation theorem for composite functions
Authors: Kowalczyk Adam
Published: June 2010

We present a simple extension of the preparation theorem of B. Malgrange and J. Mather to the case of composite functions. As a corollary we obtain a short proof of the equivariant preparation theorem of V. Poénaru.
On a zeta function associated to ternary zero forms
Authors: Cipra Barry A.
Published: June 2010

We rederive a relation due to Eie between a zeta function for ternary quadratic forms and the Riemann zeta function, correcting a factor corresponding to the prime

p
=
2

p = 2

.
Unimodular commutators
Authors: Newman Morris
Published: June 2010

Let

R
R

be a principal ideal ring and

M

k
,
n

{M_{k,n}}

the set of

k
×
n

k \times n

matrices over

R
R

. The following statments are proved: (a) If

k
≤
n

/

3

k \leq n/3

then any primitive element of

M

k
,
n

{M_{k,n}}

occurs as the first

k
k

rows of the commutator of two elements of

SL(

n
,
R

)

{\text {SL(}}n,R{\text {)}}

. (b) If every element of

SL(

3
,
R

)

{\text {SL(}}3,R{\text {)}}

is the product of at most

c
3

{c_3}

commutators, then every element of

SL(

n
,
R

)

{\text {SL(}}n,R{\text {)}}

is the product of at most

c
n

{c_n}

commutators, where

c
n

>
c
log
⁡
n
+

c
3

−
3
,
c
=
2
log
⁡
(
3

/

2
)
=
4.932
…

{c_n} > c\log n + {c_3} - 3,c = 2\log (3/2) = 4.932 \ldots

, and

n
≥
3

n \geq 3

. (c) If

n
≥
3

n \geq 3

, then every element of

SL(

n
,
Z

)

{\text {SL(}}n,Z{\text {)}}

is the product of at most

c
log
⁡
n
+
40

c\log n + 40

commutators, where

c
c

is given in (b) above
On primes 𝑝 with 𝜎(𝑝^{𝛼})=𝑚²
Authors: Chidambaraswamy J. , Krishnaiah P. V.
Published: June 2010

A. Takaku proved that for odd

α
⩾
3
,
σ
(

p
α

)
=

m
2

,
p

\alpha \geqslant 3,\sigma ({p^\alpha }) = {m^2},p

being a prime, implies that

p
>

2

2

α
+
1

p > {2^{{2^{\alpha + 1}}}}

. In this paper we extend this result to include almost all even integers

α
\alpha

.
On the structure of sets of uniqueness
Authors: Lyons Russell
Published: June 2010

We show that every

U
0

{U_0}

-set is almost a

W
W

-set.
Baire category and 𝐵ᵣ-spaces
Authors: Noll Dominikus
Published: June 2010

A topological space satisfying the open mapping theorem is called a

B
r

{B_r}

-space. We investigate the question whether completely regular

B
r

{B_r}

-spaces must be Baire spaces. The answer we obtain is twofold and surprising. On the one hand there exist first category completely regular

B
r

{B_r}

-spaces. Examples are provided in the class of Lindelöf

P
P

-spaces. On the other hand, we obtain a partial positive answer to our question. We prove that every suborderable metrizable

B
r

{B_r}

-space is in fact a Baire space. We conjecture that this is true for metrizable

B
r

{B_r}

-spaces in general. Our paper is completed by some applications. For instance, we establish the existence of a metrizable

B
r

{B_r}

-space

E
E

whose square

E
×
E

E \times E

is no longer a

B
r

{B_r}

-space.
A generalization of Smith theory
Authors: May J. P.
Published: June 2010

Using Bredon cohomology, new relations are obtained between the

b
b
m
o
d
p

bbmod p

Betti numbers of a finite

G
G

-CW complex and its singular subspace, where

G
G

is a finite

p
p

-group.
Relation between right and left involutions of a Hilbert algebra
Authors: Saworotnow P. P.
Published: June 2010

Existence of a densely defined right involution in a Hilbert algebra implies existence of a left involution.
Random sign embeddings from 𝑙ⁿᵣ,2<𝑟<∞
Authors: Figiel T. , Johnson W. B. , Schechtman G.
Published: June 2010

Estimates for any ideal norm of a "random sign embedding" from

l
r
n

l_r^n

into

l
r
m

,

2
>
r
>
∞

l_r^m,\;2 > r > \infty

, are given in terms of the corresponding ideal norm of the identity of

l
r
k

,

k
=
k
(
n
,
m
,
r
)

l_r^k,\;k = k(n,m,r)

.
A boundary analogue of Morera’s theorem in the unit ball of 𝐶ⁿ
Authors: Grinberg Eric L.
Published: June 2010

We show that the boundary values of holomorphic functions in the unit ball ofCn(n > 1){{\mathbf {C}}^n}(n{\text { > 1)}}are characterized by the vanishing of their contour integrals over certain closed curves.
A proof of Liggett’s version of the subadditive ergodic theorem
Authors: Levental Shlomo
Published: June 2010

Using a method developed by Y. Katznelson and B. Weiss we give a proof of Liggett’s improved version of the subadditive ergodic theorem.
Derivations and (hyper)invariant subspaces of a bounded operator
Authors: Zhang Shuang
Published: June 2010

Let

X
X

be a complex Banach space and

L

(
X
)

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

the set of bounded linear operators on

X
X

. For

T
∈

L

(
X
)

T \in \mathcal {L}\left ( X \right )

, a derivation

Δ
T

{\Delta _T}

is defined by

Δ
T

A
=
T
A
−
A
T

{\Delta _T}A = TA - AT

for

A
∈

L

(
X
)

A \in \mathcal {L}\left ( X \right )

. By induction,

Δ
T
m

=

Δ
T

∘

Δ
T

m
−
1

\Delta _T^m = {\Delta _T} \circ \Delta _T^{m - 1}

is defined for each integer

m
≥
2

m \geq 2

. We call the kernel of

Δ
T
m

\Delta _T^m

the

m
m

-commutant of

T
T

. For a polynomially compact operator

T
T

, we consider the (hyper)invariant subspace problem for operators in the

m
m

-commutant of

T
T

for

m
≥
1

m \geq 1

. It is easily seen that the

m
m

-commutant

m
>
1

m > 1

of

T
T

could be much larger than

Ker

(

Δ
T

)

{\operatorname {Ker}}({\Delta _T})

. So our idea is a variation of Lomonosov’s theorem in [6]. We start with several identities on derivations, and then prove our results on the existence of (hyper)invariant subspaces. Theorem 2 in [5] is generalized.
Fatou’s lemma in infinite-dimensional spaces
Authors: Yannelis Nicholas C.
Published: June 2010

Employing recent results of M. Ali Khan we provide an infinite-dimensional version of the Fatou Lemma, which includes as a special case the result of Khan and Majumdar [15].
A counterexample to an F. and M. Riesz-type theorem
Authors: Samotij Krzysztof
Published: June 2010

A premeasure is a finitely additive complex-valued function

μ
\mu

defined on the semiring of all connected subsets of

T

{\mathbf {T}}

, continuous at

∅
\emptyset

and with

μ
(
∅
)
=
μ
(

T

)
=
0

\mu (\emptyset ) = \mu ({\mathbf {T}}) = 0

. Let

κ
\kappa

be a continuous increasing concave function on

[
0
,
2
π
]

[0,2\pi ]

with

κ
(
0
)
=
0

\kappa (0) = 0

. A conjecture from [3] saying that if the Poisson integral of a premeasure

μ
\mu

is holomorphic in the open unit disk and

Var
κ

(
μ
)
>
∞

{\operatorname {Var}_\kappa }(\mu ) > \infty

then

lim

τ
→
0

Var
κ

(

μ
τ

−
μ
)
=
0

{\lim _{\tau \to 0}}{\operatorname {Var}_\kappa }({\mu _\tau } - \mu ) = 0

is disproved, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V a r Subscript kappa Baseline left-parenthesis mu right-parenthesis equals sup sigma-summation Underscript j Endscripts vertical-bar mu left-parenthesis upper I Subscript j Baseline right-parenthesis vertical-bar slash sigma-summation Underscript j Endscripts kappa left-parenthesis StartAbsoluteValue upper I Subscript j Baseline EndAbsoluteValue right-parenthesis">

Var
κ

(
μ
)
=
sup

∑
j

|

μ
(

I
j

)

|

/

∑
j

κ
(

|

I
j

|

)

{\operatorname {Var} _\kappa }(\mu ) = \sup \sum \nolimits _j {|\mu ({I_j}} )|/\sum \nolimits _j {\kappa (|{I_j}|)}

(the supremum is taken over all finite partitions of

T

{\mathbf {T}}

into connected subsets

I
j

{I_j}

) and

μ
τ

{\mu _\tau }

denotes the

τ
\tau

-translation of

μ
\mu

.
Vector-valued stochastic processes. II. A Radon-Nikodým theorem for vector-valued processes with finite variation
Authors: Dinculeanu Nicolae
Published: June 2010

Given a real-valued process

A
A

with finite variation

|
A
|

\left | A \right |

and a vector-valued process

B
B

with finite variation

|
B
|

\left | B \right |

such that for each

ω
\omega

, the Stieltjes measure

d
B

.

(
ω
)

dB{\mathbf {.}}(\omega )

is absolutely continuous with respect to

d
A
.
(
ω
)

dA.(\omega )

, there exists a vector-valued process

H
H

which, under certain separability conditions, satisfies

B
t

=

∫

[
0
,
t
]

H
s

d

A
s

{B_t} = \int _{[0,t]} {{H_s}d{A_s}}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper B EndAbsoluteValue Subscript t Baseline equals integral Underscript left-bracket 0 comma t right-bracket Endscripts double-vertical-bar upper H Subscript s Baseline double-vertical-bar d StartAbsoluteValue upper A EndAbsoluteValue Subscript s Baseline">

|
B
|

t

=

∫

[
0
,
t
]

‖

H
s

‖

d

|
A
|

s

{\left | B \right |_t} = \int _{[0,t]} {\left \| {{H_s}} \right \|d{{\left | A \right |}_s}}

for every

t
≥
0

t \geq 0

. If, moreover,

A
A

and

B
B

are optional or predictable, then so is

H
H

.
Vector measures with values in the compact operators
Authors: Ricker Werner
Published: June 2010

As a consequence of a recent result due to Kaftal and Wiess it is shown that any vector measure (for the strong operator topology) with values in the space of compact operators on a Hilbert space is

σ
\sigma

-additive for the uniform operator topology. This leads to an elegant and simple proof of a result due to Diestel and Faires on the uniform operator

σ
\sigma

-additivity of the indefinite integral induced by a compact selfadjoint operator.
Left Ore principal right ideal domains
Authors: Beauregard Raymond A.
Published: June 2010

The rings referred to in the title of this paper have long been conjectured to be principal left ideal domains. In a recent paper [6] Cohn and Schofield have produced examples (of simple rings) showing that this is not the case. As a result, interest in this type of ring has deepened (see [5], for example, where these rings are referred to as right principal Bezout domains). Our main purpose here is to prove that such rings are either principal left ideal domains or left and right primitive rings.
A reconsideration of the general parabola theorem in continued fractions
Authors: Hovstad R. M.
Published: June 2010

A direct explicit proof, using one technical lemma and the difference equations, is given of the general parabola theorem.
A Hilbert cube L-S category
Authors: Wong Raymond Y.
Published: June 2010

Let

M
M

be a compact connected Hilbert cube manifold (

Q
Q

-manifold). Define

C
z

(
M
)

{C_z}\left ( M \right )

to be the smallest integer

k
k

such that

M
M

can be covered with

k
k

open subsets each of which is homeomorphic to

Q
×

[

0
,
1

)

Q \times \left [ {0,1} \right )

. Recently L. Montejano proved that, for every compact connected polyhedron

P
,

C
z

(

P
×
Q

)

=
cat
⁡

(
P
)

+
1

P,{C_z}\left ( {P \times Q} \right ) = \operatorname {cat}\left ( P \right ) + 1

, where

cat
⁡

(
P
)

\operatorname {cat} \left ( P \right )

is the Lusternik-Schnirelmann category of

P
P

. Using a different approach, we prove a noncompact analog of the above theorem by showing that

C
z

(

P
×
Q
×

[

0
,
1

)

)

=
cat
⁡

(
P
)

{C_z}\left ( {P \times Q \times \left [ {0,1} \right )} \right ) = \operatorname {cat}\left ( P \right )

for every compact connected polyhedron

P
P

.
Complete exhaustive sieves and games
Authors: Telgársky Rastislav , Wicke Howard H.
Published: June 2010

Properties of completely regular spaces with complete exhaustive sieves are studied using the equivalent notion of partition complete spaces and associated games. Among others the following results are proved. (1) If

X
X

is the intersection of countably many partition complete subsets of

Y
Y

, then

X
X

is partition complete. (2) If

X
X

is

K
K

-scattered, where

K
K

is the class of all partition complete spaces, then

X
X

is partition complete. (3) If

X
X

is a primitive set in a

C
C

-scattered space

Y
Y

, then

X
X

is the intersection of countably many

C
C

-scattered subsets of

Y
Y

. (4) The partition completeness is perfect and preserved by open maps.
Entropy and exponential growth of 𝜋₁ in dimension two
Authors: Franks John M. , Handel Michael
Published: June 2010

The authors show that if

f
:
M
→
M

f:M \to M

is a

C

1
+
α

{C^{1 + \alpha }}

diffeomorphism of a compact surface and if the topological entropy of

f
f

is positive then there is a finite invariant set

P
P

such that the map induced by

f
f

on

π
1

(
M
−
P
)

{\pi _1}(M - P)

has exponential growth.
Answer to a problem of Miller
Authors: Chen Yue Qing
Published: June 2010

In this note we give an affirmative answer to a problem of S. Miller. Our method is very simple.
A real valued homomorphism on algebras of differentiable functions
Authors: Arias-de-Reyna Juan
Published: June 2010

In this paper we prove that, for every homomorphism

A
A

on

C
k

(
E
)

{C^k}\left ( E \right )

, there exists

x
∈
E

x \in E

such that

A

(
f
)

=
f

(
x
)

A\left ( f \right ) = f\left ( x \right )

for

f
∈

C
k

(
E
)

f \in {C^k}\left ( E \right )

. Here

C
k

(
E
)

(

k
=
1
,
2
,
…
,
∞

)

{C^k}\left ( E \right )\;\left ( {k = 1,2, \ldots , \infty } \right )

denotes the algebra of all

k
k

-times differentiable real functions on a real and separable Banach space

E
E

.
Pointwise estimates for the relative fundamental solution of \overline∂_{𝑏}
Authors: Christ Michael
Published: June 2010

Consider a compact pseudoconvex CR manifold of dimension 3 and finite type, on which the operator

∂
¯

b

{\bar \partial _b}

has closed range in

L
2

{L^2}

. The relative fundamental solution of

∂
¯

b

{\bar \partial _b}

is the distribution-kernel for that operator which inverts

∂
¯

b

{\bar \partial _b}

modulo its kernel and cokernel. We derive pointwise bounds on this fundamental solution and its derivatives.
Amenability and derivations of the Fourier algebra
Authors: Forrest Brian
Published: June 2010

It is shown that a locally compact group

G
G

is amenable if and only if every derivation of the Fourier algebra

A
(
G
)

A(G)

into a Banach

A
(
G
)

A(G)

-bimodule is continuous. Also given are necessary and sufficient conditions for

A
(
G
)

A(G)

to be weakly amenable.
A note on the cup product for pro-𝑝 groups
Authors: Würfel Tilmann
Published: June 2010

Let

G
G

be a pro-

p
p

group and

g
∈

H
1

(
G
)

g \in {H^1}(G)

. We give a group-theoretic description of the kernel of the cup product —

−
∪
g
:

H
1

(
G
)
→

H
2

(
G
)

- \cup g:{H^1}(G) \to {H^2}(G)

.
A decomposition of bounded scalarly measurable functions taking their ranges in dual Banach spaces
Authors: Bator Elizabeth M.
Published: June 2010

A decomposition of scalarly measurable functions taking their range in the dual of a Banach space into a Pettis integrable part and a weak* scalarly null part is introduced and analyzed.
A refinement of Šarkovskiĭ’s theorem
Authors: Bhatia Nam P. , Egerland Walter O.
Published: June 2010

Let

f
:
R
→
R

f:R \to R

be continuous. If

f
f

has an orbit of period

n
n

, the question of which other periods

f
f

must necessarily have was answered by A. N. Sarkovskii by giving a total ordering of the natural numbers, now called the Sarkovskii ordering. The ordering does not take into account the period types and examples show that depending on the type of the period other periods than those implied by the Sarkovskii ordering are present. Introducing the concepts of a periodic loop (a periodic orbit of a certain type) and infinite loop, we give a total ordering of loops and obtain, as a consequence, a refinement of the theorem of Sarkovskii.
The Lefschetz number of self-maps of Lie groups
Authors: Duan Hai Bao
Published: June 2010

In this note we present a simple approach to the Lefschetz number for the self-maps of Lie groups. As an application it is proved that for any map

f
:
G
→
G

f:G \to G

of a compact connected Lie group

G
G

, there is a solution to

(
f
(
x
)

)
k

=
x

{(f(x))^k} = x

for some

k
≤
⋋
+
1

k \leq \leftthreetimes + 1

, where

⋋
\leftthreetimes

is the rank of the group

G
G

.
A symmetric density property: monotonicity and the approximate symmetric derivative
Authors: Freiling C. , Rinne D.
Published: June 2010

The following is established: Let

W
W

and

B
B

be open sets of real numbers whose union has full measure. If for each

x
x

, the set

{
h
>
0

|

x
−
h
∈
W
,
x
+
h
∈
B
}

\{ h > 0|x - h \in W,x + h \in B\}

has density zero at zero, then these sets are all empty. This is then used to prove the following: If

f
f

is a continuous real valued function with a nonnegative lower approximate symmetric derivative, then

f
f

is nondecreasing.
Model-theoretic characterizations of arcs and simple closed curves
Authors: Bankston Paul
Published: June 2010

Two compact Hausdorff spaces areco-elementarily equivalentif they have homeomorphic ultracopowers; equivalently if their Banach spaces of continuous real-valued functions have isometrically isomorphic Banach ultrapowers (or, approximately satisfy the same positive-bounded sentences). We prove here that any locally connected compact metrizable space co-elementarily equivalent with an arc (resp. a simple closed curve) is itself an arc (resp. a simple closed curve). The hypotheses of metrizability and local connectedness cannot be dropped.
Alternating sequences with nonpositive operators
Authors: Akcoglu M. A.
Published: June 2010

Let

(

T
n

)

({T_n})

be a sequence of linear operators acting on complex valued functions on a

σ
\sigma

-finite measure space. Assume that each

T
n

{T_n}

contracts all the

p
p

-norms,

1
≤
p
≤
∞
(

i

.e

.

‖

T
n

‖

p

≤
1
)

1 \leq p \leq \infty ({\text {i}}{\text {.e}}{\text {.}}{\left \| {{T_n}} \right \|_p} \leq 1)

. It is shown that a.e.

lim
n

T
1
∗

⋯

T
n
∗

T
n

⋯

T
1

f

{\lim _n}T_1^ * \cdots T_n^ * {T_n} \cdots {T_1}f

exists for each

L
p

{L_p}

function

f
,
1
>
p
>
∞

f,1 > p > \infty

.
The use of extended valued normal forms in removing a boundedness condition on an operator inequality
Authors: Brown C. C.
Published: June 2010

The cone of nonnegative extended valued normal forms, defined on the nonnegative bounded symmetric operators in a Hilbert space, is discussed in connection with generalising a known inequality to the unbounded case. The cone of normal forms seems to offer a possibility for seeing unbounded operators in a more general perspective.
Intersections of reflective subcategories
Authors: Adámek J. , Rosický J.
Published: June 2010

It is shown that the class of full reflective subcategories of

F
q

\mathcal {Fq}

(and of other concrete categories) is not closed under intersections. This answers a question raised by Herrlich in 1967. A natural example of nonreflective intersection is presented in the category of bitopological spaces.
Near-fields associated with invariant linear 𝜅-relations
Authors: Fuchs Peter , Maxson C. J.
Published: June 2010

In this paper we investigate a construction method for subnearrings of

M

(
G
)

M\left ( G \right )

proposed by H. Wielandt using subgroups of direct powers

G
κ

{G^\kappa }

of

G
G

called invariant linear

κ
\kappa

-relations. If

κ
=
2

\kappa = 2

we characterize, in terms of properties of these subgroups, when the associated near-rings are near-fields and prove that every near-field arising from an invariant linear

2
2

-relation must be a field.
Generating functions for relatives of classical polynomials
Authors: Barry P. D. , Hurley D. J.
Published: June 2010

For several classical polynomials

u
n

(
x
)

{u_n}(x)

satisfying a second order linear differential equation

D
n

(
x
)

{D_n}(x)

, there is a generating function

u
(
x
,
t
)
=

∑

n
=
0

∞

u
n

(
x
)

t
n

u(x,t) = \sum \nolimits _{n = 0}^\infty {{u_n}(x){t^n}}

. We provide expansions

υ
(
x
,
t
)
=

∑

n
=
0

∞

υ
n

(
x
)

t
n

\upsilon (x,t) = \sum \nolimits _{n = 0}^\infty {{\upsilon _n}(x){t^n}}

where

υ
n

(
x
)

{\upsilon _n}(x)

is a second solution of

D
n

(
x
)

{D_n}(x)

.
A new proof of an inequality of Littlewood and Paley
Authors: Luecking Daniel H.
Published: June 2010

A fairly elementary new proof is presented of the inequality

(
p
≥
2
)

(p \geq 2)

:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue h Superscript prime Baseline EndAbsoluteValue Superscript p Baseline left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue right-parenthesis Superscript p minus 1 Baseline d x d y less-than-or-equal-to double-vertical-bar h double-vertical-bar Subscript upper H Sub Superscript p Subscript Superscript p Baseline comma f element-of upper H Superscript p Baseline period">

∫

|

h
′

|

p

(

1
−

|
z
|

)

p
−
1

d
x
d
y
≤

‖
h
‖

H
p

p

,

f
∈

H
p

.

\int {{{\left | {h’} \right |}^p}{{\left ( {1 - \left | z \right |} \right )}^{p - 1}}dxdy \leq \left \| h \right \|_{{H^p}}^p} ,\quad f \in {H^p}.

\]
In addition, the inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartAbsoluteValue h EndAbsoluteValue Superscript p minus s Baseline StartAbsoluteValue h prime EndAbsoluteValue Superscript s Baseline left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue right-parenthesis Superscript s minus 1 Baseline d x d y less-than-or-equal-to double-vertical-bar h double-vertical-bar Subscript upper H Sub Superscript p Superscript p Baseline">

∫

|
h
|

p
−
s

|

h
′

|

s

(
1
−

|
z
|

)

s
−
1

d
x
d
y
≤

‖
h
‖

H
p

p

\int {{{\left | h \right |}^{p - s}}{{\left | {h’} \right |}^s}{{(1 - \left | z \right |)}^{s - 1}}dxdy \leq \left \| h \right \|_{{H^p}}^p}

\]
is shown to hold for

h
∈

H
p

,
p
>
0

h \in {H^p},p > 0

, if and only if

2
≤
s
>
p
+
2

2 \leq s > p + 2

, generalizing the known case

s
=
2

s = 2

.
On functions that are trivial cocycles for a set of irrationals
Authors: Baggett Larry
Published: June 2010

The main result of this paper is that the set of irrationals, for which a given function is a trivial cocycle, must be of the first category, unless the function is the exponential of a trigonometric polynomial.
Completeness theorems for universal and implicational logics of algebras via congruences
Authors: Quackenbush Robert W.
Published: June 2010

In this paper, simple algebraic proofs are given for the completeness theorems for the implicational and universal logics of algebras. The proofs are obtained by examining congruences,θ\theta, on the algebra of terms,F(ω)F(\omega ), such thatF(ω)/θF(\omega )/\thetabelongs to the given class of algebras. Thus, they are direct analogs of G. Birkhoff’s proof of the completeness theorem for equational logic.
Finite extensions of rings
Authors: Cortzen Barbara , Small Lance W.
Published: June 2010

The paper concerns some cases of ring extensions

R
⊂
S

R \subset S

, where

S
S

is finitely generated as a right

R
R

-module and

R
R

is right Noetherian. In

\S 

1

{\text {\S }}1

it is shown that if

R
R

is a Jacobson ring, then so is

S
S

, with the converse true in the

PI

{\text {PI}}

case. In

\S 

2

{\text {\S }}2

we show that if

S
S

is semiprime

PI

{\text {PI}}

,

R
R

must also be left (as well as right) Noetherian and

S
S

is finitely generated as a left .

R
R

-module.

\S 

3

{\text {\S }}3

contains a result on

E
E

-rings.
An extension of Ando-Krieger’s theorem to ordered Banach spaces
Authors: Caselles V.
Published: June 2010

In this paper it is shown that an operator defined on a suitable ordered Banach space of measurable functions by a positive, irreducible kernel is never quasi-nilpotent, thus giving an extension of Ando-Krieger’s theorem for operators defined on ordered Banach spaces.
A rigidity theorem for quaternionic-Kähler manifolds
Authors: LeBrun Claude
Published: June 2010

Let

(
M
,
g
)

(M,g)

be a compact quaternionic-Kähler manifold of dimension

≥
8

\geq 8

and positive scalar curvature. It is shown that

(
M
,
g
)

(M,g)

has no nontrivial deformations through quaternionic Kähler manifolds.
Undefinable classes and definable elements in models of set theory and arithmetic
Authors: Enayat Ali
Published: June 2010

Every countable model

M

{\mathbf {M}}

of PA or ZFC, by a theorem of S. Simpson, has a "class"

X
X

which has the curious property: Every element of the expanded structure

(

M

,
X
)

({\mathbf {M}},X)

is definable. Here we prove: Theorem A. Every completion

T
T

of PA has a countable model

M

{\mathbf {M}}

(indeed there are

2

ω

{{\mathbf {2}}^\omega }

many such

M

{\mathbf {M}}

’s for each

T
T

) which is not pointwise definable and yet becomes pointwise definable upon adjoining any undefinable class

X
X

to

M

{\mathbf {M}}

. Theorem B. Let

M

⊨

ZF

+
V
=

HOD

{\mathbf {M}} \vDash {\text {ZF}} + V = {\text {HOD}}

be a well-founded model of any cardinality. There exists an undefinable class

X
X

such that the definable points of

M

{\mathbf {M}}

and

(

M

,
X
)

({\mathbf {M}},X)

coincide. Theorem C. Let

M

⊨

PA

{\mathbf {M}}\, \vDash {\text {PA}}

or

ZF

+
V
=

HOD

{\text {ZF}} + V = {\text {HOD}}

. There exists an undefinable class

X
X

such that the definable points of

M

{\mathbf {M}}

and

(

M

,
X
)

({\mathbf {M}},X)

coincide if one of the conditions below is satisfied. (A) The definable elements of

M

{\mathbf {M}}

are cofinal in

M

{\mathbf {M}}

. (B)

M

{\mathbf {M}}

is recursively saturated and

cf
⁡
(

M

)
=
ω

\operatorname {cf}(\mathbf {M}) = \omega

.
A comparison principle for large deviations
Authors: Baxter John R. , Jain Naresh C.
Published: June 2010

If

{

μ
n

}

\{ {\mu _n}\}

and

{

ν
n

}

\left \{ {{\nu _n}} \right \}

are two sequences of probability measures on a separable metric space, we give conditions under which

{

μ
n

}

\left \{ {{\mu _n}} \right \}

satisfies a large deviation principle if and only if

{

ν
n

}

\left \{ {{\nu _n}} \right \}

does. A known and a new theorem follow immediately from the application of this comparison principle to standard results in large deviation theory.
Congruence and one-dimensionality of metric spaces
Authors: Janos Ludvik
Published: June 2010

Two subsets

A
A

and

B
B

of a metric space

(
X
,
d
)

(X,d)

are said to be congruent if there is a bijection between them which preserves the distance

d
d

. We show that if a separable locally compact metric space is such that no distinct subsets of cardinality 3 are congruent then its dimension is

≤
1

\leq 1

. We also show that the real line

R

\mathbb {R}

can be given a compatible metric with this property.
Fixed point sets of homeomorphisms of metric products
Authors: Martin John R.
Published: June 2010

In this paper it is investigated as to when a nonempty closed subset

A
A

of a metric product

X
X

containing intervals or spheres as factors can be the fixed point set of an autohomeomorphism of

X
X

. It is shown that if

X
X

is the Hilbert cube

Q
Q

or contains either the real line

R
R

or a

(
2
n
−
1
)

(2n - 1)

-sphere

S

2
n
−
1

{S^{2n - 1}}

as a factor, then

A
A

can be any nonempty closed subset. In the case where

A
A

is in

Int
⁡
(

B

2
n
+
1

)

\operatorname {Int}({B^{2n + 1}}{\text {)}}

, the interior of the closed unit

(
2
n
+
1
)
−

ball 

B

2
n
+
1

,

(2n + 1) - {\text {ball }}{B^{2n + 1}},

, a strong necessary condition is given. In particular, for

B
3

,
A

{B^3},A

can neither be a nonamphicheiral knot nor a standard closed or nonplanar bordered surface.
A range theorem for the Radon transform
Authors: Madych W. R. , Solmon D. C.
Published: June 2010

Conditions are prescribed for a function

g
g

which are sufficient to ensure that it is the Radon transform of a continuous function

f
f

on

R

n

{{\mathbf {R}}^n}

such that

f
(
x
)
=
O
(

|
x
|

−
n
−
k
−
1

)

f(x) = O({\left | x \right |^{ - n - k - 1}})

as

|
x
|

→
∞

\left | x \right | \to \infty

. Roughly speaking, these criteria involve smoothness and the classical polynomial consistency conditions up to order

k
k

on

g
g

. In particular, the result implies Helgason’s Schwartz theorem for the Radon transform [Acta Math. 113 (1965)].
Estimates of positive linear operators on 𝐿^{𝑝}
Authors: Assani I.
Published: June 2010

Let

0
>
α
>
1

0 > \alpha > 1

and

T
T

a positive linear operator on

L
p

,
1
>
p
>
+
∞

{L^p},1 > p > + \infty

, such that

|

|

(
1
−
α
)
I
+
α
T

|

|

p

≤
1

{|| {(1 - \alpha )I + \alpha T} ||_p} \leq 1

(

I
I

= identity). For such operators, (which do not necessarily satisfy (i)

|

|

T

|

|

p

≤
1

{|| T ||_p} \leq 1

(contraction), (ii)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript n greater-than-or-equal-to 1 Endscripts double-vertical-bar left-parenthesis upper I plus upper T plus midline-horizontal-ellipsis plus upper T Superscript n minus 1 Baseline right-parenthesis slash n double-vertical-bar Subscript p Baseline less-than-or-equal-to 1">

sup

n
≥
1

‖

(
I
+
T
+
⋯
+

T

n
−
1

)

/

n

‖

p

≤
1

{\sup _{n \geq 1}}{\left \| {(I + T + \cdots + {T^{n - 1}})/n} \right \|_p} \leq 1

(Cesàro mean bounded by one)) [1] we show, using M. A. Akcoglu’s estimate, that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar sup Underscript StartLayout 1st Row n greater-than-or-equal-to 1 2nd Row n element-of upper N EndLayout Endscripts StartFraction f plus upper T f plus midline-horizontal-ellipsis plus upper T Superscript n minus 1 Baseline f Over n EndFraction double-vertical-bar Subscript p Baseline less-than-or-equal-to gamma left-parenthesis alpha right-parenthesis StartAbsoluteValue EndAbsoluteValue f StartAbsoluteValue EndAbsoluteValue Subscript p Baseline for any f element-of upper L Superscript p Baseline period">

‖

sup

n
≥
1

n
∈
N

f
+
T
f
+
⋯
+

T

n
−
1

f

n

‖

p

≤
γ
(
α
)

|

|

f

|

|

p

for any 

f
∈

L
p

.

{\left \| {\sup \limits _{\begin {array}{*{20}{c}} {n \geq 1} \\ {n \in N} \\ \end {array} } \frac {{f + Tf + \cdots + {T^{n - 1}}f}}{n}} \right \|_p} \leq \gamma (\alpha )||f|{|_p}\;{\text {for any }}f \in {L^p}.

\]
We also obtain the pointwise ergodic theorem in

L
p

{L^p}

.
Inequalities for sums of independent random variables
Authors: Carothers N. L. , Dilworth S. J.
Published: June 2010

A moment inequality is proved for sums of independent random variables in the Lorentz spaces

L

p
,
q

{L_{p,q}}

, thus extending an inequality of Rosenthal. The latter result is used in combination with a square function inequality to give a proof of a Banach space isomorphism theorem. Further moment inequalities are also proved.
Toeplitz spectral inclusion and generalized in modulus property
Authors: Janas J.
Published: June 2010

Let

A
⊂

C
b

(
X
)

A \subset {C_b}(X)

be a function algebra and let

μ
\mu

be a Borel measure on

X
X

and

H
2

(
μ
)
=

A
¯

{H^2}(\mu ) = \bar A

—the closure in

L
2

(
μ
)

{L^2}(\mu )

. It turns out that the spectral inclusion theorem for Toeplitz operators, defined in the above context, implies the density of finite sums

∑
i

|

u
i

|

2

,

u
i

∈
A

\sum \nolimits _i {{{\left | {{u_i}} \right |}^2},{u_i} \in A}

, in (i) the cone of positive functions in

L
1

(
μ
)

{L^1}(\mu )

, (ii) the cone of positive functions in

L
p

(
μ
)
,
p
≤
2

{L^p}(\mu ),p \leq 2

, if

μ
(
X
)

 > 

∞

\mu (X){\text { > }}\infty

, (iii) the cone of positive functions in

C
(
X
)

C(X)

, if

X
X

is compact.