bluuebunny/crossref_metadata_2025_split huggingface dataset:
Properties of weak 𝜃̄-refinable spaces
Authors: Smith J. C. 
Published: June 2010

A space

X
X

is called weak

θ
¯

\overline \theta

-refinable if every open cover of

X
X

has a refinement

⋃

i
=
1

∞

G

i

\bigcup \nolimits _{i = 1}^\infty {{\mathcal {G}_i}}

satisfying (1)

G

i

=
{

G
α

:
α
ϵ

A
i

}

{\mathcal {G}_i} = \{ {G_\alpha }:\alpha \epsilon {A_i}\}

is an open collection for each

i
i

, (2) each

x
ϵ
X

x\epsilon X

has finite positive order with respect to some

G

i

{\mathcal {G}_i}

, (3) the open cover

{

G
i

=
⋃

[

G
α

:
α
ϵ

A
i

]

}

i
=
1

∞

\{ {G_i} = \bigcup {[{G_\alpha }:\alpha \epsilon } {A_i}]\} _{i = 1}^\infty

is point finite. In this paper the author shows that the above property lies between the properties of

θ
\theta

-refinable and weak

θ
\theta

-refinable. The main result is the fact that if

X
X

is countably metacompact and satisfies property

(
δ
)

(\delta )

, every weak

θ
¯

\overline \theta

-cover of

X
X

has a countable subcover. Results concerning paracompactness, metacompactness and the star-finite property are also obtained.
Applications of cluster sets in minimal topological spaces
Authors: Hamlett T. R. 
Published: June 2010

Given a function

f
f

from a topological space

X
X

into a topological space

Y
Y

and a point

x
ϵ
X

x\epsilon X

, the cluster set of

f
f

at

x
x

is

C

(
f
;
x
)
=
∩
{
Cl
⁡
(
f
(
U
)
)
:

U

is a neighborhood of

x
}

\mathcal {C}(f;x) = \cap \{ \operatorname {Cl} (f(U)):\;U\;{\text {is a neighborhood of }}x\}

, where

Cl
⁡
(
U
)

\operatorname {Cl} (U)

denotes the closure of

U
U

. In this paper,

Y
Y

is taken to be a minimal topological space and

C

(
f
;
x
)

\mathcal {C}(f;x)

is used as a tool to obtain information about the continuity of

f
f

.
Bilateral random walks on compact semigroups
Authors: Mukherjea A. , Tserpes N. A. 
Published: June 2010

Let

μ
\mu

be a regular Borel probability measure with support

F
F

on a compact semigroup

S
S

. Let

X
0

,

X

±
1

,

X

±
2

,
⋯

{X_0},{X_{ \pm 1}},{X_{ \pm 2}}, \cdots

be a sequence of independent random variables with values in

S
S

, having identical distribution

P
(

X
n

∈
B
)
=
μ
(
B
)

P({X_n} \in B) = \mu (B)

. The random walk

W
n

=

X

−
n

⋯

X

−
1

X
0

X
1

⋯

X
n

{W_n} = {X_{ - n}} \cdots {X_{ - 1}}{X_0}{X_1} \cdots {X_n}

is studied in this paper. Let

D
D

be the closed semigroup generated by

F
F

and let

K
K

be the kernel of

D
D

. An element

x
∈
D

x \in D

is called recurrent iff

P
x

(

W
n

∈

N
x

i

.o

.

)
=
1

{P_x}({W_n} \in {N_x}{\text {i}}{\text {.o}}{\text {.}}) = 1

for every open neighborhood

N
x

{N_x}

of

x
x

. We prove:

x
x

is essential for

W
n

{W_n}

if and only if

x
∈
K

x \in K

if and only if

x
x

is recurrent if and only if

Σ

P
x

(

W
n

∈

N
x

)
=
Σ

μ
n

∗
[

μ
n

(

x

−
1

⋅
)
]
(

N
x

)
=
∞

\Sigma {P_x}({W_n} \in {N_x}) = \Sigma {\mu ^n} \ast [{\mu ^n}({x^{ - 1}} \cdot )]({N_x}) = \infty

for every

N
x

{N_x}

. Moreover all states in

K
K

are recurrent positive. These results extend results of the authors for the unilateral random walks (using different methods) and recent results of Larisse for the discrete case.
A characterization of discreteness for locally compact groups in terms of the Banach algebras 𝐴_{𝑝}(𝐺)
Authors: Granirer Edmond E. 
Published: June 2010

The Banach algebra

A
A

is said to have the bounded power property if for any

x
∈
A

x \in A

, with

|

|

x

|

|

s
p

=

lim

n
→
∞

|

|

x
n

|

|

1

/

n

⩽
1

||x|{|_{sp}} = {\lim _{n \to \infty }}||{x^n}|{|^{1/n}} \leqslant 1

, one has

sup
n

|

|

x
n

|

|

>
∞

{\sup _n}||{x^n}|| > \infty

. It has been shown by B. M. Schreiber [9, Theorem (8.6)] that, if

G
G

is a locally compact abelian group, then the Fourier algebra

A
(
G
)
=

L
1

(
Γ

)
∧

A(G) = {L^1}{(\Gamma )^ \wedge }

has the bounded power property, if and only if

G
G

is discrete. We improve this result in the Theorem. Let

G
G

be an arbitrary locally compact group and

1
>
p
>
∞

1 > p > \infty

. Then

A
p

(
G
)

{A_p}(G)

has the bounded power property if and only if

G
G

is discrete. Our proof, even for abelian

G
G

and

p
=
2

p = 2

(then

A
2

(
G
)
=
A
(
G
)

{A_2}(G) = A(G)

is the usual Fourier algebra of

G
G

), is much simpler and entirely different from that of [9].
A characterization of 𝐵-slowly varying functions
Authors: Bloom Steven 
Published: June 2010

A measurable function

φ
>
0

\varphi > 0

that satisfies the limit condition

lim

x
→
∞

(
φ
(
x
+
t
φ
(
x
)
)

/

φ
(
x
)
)
=
1

{\lim _{x \to \infty }}(\varphi (x + t\varphi (x))/\varphi (x)) = 1

for all

t
t

is said to be

B
B

-slowly varying. If

φ
\varphi

is continuous, this limit is shown to hold uniformly for

t
t

in compact sets, and an integral representation is derived.
Functions with different strong and weak 𝐹-variations
Authors: Yoder Lane 
Published: June 2010

This paper shows by example how different the strong

Φ
\Phi

-variation can be from the weak

Φ
\Phi

-variation. Let

Φ
\Phi

be a convex function on

[
0
,
∞
)

[0,\infty )

with

Φ
(
0
)
=
0

\Phi (0) = 0

. A continuous function

f
f

on

[
a
,
b
]

[a,b]

is of bounded strong

Φ
\Phi

-variation if

sup
Σ
Φ
(

|

f
(

x
i

)
−
f
(

x

i
−
1

)

|

)
>
∞

\sup \Sigma \Phi (|f({x_i}) - f({x_{i - 1}})|) > \infty

for the partitions of

[
a
,
b
]

[a,b]

. Since

inf
Σ
Φ
(

|

f
(

x
i

)
−
f
(

x

i
−
1

)

|

)
=
0

\inf \Sigma \Phi (|f({x_i}) - f({x_{i - 1}})|) = 0

if

lim

x
→
0

x

−
1

Φ
(
x
)
=
0

{\lim _{x \to 0}}{x^{ - 1}}\Phi (x) = 0

, the weak

Φ
\Phi

-variation is defined as

inf
Σ
Φ
(
ω
(
f
;

x

i
−
1

,

x
i

)
)

\inf \Sigma \Phi (\omega (f;{x_{i - 1}},{x_i}))

, where

ω
(
f
;
c
,
d
)

\omega (f;c,d)

is the oscillation of

f
f

on

[
c
,
d
]

[c,d]

. Of special interest is the case

Φ
(
x
)
=

x
p

,
p
⩾
1

\Phi (x) = {x^p},p \geqslant 1

, in terms of which strong and weak variation dimensions are defined. They are denoted by

dim

s

(
f
)

{\dim _{\text {s}}}(f)

and

dim

w

(
f
)

{\dim _{\text {w}}}(f)

, respectively. By a lemma of Goffman and Loughlin, the Hausdorff dimension

d
d

of the graph of

f
f

provides a lower bound for

dim
w

(
f
)
:
1

/

(
2
−
d
)
⩽

dim
w

(
f
)

{\dim _w}(f):1/(2 - d) \leqslant {\dim _w}(f)

. A Lipschitz condition of order a provides an upper bound for

dim
s

(
f
)
:

dim
s

(
f
)
⩽
1

/

α

{\dim _s}(f):{\dim _s}(f) \leqslant 1/\alpha

. Besicovitch and Ursell showed that

1
⩽
d
⩽
2
−
α

1 \leqslant d \leqslant 2 - \alpha

and gave examples to show that

d
d

can take on any value in this interval. It turns out that these examples provide the extreme cases for variation dimensions; i.e.,

dim
w

(
f
)
=
1

/

(
2
−
d
)

{\dim _w}(f) = 1/(2 - d)

and

dim
s

(
f
)
=
1

/

α

{\dim _s}(f) = 1/\alpha

.
Asymptotic stability for abstract nonlinear functional differential equations
Authors: Webb G. F. 
Published: June 2010

The nonlinear autonomous functional differential equation

x
˙

(
t
)
=
f
(
x
(
t
)
)
+
g
(

x
t

)
,
t
⩾
0
,

x
0

=
ϕ

\dot x(t) = f(x(t)) + g({x_t}),t \geqslant 0,{x_0} = \phi

is investigated by means of the theory of semigroups of nonlinear operators. The properties of the semigroup associated with this equation provide stability information about the solutions.
A best constant for Zygmund’s conjugate function inequality
Authors: Bennett Colin 
Published: June 2010

When the space

L

log
+

L

L{\log ^ + }L

is given the Hardy-Littlewood norm the best constant in the corresponding version of Zygmund’s conjugate function inequality is shown to be
\[

K

=

1

−
2

−

3

−
2

+

5

−
2

−

7

−
2

+
⋯

1

−
2

+

3

−
2

+

5

−
2

+

7

−
2

+
⋯

.

{\mathbf {K}} = \frac {{{1^{ - 2}} - {3^{ - 2}} + {5^{ - 2}} - {7^{ - 2}} + \cdots }}{{{1^{ - 2}} + {3^{ - 2}} + {5^{ - 2}} + {7^{ - 2}} + \cdots }}.

\]
This complements the recent result of Burgess Davis that the best constant in Kolmogorov’s inequality is

K

−
1

{{\mathbf {K}}^{ - 1}}

.
Extreme operators in the unit ball of 𝐿(𝐶(𝑋),𝐶(𝑌)) over the complex field
Authors: Gendler Alan 
Published: June 2010

Assume that

X
X

and

Y
Y

are compact Hausdorff spaces and that

C
(
X
)

C(X)

and

C
(
Y
)

C(Y)

are the Banach spaces of continuous complex-valued functions on

X
X

and

Y
Y

, respectively.

L
(
C
(
X
)
,
C
(
Y
)
)

L(C(X),C(Y))

is the space of bounded linear operators from

C
(
X
)

C(X)

to

C
(
Y
)

C(Y)

. If

E
E

is a Banach space, then

S
(
E
)

S(E)

is the closed unit ball in

E
E

. An operator

T
T

in

S
(
L
(
C
(
X
)
,
C
(
Y
)
)
)

S(L(C(X),C(Y)))

is nice if

T
∗

(
ext
⁡
S
(
C

(
Y

)
∗

)
)
⊂
ext
⁡
S
(
C

(
X

)
∗

)

{T^ \ast }(\operatorname {ext} S(C{(Y)^ \ast })) \subset \operatorname {ext} S(C{(X)^ \ast })

. For each

y
∈
Y
,

ε
y

y \in Y,{\varepsilon _y}

denotes point mass at

y
y

. The main theorem states that if

T
T

is extreme in

S
(
L
(
C
(
X
)
,
C
(
Y
)
)
)

S(L(C(X),C(Y)))

and

|

|

T
∗

(

ε
y

)

|

|

=
1

||{T^ \ast }({\varepsilon _y})|| = 1

for all

y
∈
Y

y \in Y

, then

T
T

is nice. Other theorems are proved by using the same techniques as in the proof of the main theorem.
Roth’s equivalence problem in unit regular rings
Authors: Hartwig Robert E. 
Published: June 2010

It is shown that for matrices over a unit regular ring

[

A

a
m
p
;
C

0

a
m
p
;
D

]

∼

[

A

a
m
p
;
0

0

a
m
p
;
D

]

\left [ {\begin {array}{*{20}{c}} A & C \\ 0 & D \\ \end {array} } \right ]\sim \left [ {\begin {array}{*{20}{c}} A & 0 \\ 0 & D \\ \end {array} } \right ]

if and only if there exist solutions

X
X

and

Y
Y

to

A
X
−
Y
D
=
C

AX - YD = C

, thus providing a partial generalization to Roth’s theorem.
Cesàro summability of the conjugate series and the double Hilbert transform
Authors: Basinger John O. 
Published: June 2010

If

f
(
x
,
y
)

f(x,y)

, a

2
π

2\pi

periodic function in each variable, has a modulus of continuity

w
f

(
δ
)
=
o
(
1

/

log
⁡
(
1

/

δ
)
)

{w_f}(\delta ) = o(1/\log (1/\delta ))

then
\[

σ
~

n

(
x
,
y
,
f
)
−

∫

1

/

n

π

∫

1

/

n

π

[
f
(
x
+
u
,
y
+
v
)
−
f
(
x
−
u
,
y
+
v
)
−
f
(
x
+
u
,
y
−
v
)
+
f
(
x
−
u
,
y
−
v
)
]

4
tan
⁡
(
u

/

2
)
tan
⁡
(
v

/

2
)

d
u
d
v
→
0

uniformly

in

(
x
,
y
)

{\tilde \sigma _n}(x,y,f) - \int _{1/n}^\pi {\int _{1/n}^\pi {\frac {{[f(x + u,y + v) - f(x - u,y + v) - f(x + u,y - v) + f(x - u,y - v)]}}{{4\tan (u/2)\tan (v/2)}}} } dudv \to 0\quad {\text {uniformly}}\;{\text {in}}\;(x,y)

\]
where

σ
~

n

(
x
,
y
,
f
)

{\tilde \sigma _n}(x,y,f)

is the first arithmetic mean of the conjugate series. This theorem is best possible in that

o
(
1

/

log
⁡
(
1

/

δ
)
)

o(1/\log (1/\delta ))

cannot be replaced by

O
(
1

/

log
⁡
(
1

/

δ
)
)

O(1/\log (1/\delta ))

.
Deficiencies of the associated curves of a holomorphic curve in the projective space
Authors: Niinō Kiyoshi 
Published: June 2010

Let

k

x

_kx

be the nonconstant associated holomorphic curve of rank

k
(
1
≦
k
≦
n
−
1
)

k(1 \leqq k \leqq n - 1)

of a transcendental holomorphic curve

x
:

C

→

P
n

C

x:{\mathbf {C}} \to {P_n}{\mathbf {C}}

. It is proved that if

1
≦
k
≦
n
−
2

1 \leqq k \leqq n - 2

and

A
j
k

∈

P

l
(
k
)
−
1

C

,
j
=
1
,
…
,
2
l
(
k
)
−
2
(
l
(
k
)
=

(

k
+
1

n
+
1

)
)

A_j^k \in {P_{l(k) - 1}}{\mathbf {C}},j = 1, \ldots ,2l(k) - 2(l(k) = (_{k + 1}^{n + 1}))

are in general position and

⟨
k

x
,

A
j
k

⟩
≢
0

{\langle _k}x,A_j^k\rangle \not \equiv 0

for all

A
j
k

A_j^k

, then

∑

j
=
1

2
l
(
k
)
−
2

δ
k

(

A
j
k

)
≦
2
l
(
k
)
−
3

\sum \nolimits _{j = 1}^{2l(k) - 2} {{\delta _k}(A_j^k) \leqq 2l(k) - 3}

and that in the case when

k
=
n
−
1
,

∑

A

n
−
1

δ

n
−
1

(

A

n
−
1

)
≦
l
(
n
−
1
)

k = n - 1,\sum \nolimits _{{A^{n - 1}}} {{\delta _{n - 1}}({A^{n - 1}}) \leqq l(n - 1)}

, where

{

A

n
−
1

}

\{ {A^{n - 1}}\}

is a finite subset of

P

l
(
n
−
1
)
−
1

C

{P_{l(n - 1) - 1}}{\mathbf {C}}

in general position such that

⟨

n
−
1

x
,

A

n
−
1

⟩
≢
0

{\langle _{n - 1}}x,{A^{n - 1}}\rangle \not \equiv 0

for all

A

n
−
1

{A^{n - 1}}

. These are sharp.
The meaning of the Cauchy-Schwarz-Buniakovsky inequality
Authors: Zarantonello Eduardo H. 
Published: June 2010

It is proved that a mapping

T
:
X
→

X

∗

T:X \to {X^{\ast }}

from a topological real vector space into its dual satisfies the inequality

⟨
T
y
,
x
⟩
⩽

⟨
T
x
,
x

⟩

1

/

2

⋅

⟨
T
y
,
y

⟩

1

/

2

\langle Ty,x\rangle \leqslant {\langle Tx,x\rangle ^{1/2}} \cdot {\langle Ty,y\rangle ^{1/2}}

if and only if it is the restriction of a positively homogeneous subdifferential operator.
Nonstandard measure theory–Hausdorff measure
Authors: Wattenberg Frank 
Published: June 2010

In this paper it is shown that the Hausdorff measures

λ
t

{\lambda ^t}

for

t
∈
[
0
,
∞
)

t \in [0,\infty )

can be simultaneously represented as

∗

^\ast

finite counting measures in an appropriate nonstandard model. That is, the following theorem is proved. Theorem. Suppose X is a metric space and

ν
\nu

is an infinite positive

∗

^\ast

integer. Then there is a

∗

^\ast

finite set G such that for every standard

t
∈
[
0
,
∞
)

t \in [0,\infty )

and every

λ
t

{\lambda ^t}

-integrable Borel function,

f
:
X
→

R

f:X \to {\mathbf {R}}

,
\[

∫

f

d

λ
t

=

St

(

1

ν
t

∑

x
∈
G

∗
f
(
x
)

)

.

\int {f\;d{\lambda ^t}} = {\text {St}}\left ( {\frac {1}{{{\nu ^t}}}\sum \limits _{x \in G} {*f(x)} } \right ).

\]
Generalizations of l’Hôpital’s rule
Authors: Lee Cheng Ming 
Published: June 2010

An essential limit, similar to the concept of essential bounded functions, is defined and briefly discussed. Using the essential limit, l’Hôpital’s rule is generalized to include the following theorem as a special case. Theorem. Let F, G be real-valued functions defined on the open interval (a, b). Suppose that the approximate derivatives

F

ap

′

(
x
)

{F’_{{\text {ap}}}}(x)

and

G

ap

′

(
x
)

{G’_{{\text {ap}}}}(x)

exist finitely,

G

ap

′

(
x
)
>
0

{G’_{{\text {ap}}}}(x) > 0

for almost all x in (a, b), and the extreme approximate derivates of both F and G are finite nearly everywhere in (a, b). Then

lim

x
→
a
+

[
F
(
x
)

/

G
(
x
)
]
=

ess

lim

x
→
a
+

[

F

ap

′

(
x
)

/

G

ap

′

(
x
)
]

{\lim _{x \to a + }}[F(x)/G(x)] = {\text {ess}}\;{\lim _{x \to a + }}[{F’_{{\text {ap}}}}(x)/{G’_{{\text {ap}}}}(x)]

provided that the essential limit in the right-hand side exists and that

lim

x
→
a
+

F
(
x
)
=

lim

x
→
a
+

G
(
x
)
=
0

{\lim _{x \to a + }}F(x) = {\lim _{x \to a + }}G(x) = 0

or

lim

x
→
a
+

G
(
x
)
=
−
∞

{\lim _{x \to a + }}G(x) = - \infty

.
A note on generators of semigroups
Authors: Tsui Sze Kai 
Published: June 2010

The generator T of a norm-continuous semigroup of identity preserving positive linear mappings on a

C
∗

{C^\ast }

-algebra

A

\mathfrak {A}

is characterized as one that satisfies

T
(

u
∗

u
)
⩾

u
∗

T
(
u
)
+
T
(

u
∗

)
u

T({u^\ast }u) \geqslant {u^\ast }T(u) + T({u^\ast })u

for all unitary elements u in

A

\mathfrak {A}

.
Schur multipliers of some finite nilpotent groups
Authors: Jackson David A. 
Published: June 2010

Let B denote the Burnside group,

B
(

p
α

,
d
)

B({p^\alpha },d)

and let

G
=
B

/

B
k

G = B/{B_k}

where p is a prime and

1
>
k
>
p

1 > k > p

. We show that the Schur multiplier,

M
(
G
)

M(G)

, is a direct power of

Ψ
(
k
,
d
)

\Psi (k,d)

cyclic groups, each having order

p
α

{p^\alpha }

, where

Ψ
(
k
,
d
)
=

k

−
1

Σ

n

|

k

μ
(
k

/

n
)

d
n

\Psi (k,d) = {k^{ - 1}}{\Sigma _{n|k}}\mu (k/n){d^n}

. (This is Witt’s formula for the rank of

F
k

/

F

k
+
1

{F_k}/{F_{k + 1}}

where F is free on d generators.) In addition we can show that

M
(
B
(
3
,
d
)
)

M(B(3,d))

is elementary abelian of exponent 3 and rank

2

(
2
d

)
+
4

(
3
d

)
+
3

(
4
d

)

2(_2^d) + 4(_3^d) + 3(_4^d)

.
The automorphism group of 2_{𝐹₄}(2)’
Authors: Weiss Richard 
Published: June 2010

We give a new proof that

Aut

(
2

F
4

(
2

)
′

)
≅

Aut

(
2

F
4

(
2
)
)
≅

2

F
4

(
2
)

{\operatorname {Aut}}{(^2}{F_4}(2)’) \cong {\operatorname {Aut}}{(^2}{F_4}(2)) \cong {\;^2}{F_4}(2)

(see [4]).
A ternary function for distributivity and permutability of an equivalence lattice
Authors: Korec Ivan 
Published: June 2010

The main result of the paper is Theorem 1. Let A be a countable set and L be a complete sublattice of the equivalence lattice on A. The following are equivalent (i) L is a distributive lattice of permutable equivalence relations. (ii) There is an algebra with congruence lattice L among the fundamental operations of which is a ternary function f with the property

(

1

)

f
(
a
,
b
,
b
)
=
f
(
a
,
b
,
a
)
=
f
(
b
,
b
,
a
)
=
a

\begin{equation}\tag {$1$} \quad f(a,b,b) = f(a,b,a) = f(b,b,a) = a\end{equation}

for all

a
,
b
∈
A

a, b \in A

.
Local ergodic theorems for noncommuting semigroups
Authors: McGrath S. A. 
Published: June 2010

Let

(
X
,
μ
)

(X,\mu )

be a

σ
\sigma

-finite measure space and

L
p

(
μ
)
,
1
⩽
p
⩽
∞

{L_p}(\mu ),1 \leqslant p \leqslant \infty

, the usual Banach spaces of complex-valued functions. For

k
=
1
,
2
,
…
,
n

k = 1,2, \ldots ,n

, let

{

T
k

(
t
)
:
t
⩾
0
}

\{ {T_k}(t):t \geqslant 0\}

be a strongly continuous semigroup of Dunford-Schwartz operators. If
\[

f
∈

R

n
−
1

=

{

g
:

∫

|

g

|

>
t

|

g

/

t

|

(
log
⁡

|

g

/

t

|

)

n
−
1

d
μ
>
∞

for

all

t
>
0

}

,

f \in {R_{n - 1}} = \left \{ {g:\int _{|g| > t} {|g/t|{{(\log |g/t|)}^{n - 1}}d\mu > \infty {\text {for}}\;{\text {all}}\;t > 0} } \right \},

\]
then
\[

1

α
1

α
2

⋯

α
n

∫
0

α
n

⋯

∫
0

α
i

T
n

(

t
n

)
⋯

T
1

(

t
1

)
f
(
x
)
d

t
1

⋯
d

t
n

→
f
(
x
)

\frac {1}{{{\alpha _1}{\alpha _2} \cdots {\alpha _n}}}\int _0^{{\alpha _n}} \cdots \int _0^{{\alpha _i}} {{T_n}({t_n}) \cdots {T_1}({t_1})f(x)d{t_1} \cdots d{t_n} \to f(x)}

\]

μ
\mu

-a.e. as

α
1

↘
0
,
…
,

α
n

↘
0

{\alpha _1} \searrow 0, \ldots ,{\alpha _n} \searrow 0

independently. If

f
∈

L
p

(
μ
)
,
1
>
p
>
∞

f \in {L_p}(\mu ),1 > p > \infty

, then the limit exists in norm as well as pointwise.
Taylor-Dirichlet series and algebraic differential-difference equations
Authors: Wadleigh Frank 
Published: June 2010

It is proved that if a convergent Taylor-Dirichlet series
\[

∑

k
=
0

∞

P
k

(
s
)

e

−

λ
k

s

,

s
=
σ
+
i
t
,

λ
k

∈

C

,

P
k

(
s
)
∈

C

[
s
]
,

R

(

λ
k

)
↑
∞
,

\sum \limits _{k = 0}^\infty {{P_k}(s){e^{ - {\lambda _k}s}},\quad s = \sigma + it,{\lambda _k} \in {\mathbf {C}},{P_k}} (s) \in {\mathbf {C}}[s],\mathcal {R}({\lambda _k}) \uparrow \infty ,

\]
satisfies an algebraic differential-difference equation then the set of its exponents

{

λ
k

}

k
=
0

∞

\{ {\lambda _k}\} _{k = 0}^\infty

has a finite, linear, integral basis. This generalizes a theorem of A. Ostrowski. An application of the theorem to a problem of oscillation in neuro-muscular systems is indicated.
Cyclic vectors of Lambert’s weighted shifts
Authors: Yadav B. S. , Chatterjee S. 
Published: June 2010

Let

B
(
H
)

B(H)

denote the Banach algebra of all bounded linear operators on an infinite-dimensional separable complex Hilbert space H, and let

l
2

{l^2}

be the Hilbert space of all square-summable complex sequences

x
=
{

x
0

,

x
1

,

x
2

,
…
}

x = \{ {x_0},{x_1},{x_2}, \ldots \}

. For an injective operator A in

B
(
H
)

B(H)

and a nonzero vector f in H, put

w
m

=

‖

A
m

f

‖

/

‖

A

m
−
1

f

‖

,
m
=
1
,
2
,
…
.

{w_m} = \left \| {{A^m}f} \right \| / \left \| {{A^{m - 1}}f} \right \|,m = 1,2, \ldots .

The operator

T

A
,
f

{T_{A,f}}

on

l
2

{l^2}

, defined by

T

A
,
f

(
x
)
=
{

w
1

x
1

,

w
2

x
2

,
…
}

{T_{A,f}}(x) = \{ {w_1}{x_1},{w_2}{x_2}, \ldots \}

, is called a weighted (backward) shift with the weight sequence

{

w
m

}

m
=
1

∞

\{ {w_m}\} _{m = 1}^\infty

. This paper is concerned with the investigation of the existence of cyclic vectors of

T

A
,
f

{T_{A,f}}

. Also it is shown that if A satisfies certain nice conditions, then every transitive subalgebra of

B
(
H
)

B(H)

containing

T

A
,
f

{T_{A,f}}

coincides with

B
(
H
)

B(H)

.
Variability regions for bounded analytic functions with applications to families defined by subordination
Authors: Abu-Muhanna Yusuf , MacGregor Thomas H. 
Published: June 2010

We examine the set of points

(
φ
(
ζ
)
,

φ
′

(
ζ
)
,
…
,

φ

(
n
)

(
ζ
)
)

(\varphi (\zeta ),\varphi ’(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))

where

|

ζ

|

>
1

|\zeta | > 1

and

φ
\varphi

varies over the class of functions analytic in the open unit disk and is either (1) uniformly bounded or (2) subordinate to a given univalent function. In each case boundary points of the set correspond to unique functions associated with finite Blaschke products. This yields information about the form of solutions to extremal problems over the classes, including the problem
\[

max
Re
⁡
F
(
φ
(
ζ
)
,

φ
′

(
ζ
)
,
…
,

φ

(
n
)

(
ζ
)
)

\max \operatorname {Re} F(\varphi (\zeta ),\varphi ’(\zeta ), \ldots ,{\varphi ^{(n)}}(\zeta ))

\]
where

|

ζ

|

>
1

|\zeta | > 1

and F is analytic.
Some inequalities for entire functions
Authors: Saitoh Saburou 
Published: June 2010

For any entire functions

φ
(
z
)

\varphi (z)

and

ψ
(
z
)

\psi (z)

on C with finite norm
\[

{

1
π

∫

∫

C

|

f
(
z
)

|

2

exp
⁡
(
−

|

z

|

2

)
d
x

d
y

}

1

/

2

>
∞
,

{\left \{ {\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|f(z){|^2}\exp ( - |z{|^2})dx\;dy} } } \right \}^{1/2}} > \infty ,

\]
we show that the inequality
\[

2
π

∫

∫

C

|

φ
(
z
)
ψ
(
z
)

|

2

exp
⁡
(
−
2

|

z

|

2

)

d
x

d
y

⩽

1
π

∫

∫

C

|

φ
(
z
)

|

2

exp
⁡
(
−

|

z

|

2

)

d
x

d
y

1
π

∫

∫

C

|

ψ
(
z
)

|

2

exp
⁡
(
−

|

z

|

2

)

d
x

d
y

\begin {array}{*{20}{c}} {\frac {2}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z)\psi (z){|^2}\exp ( - 2|z{|^2})\;dx\;dy} } } \hfill \\ { \leqslant \frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\varphi (z){|^2}\exp ( - |z{|^2})\;dx\;dy\frac {1}{\pi }\int {\int \limits _{\mathbf {C}} {|\psi (z){|^2}\exp ( - |z{|^2})\;dx\;dy} } } } } \hfill \\ \end {array}

\]
holds. This inequality is obtained as a special case of a general result. We also refer to some properties of a tensor product of spaces of entire functions.
An improved estimate for the Bloch norm of functions in Doob’s class
Authors: Hwang J. S. , Rung D. C. 
Published: June 2010

For any fixed

0
>
ρ
>
2
π
,

D

(
ρ
)

0 > \rho > 2\pi ,\mathcal {D}(\rho )

is the family of all holomorphic functions in

Δ
\Delta

which satisfy (i)

f
(
0
)
=
0

f(0) = 0

, and (ii)

lim
_

z
→
τ

|

f
(
z
)

|

⩾
1

{\underline {\lim } _{z \to \tau }}|f(z)| \geqslant 1

, for all

τ
\tau

lying on some arc

Γ
f

⊆
∂
Δ

{\Gamma _f} \subseteq \partial \Delta

with arclength

|

Γ
f

|

⩾
ρ

|{\Gamma _f}| \geqslant \rho

. We showed that for each

f
∈

D

(
ρ
)

f \in \mathcal {D}(\rho )

there exists a point

z
f

∈
Δ

{z_f} \in \Delta

at which
\[

|

f
′

(

z
f

)

|

(
1
−

|

z
f

|

2

)
⩾

2
e

sin
⁡
(
π
−
(
ρ

/

2
)
)

(
π
−
(
ρ

/

2
)
)

.

|f’({z_f})|(1 - |{z_f}{|^2}) \geqslant \frac {2}{e}\frac {{\sin (\pi - (\rho /2))}}{{(\pi - (\rho /2))}}.

\]
In this paper we improve this estimate by replacing the quantity

π
−
(
ρ

/

2
)

\pi - (\rho /2)

with a value

θ
(
ρ
)

\theta (\rho )

which lies between 0 and

π
−
(
ρ

/

2
)

\pi - (\rho /2)

and so improves the estimate. The value

θ
(
ρ
)

\theta (\rho )

is defined as the (unique) solution in this interval of the equation

F
ρ

(
θ
)
=
log
⁡
(
cot
⁡
(
ρ

/

4
)
cot
⁡
(
θ

/

2
)
)
−
θ

/

sin
⁡
θ
=
0.

{F_\rho }(\theta ) = \log (\cot (\rho /4)\cot (\theta /2)) - \theta /\sin \theta = 0.

.
Exponential sums related to binomial coefficient parity
Authors: Stein Alan H. 
Published: June 2010

Let

α
(
n
)

\alpha (n)

be the number of 1’s in the binary expansion of n,

z
>
0

z > 0

and

ϕ
z

(
x
)
=

Σ

n
>
x

z

α
(
n
)

{\phi _z}(x) = {\Sigma _{n > x}}{z^{\alpha (n)}}

. Let

θ
z

=
(
log
⁡
(
1
+
z
)
)

/

log
⁡
2
,
a
(
z
)
=
lim
inf

x

−

θ
z

ϕ
z

(
x
)
,
b
(
z
)
=
lim
sup

x

−

θ
z

ϕ
z

(
x
)

{\theta _z} = (\log (1 + z))/\log 2,a(z) = \lim \inf {x^{ - {\theta _z}}}{\phi _z}(x),b(z) = \lim \sup {x^{ - {\theta _z}}}{\phi _z}(x)

. Then

0
>
a
(
z
)
⩽
1
⩽
b
(
z
)
>
2

0 > a(z) \leqslant 1 \leqslant b(z) > 2

. Furthermore, if

z
≠
1

z \ne 1

, then

a
(
z
)
>
b
(
z
)

a(z) > b(z)

.