bluuebunny/crossref_metadata_2025_split huggingface dataset:
Reproduction and Resistance to Stress: When and How
Authors: Wingfield J. C. , Sapolsky R. M. 
Published: July 2010

AbstractEnvironmental and social stresses have deleterious effects on reproductive function in vertebrates. Global climate change, human disturbance and endocrine disruption from pollutants are increasingly likely to pose additional stresses that could have a major impact on human society. Nonetheless, some populations of vertebrates (from fish to mammals) are able to temporarily resist environmental and social stresses, and breed successfully. A classical trade‐off of reproductive success for potential survival is involved. We define five examples. (i) Aged individuals with minimal future reproductive success that should attempt to breed despite potential acute stressors. (ii) Seasonal breeders when time for actual breeding is so short that acute stress should be resisted in favour of reproductive success. (iii) If both members of a breeding pair provide parental care, then loss of a mate should be compensated for by the remaining individual. (iv) Semelparous species in which there is only one breeding period followed by programmed death. (v) Species where, because of the transience of dominance status in a social group, individuals may only have a short window of opportunity for mating. We suggest four mechanisms underlying resistance of the gonadal axis to stress. (i) Blockade at the central nervous system level, i.e. an individual no longer perceives the perturbation as stressful. (ii) Blockade at the level of the hypothalamic‐pituitary‐adrenal axis (i.e. failure to increase secretion of glucocorticosteroids). (iii) Blockade at the level of the hypothalamic‐pituitary‐gonad axis (i.e. resistance of the reproductive system to the actions of glucocorticosteroids). (iv) Compensatory stimulation of the gonadal axis to counteract inhibitory glucocorticosteroid actions. Although these mechanisms are likely genetically determined, their expression may depend upon a complex interaction with environmental factors. Future research will provide valuable information on the biology of stress and how organisms cope. Such mechanisms would be particularly insightful as the spectre of global change continues to unfold.
Central Blockade of Oxytocin Receptors During Late Gestation Disrupts Systemic Release of Oxytocin During Suckling In Rats
Authors: Lipschitz D. L. , Crowley W. R. , Bealer S. L. 
Published: July 2010

AbstractThere is evidence that the central oxytocin system is activated and undergoes reorganization before parturition. The present study was designed to determine the effects of central oxytocin receptor blockade during late gestation on parturition, pup growth, and oxytocin release during suckling. Female Sprague‐Dawley rats were implanted on gestation day 12–14 with Alzet® osmotic minipumps containing an oxytocin receptor antagonist (d(CH2)5, Tyr (Me)2, Orn8–vasotocin; OT‐X) or artificial cerebrospinal fluid (VEH), which was infused into the third cerebral ventricle. Pumps were removed within 24 h of parturition. Daily maternal body weight and food intake were monitored during gestation and lactation. The length of gestation, duration of parturition, pup number, litter weight and interbirth interval were recorded. Subsequently, pup number and litter weights were recorded daily until lactation day 10 or 11, when maternal and pup behaviour, and plasma oxytocin concentration before and during suckling were measured. Central oxytocin blockade had no effect on the timing of parturition, maternal behaviour, litter size, still births, or litter weights at birth. However, beginning on day 3 of lactation, average weights of litters of OT‐X females were significantly lower than litters of VEH‐treated females. Furthermore, while basal plasma oxytocin concentrations, oxytocin increases in response to suckling and dam/pup interactions did not differ between groups, a significant delay in suckling‐induced systemic oxytocin release was observed in OT‐X females. Finally, OT‐X dams weighed less than VEH dams during the postpartum observation period, although food intakes were similar. These data suggest that central actions of oxytocin during late gestation are necessary for the normal timing of systemic release of oxytocin during suckling, normal pup weight gain, and maintenance of maternal body weight.
MT1 Melatonin Receptor mRNA Expressing Cells in the Pars Tuberalis of the European Hamster: Effect of Photoperiod
Authors: Dardente H. , Klosen P. , Pévet P. , Masson‐Pévet M. 
Published: July 2010

AbstractMelatonin, secreted only during the night by the pineal gland, transduces the photoperiodic message to the organism. One important target for the hormone is the pars tuberalis (PT) of the adenohypophysis which displays a very high number of melatonin binding sites in mammals and is implicated in the seasonal regulation of prolactin secretion. To gain insight into the mechanism by which the melatonin signal is decoded in the PT, we studied the effect of photoperiod on the PT cells expressing the MT1 melatonin receptor in a highly photoperiodic species, the European hamster. Recently, we showed that, in the rat, the MT1 receptor mRNA is expressed in PT‐specific cells characterized by their expression of β‐thyroid stimulating hormone (β‐TSH) along with the α‐glycoprotein subunit (α‐GSU). As the cellular composition of the PT shows variability among species, we first identified the cell type expressing the MT1 receptor in the European hamster by combining immunocytochemistry and nonradioactive in situ hybridization for the MT1 receptor mRNA. Our results show that, in the European hamster, as in the rat, the MT1 receptor is only expressed by the PT‐specific‐cells, β‐TSH and α‐GSU positive. In a second step, we analysed the effects of photoperiod on the MT1 mRNA, and on β‐TSH and α‐GSU both at the mRNA and protein levels. Our data show that, compared to long photoperiod, short photoperiod induces a dramatic decrease of MT1, β‐TSH and α‐GSU expression. Protein levels of β‐TSH and α‐GSU were also dramatically reduced in short photoperiod. Together, our data suggest that melatonin exerts its seasonal effects in the PT by signalling to PT specific‐cells through the MT1 receptor subtype.
Relations among characteristic classes of 𝑛-manifolds imbedded in 𝑅^{𝑛+𝑘}
Authors: Papastavridis Stavros 
Published: July 2010

Let

I
n

⊆

H
∗

(
B
O
;

Z
2

)

{I_n} \subseteq {H^ \ast }(BO;{Z_2})

be the (graded) set of those normal characteristic classes which are zero on all compact, closed

C
∞

{C^\infty }

manifolds. Let

I

n
,
k

⊆

H
∗

(
B
O
;

Z
2

)

{I_{n,k}} \subseteq {H^ \ast }(BO;{Z_2})

be the set of those characteristic classes which are zero on all n-manifolds which imbed in

R

n
+
k

{R^{n + k}}

. Let K be the (graded) ideal in

H
∗

(
B
O
;

Z
2

)

{H^ \ast }(BO;{Z_2})

generated by the Stiefel-Whitney classes

w
k

,

w

k
+
1

,

w

k
+
2

,

w

k
+
3

,
…

{w_k},{w_{k + 1}},{w_{k + 2}},{w_{k + 3}}, \ldots

. We will prove the following result: If

1
⩽
i
⩽
min
{
(
2
k
−
2
)
,
(
n
+
k
−
1
)

/

2
}

1 \leqslant i \leqslant \min \{ (2k - 2),(n + k - 1)/2\}

, then

I

n
,
k

i

=

I
n
i

+

K
i

I_{n,k}^i = I_n^i + {K^i}

. Also, we will prove an analogous result for manifolds with an extra structure.
Foliation preserving Lie group actions and characteristic classes
Authors: Suzuki Haruo 
Published: July 2010

Let

F

~

\tilde {\mathcal {F}}

be a codimension

k
k

foliation of a manifold

M
M

and

F

\mathcal {F}

a subfoliation of

F

~

\tilde {\mathcal {F}}

with codimension

q
q

. Let a Lie group

G
G

of dimension

k
k

act on

M
M

transversally locally freely to

F

~

\tilde {\mathcal {F}}

and preserving

F

\mathcal {F}

. Let

F

′

\mathcal {F}’

be the foliation determined by

F

\mathcal {F}

and the

G
G

-action. Then we have the following relations between exotic classes of

F

\mathcal {F}

and

F

′

:

α

F

(
[

c
^

I

c
J

]
)
=

α

F

′

(
[

c
^

I

c
J

]
)

\mathcal {F}’:{\alpha _\mathcal {F}}([{\hat c_I}{c_J}]) = {\alpha _{\mathcal {F}’}}([{\hat c_I}{c_J}])

for

I
=
(

i
1

,
…
,

i
λ

)

I = ({i_1}, \ldots ,{i_\lambda })

,

J
=
(

j
1

,
…
,

j
μ

)

J = ({j_1}, \ldots ,{j_\mu })

,

1
⩽

j
γ

,

j
l

⩽
q
−
k

1 \leqslant {j_\gamma },{j_l} \leqslant q - k

, and

α

F

(
[

c
^

I

c
J

]
)
=
0

{\alpha _\mathcal {F}}([{\hat c_I}{c_J}]) = 0

otherwise.
Lebesgue constants for Jacobi expansions
Authors: Cartwright Donald I. 
Published: July 2010

Sharp estimates are given for the Lebesgue constants

|

|

|

s
n

|

|

|

p

=
sup

{

‖

s
n

f

‖

p

:
f
∈

L
w
p

,

‖
f
‖

p

⩽
1

}

|||{s_n}||{|_p} = \sup \left \{ {{{\left \| {{s_n}f} \right \|}_p}:f \in L_w^p,{{\left \| f \right \|}_p} \leqslant 1} \right \}

for

p
p

outside the Pollard interval

(

p
0
′

,

p
0

)

({p’_0},{p_0})

, where

s
n

f

{s_n}f

is the

n
n

th partial sum of the Jacobi expansion of a function

f
f

which is in the

L
p

{L^p}

space with respect to the weight

w
(
x
)
=

(
1
−
x

)
α

(
1
+
x

)
β

w(x) = {(1 - x)^\alpha }{(1 + x)^\beta }

on

[
−
1
,
1
]

[ - 1,1]

.
On inequalities of periodic functions and their derivatives
Authors: Ditzian Z. 
Published: July 2010

The inequality

‖
f
‖

B

⩽

a
k

‖

f

(
k
)

‖

B

{\left \| f \right \|_B} \leqslant {a_k}{\left \| {{f^{(k)}}} \right \|_B}

is proved for many spaces of periodic functions. An analogue for sequences is also given.
Distortion and coefficient estimation of schlicht functions
Authors: Hu Ke 
Published: July 2010

Let
\[

S
(
k
)
=

{

f
(
z
)
=
z
+

∑

n
=
2

∞

a
n

z
n

∈
S
,

lim
¯

ρ
→

1
−

(
1
−
ρ
)

2

ρ

max

|
z
|

=
ρ

|

f
(
z
)

|

=

α
f

⩾
k
>
0

}

.

S(k) = \left \{ {f(z) = z + \sum \limits _{n = 2}^\infty {{a_n}{z^n} \in S,} \;{{\overline {\lim } }_{\rho \to {1^ - }}}\frac {{{{(1 - \rho )}^2}}}{\rho }\max \limits _{\left | z \right | = \rho } \left | {f(z)} \right | = {\alpha _f} \geqslant k > 0} \right \}.

\]
In this paper, we prove that for

f
(
z
)
∈
S
(
k
)

f(z) \in S(k)

the inequality
\[

|

a
n

|

>
n

P
+

(
1
+
p
)

A

1

/

2

−
p

1
+
P

\left | {{a_n}} \right | > n\sqrt {\frac {{P + \sqrt {(1 + p){A^{1/2}} - p} }}{{1 + P}}}

\]
holds where

p
=

k
2

/

(
1
−

k
2

)

p = {k^2}/(1 - {k^2})

and

1
⩽
A
>

(
1.0657

)
8

1 \leqslant A > {(1.0657)^8}

. This strengthens a recent result of Horowitz.
The Lévy-Lindeberg central limit theorem in 𝐿_{𝑝}, 0<𝑝<1
Authors: Giné Evarist 
Published: July 2010

A

L
p

(
T
,
Σ
,
μ
)

{L_p}(T,\Sigma ,\mu )

-valued r.v.

X
X

,

0
>
p
>
1

0 > p > 1

, satisfies the Lévy-Lindeberg central limit theorem if and only if it is centered and pregaussian, that is, if and only if

∫
T

(
E

X
2

(
t
)
)

p

/

2

d
μ
(
t
)
>
∞

\int \limits _T {{{(E{X^2}(t))}^{p/2}}d\mu (t) > \infty }

E
X
(
t
)
=
0

EX(t) = 0

-a.e. and

μ
\mu

.
Probability measure representation of norms associated with the notion of entropy
Authors: Dąbrowski Romuald 
Published: July 2010

One of the applications of Banach spaces introduced by B. Korenblum [1,2] is a new convergence test [2] for Fourier series including both Dirichlet-Jordan and the Dini-Lipschitz tests [3], The norms of the spaces are given in terms of

κ
\kappa

-entropy where

κ

(
s
)

⩾
0

\kappa \left ( s \right ) \geqslant 0

,

0

>

s
⩽
1

0{\text { > }}s \leqslant 1

, is a nondecreasing concave function such that

κ

(
1
)

=
1

\kappa \left ( 1 \right ) = 1

. The

κ
\kappa

-norms fill the gap between the uniform and the variation norms. The original proof of the general properties of

κ
\kappa

-norms uses both combinatorial and approximation arguments which are rather complicated. We give a simple proof introducing a probabilistic representation of the norms so that the

κ
\kappa

-norm of a real function

f
f

on

T
=
R

/

2
π
Z

T = R / 2\pi Z

is the expectation of the mean oscillation of

f
f

on a subinterval of

T
T

, chosen in a suitable random process.
Stability of typical continuous functions with respect to some properties of their iterates
Authors: Smítal J. , Neubrunnová K. 
Published: July 2010

Let

I
I

be a real compact interval, and let

C
C

be the space of continuous functions

I
→
I

I \to I

with the uniform metric. For

f
∈
C

f \in C

denote

ν
(
f
)
=

sup

x
∈
I

(
lim

sup

n
→
x

f
n

(
x
)
−
lim

inf

n
→
x

f
n

(
x
)
)

\nu (f) = {\sup _{x \in I}}(\lim {\sup _{n \to x}}{f^n}(x) - \lim {\inf _{n \to x}}{f^n}(x))

, where

f
n

{f^n}

is the

n
n

th iterate of

f
f

. Then for each positive

d
d

there is an open set

C
∗

{C^*}

dense in

C
C

such that the oscillation of

v
v

at each point of

C
∗

{C^*}

is less than

d
d

. Consequently,

ν
\nu

is continuous in

C
C

except of the points of a first Baire category set.
Weak convergence of the empirical characteristic function
Authors: Yukich J. E. 
Published: July 2010

Let

P
P

be a probability measure on

R

{\mathbf {R}}

such that the density

f
(
x
)

f(x)

for

P
P

exists and there exists

x
0

>
0

{x_0} > 0

such that

f
(
x
)
+
f
(
−
x
)

f(x) + f( - x)

is decreasing for all

|
x
|

⩾

x
0

\left | x \right | \geqslant {x_0}

. Let

c
(
t
)

c(t)

be the characteristic function for

P
P

,

c
n

(
t
)

{c_n}(t)

the empirical characteristic function, and let

C
n

(
t
)
:=

n

1

/

2

(

c
n

(
t
)
−
c
(
t
)
)

{C_n}(t): = {n^{1/2}}({c_n}(t) - c(t))

. New necessary and sufficient metric entropy conditions are obtained for the weak convergence of

C
n

(
t
)

{C_n}(t)

on the space of continuous complex valued functions on

[
−

1
2

,

1
2

]

[ - \tfrac {1}{2},\tfrac {1}{2}]

. The result is used to characterize the weak convergence of

C
n

(
t
)

{C_n}(t)

in terms of the tail behavior of

P
P

and it also provides the first step towards a generalization of the Borisov-Dudley-Durst theorem. It also provides a partial response to a challenge raised by Dudley.
Cyclic vectors for backward hyponormal weighted shifts
Authors: Walsh Shelley 
Published: July 2010

A unilateral weighted shift

T
T

on a Hilbert space

H
H

is an operator such that

T

e
n

=

w
n

e

n
+
1

T{e_n} = {w_n}{e_{n + 1}}

for some orthonormal basis

{

e
n

}

n
=
0

∞

\left \{ {{e_n}} \right \}_{n = 0}^\infty

and weight sequence

{

w
n

}

n
=
0

∞

\left \{ {{w_n}} \right \}_{n = 0}^\infty

. If we assume

w
n

>
0

{w_n} > 0

, for all

n
n

, and let

β

(
n
)

=

w
0

⋯

w

n
−
1

\beta \left ( n \right ) = {w_0} \cdots {w_{n - 1}}

for

n
>
0

n > 0

and

β

(
0
)

=
1

\beta \left ( 0 \right ) = 1

, then

T
T

is unitarily equivalent to

f
↦
z
f

f \mapsto zf

on the weighted space

H
2

(
β
)

{H^2}\left ( \beta \right )

of formal power series

∑

n
=
0

∞

f
^

(
n
)

z
n

\sum \nolimits _{n = 0}^\infty {\hat f\left ( n \right ){z^n}}

such that

∑

n
=
0

∞

|

f
^

(
n
)

|

2

[

β

(
n
)

]

2

>
∞

\sum \nolimits _{n = 0}^\infty {{{\left | {\hat f\left ( n \right )} \right |}^2}{{\left [ {\beta \left ( n \right )} \right ]}^2} > \infty }

. Regarding

T
T

as multiplication for

z
z

on the space

H
2

(
β
)

{H^2}\left ( \beta \right )

, it is shown that, if

w
n

↑
1

{w_n} \uparrow 1

and

f
f

is analytic in a neighborhood of the unit disk, then either

f
f

is cyclic for

T
∗

{T^ * }

or

f
f

is contained in a finite-dimensional

T
∗

{T^ * }

-invariant subspace. This was shown—by different methods— for the unweighted shift operator by Douglas, Shields, and Shapiro [2]. It is also shown that every finite-dimensional

T
∗

{T^ * }

-invariant subspace is of the form
\[

(

(

z
−

α
1

)

n
1

⋯

(

z
−

α
k

)

n
k

H
2

(
β
)

)

⊥

,

{\left ( {{{\left ( {z - {\alpha _1}} \right )}^{{n_1}}} \cdots {{\left ( {z - {\alpha _k}} \right )}^{{n_k}}}{H^2}\left ( \beta \right )} \right )^ \bot },

\]
for some

α
1

,
…
,

α
k

{\alpha _1}, \ldots ,{\alpha _k}

in the unit disk and

n
1

,
…
,

n
k

{n_1}, \ldots ,{n_k}

positive integer.
Tightness in product spaces
Authors: Dissanayake U. N. B. , Willard S. W. 
Published: July 2010

A product

∏

X
i

\prod {X_i}

of topological spaces

X
i

,
i
∈
I

{X_i},i \in I

will be said to preserve tightness if
\[

∂

(

∏

X
i

)

≤

|
I
|

⋅

sup

{

∂

(

X
i

)

|

i
∈
I

}

\partial \left ( {\prod {X_i}} \right ) \leq \left | I \right | \cdot {\text {sup}}\left \{ {\partial \left ( {{X_i}} \right )\left | {i \in I} \right .} \right \}

\]
where

∂

(
X
)

\partial \left ( X \right )

denotes the tightness of

X
X

. We show

∏

X
i

\prod {X_i}

preserves tightness whenever each finite subproduct does. It is further shown that this is the case whenever each

X
i

{X_i}

is a locally compact

T
2

{T_2}

-space, and whenever each

X
i

{X_i}

is a locally Lindelöf

T
3

{T_3}

P
P

-space, extending 5.9 in [J].
Coefficients of odd univalent functions
Authors: Hu Ke 
Published: July 2010

Let
\[

S
2

=

{

f
2

(
z
)
=
z
+

∑

n
=
1

∞

b
n

z

2
n
+
1

∈
S

}

.

{S_2} = \left \{ {{f_2}(z) = z + \sum \limits _{n = 1}^\infty {{b_n}{z^{2n + 1}}} \in S} \right \}.

\]
In this note we prove

|

b
n

|

>
1.1305

\left | {{b_n}} \right | > 1.1305

. This is an improvement of V. I. Milin’s result [1].
Growth of harmonic conjugates in the unit disc
Authors: Jevtić Miroljub 
Published: July 2010

Assuming some mild regularity conditions on a positive nondecreasing function

ψ
(
x
)
=
O
(

x
a

)

\psi (x) = O({x^a})

(for some

a
>
0
,
x
→
∞

a > 0,x \to \infty

), we show that
\[

M
p

(
r
,
u
)
=
O

(

ψ

(

1

1
−
r

)

)

(
r
→
1
,
0
>
p
>
1
)

{M_p}(r,u) = O\left ( {\psi \left ( {\frac {1} {{1 - r}}} \right )} \right )(r \to 1,0 > p > 1)

\]
implies

M
p

(
r
,
v
)
=
O

(

ψ
~

p

(
1

/

(
1
−
r
)
)

)

1

/

p

{M_p}(r,v) = O{({\tilde \psi ^p}(1/(1 - r)))^{1/p}}

, where

u
(
z
)
+
i
v
(
z
)

u(z) + iv(z)

is holomorphic in the open unit disc and
\[

ψ
~

p

(
x
)
=

∫

1

/

2

x

ψ
p

(
t
)

t

d
t
,

x
⩾

1
2

.

{\tilde \psi ^p}(x) = \int _{1/2}^x {\frac {{{\psi ^p}(t)}}{t}dt,\quad x \geqslant \frac {1}{2}.}

\]
Limit boundary value problems of retarded functional-differential equations
Authors: Chen Shao Zhu 
Published: July 2010

We establish necessary and sufficient conditions assuring the existence and uniqueness of solutions of the limit boundary value problems on a half-line

[
a
,
∞
)

[a,\infty )

for the retarded functional equation
\[

x
¨

(
t
)
+
f
(
t
,
x
(

g
1

(
t
)
)
,
…
,
x
(

g
m

(
t
)
)
)
h
(

x
˙

(
t
)
)
=
0

\ddot x(t) + f(t,x({g_1}(t)), \ldots ,x({g_m}(t)))h(\dot x(t)) = 0

\]
.
An oscillation criterion for second order nonlinear differential equations
Authors: Wong James S. W. 
Published: July 2010

An oscillation crierion is given for the second order nonlinear differential equation

x
+
a
(
t
)

|

x

|

γ

sgn
⁡
x

= 0

x + a(t)|x{|^\gamma }\operatorname {sgn} x{\text { = 0}}

,

γ
>
0

\gamma > 0

, where the coefficient

a
(
t
)

a(t)

is not assumed to be nonnegative for all large values of

t
t

. The result involves a condition obtained by Kamenev for the linear differential equation.
Boundedness properties in Volterra integro-differential systems
Authors: Mahfoud W. E. 
Published: July 2010

Sufficient conditions are given to insure that all solutions of the integrodifferential system
\[

x
′

=
A
(
t
)
x
+

∫
0
t

C
(
t
,
s
)
x
(
s
)
d
s
+
f
(
t
)

x’ = A(t)x + \int _0^t {C(t,s)x(s)ds + f(t)}

\]
are uniform bounded.
Two mixed Hadamard type generalizations of Heinz inequality
Authors: Furuta Takayuki 
Published: July 2010

We give two types of mixed Hadamard inequalities containing the terms

T
,

|
T
|

T,\left | T \right |

, and

|

T
∗

|

\left | {{T^ * }} \right |

, where

T
T

is a bounded linear operator on a complex Hilbert space. As an immediate consequence of these results, we can easily show some extensions of the Hadamard inequality and also the Heinze inequality:
\[

(
∗
)

|

(

T
x
,
y

)

|

2

≤

(

|
T
|

2
α

x
,
x

)

(

|

T
∗

|

2

(

1
−
α

)

y
,
y

)

\left ( * \right )\quad {\left | {\left ( {Tx,y} \right )} \right |^2} \leq \left ( {{{\left | T \right |}^{2\alpha }}x,x} \right )\left ( {{{\left | {{T^ * }} \right |}^{2\left ( {1 - \alpha } \right )}}y,y} \right )

\]
for any

T
T

, any

x
,
y

x,y

in

H
H

, and any real number

α
\alpha

with

0
≤
α
≤
1

0 \leq \alpha \leq 1

. And the following conditions are equivalent in case

0
>
α
>
1

0 > \alpha > 1

: (1) the equality in (*) holds; (2)

|
T
|

2
α

x

{\left | T \right |^{2\alpha }}x

and

T
∗

y

{T^ * }y

are linearly dependent; (3)

T
x

Tx

and

|

T
∗

|

2

(

1
−
α

)

y

{\left | {{T^ * }} \right |^{2\left ( {1 - \alpha } \right )}}y

are linearly dependent. Results in this paper would remain valid for unbounded operators under slight modifications.
Martin boundaries of Denjoy domains
Authors: Segawa Shigeo 
Published: July 2010

Let

E
(
⊂

C
^

)

E( \subset {\mathbf {\hat C}})

be a compact set in the real axis. It is shown that there exists an

E
E

with zero linear measure such that Martin compactification of the domain

C
^

−
E

{\mathbf {\hat C}} - E

is not homeomorphic to

C
^

{\mathbf {\hat C}}

. Moreover, it is shown that if for some

λ
>

1
2

\lambda > \tfrac {1}{2}

\[

|

E
c

∩
[
−
t
,
t
]

|

t

=
O

(

(

1

log
⁡

t

−
1

)

λ

)

(
t
→
0
)
,

\frac {{|{E^c} \cap [ - t,t]|}}{t} = O\left ( {{{\left ( {\frac {1}{{\log {t^{ - 1}}}}} \right )}^\lambda }} \right )\quad (t \to 0),

\]
the set of minimal Martin boundary points of

C
^

−
E

{\mathbf {\hat C}} - E

’over 0’ consists of two points. This assertion is not valid for

λ
=

1
2

\lambda = \tfrac {1}{2}

.
Polynomially moving ergodic averages
Authors: Schwartz Mark 
Published: July 2010

Given an increasing sequence of positive integers

{

m
n

}

\left \{ {{m_n}} \right \}

, a non-decreasing sequence of positive integers

{

b
n

}

\left \{ {{b_n}} \right \}

, and a measurable, measure-preserving ergodic transformation

τ
\tau

on a probability space

(

Ω
,

F

,
μ

)

\left ( {\Omega ,\mathcal {F},\mu } \right )

, the a.s. convergence of the moving averages

T
n

(
f
)

=

b
n

−
1

∑

k
=

m
n

+
1

m
n

+

b
n

f

(

τ
k

)

{T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )}

is considered, for

f
∈

L
p

(
Ω
)

f \in {L_p}\left ( \Omega \right )

. A counterexample is constructed in the case of polynomial-like

{

m
n

}

\left \{ {{m_n}} \right \}

.
Compact homomorphisms of 𝐶*-algebras
Authors: Ghahramani F. 
Published: July 2010

Suppose

A
A

is a

C
∗

{C^*}

-algebra and

B
B

is a Banach algebra such that it can be continuously imbedded in

B
(
H
)

B(H)

, the Banach algebra of bounded linear operators on some Hilbert space

H
H

. It is shown that if

θ
\theta

is a compact algebra homomorphism from

A
A

into

B
B

, then

θ
\theta

is a finite rank operator, and the range of

θ
\theta

is spanned by a finite number of idempotents. If, moreover,

B
B

is commutative, then

θ
\theta

has the form

θ
(
x
)
=

X

1

(
x
)

E
1

+
⋯
+

X

k

(
x
)

E
k

\theta (x) = {\mathcal {X}_1}(x){E_1} + \cdots + {\mathcal {X}_k}(x){E_k}

, where

E
1

,
…
,

E
k

{E_1}, \ldots ,{E_k}

are fixed mutually orthogonal idempotents in

B
B

and

X

1

,
…
,

X

k

{\mathcal {X}_1}, \ldots ,{\mathcal {X}_k}

are fixed multiplicative linear functionals on

A
A

.