bluuebunny/crossref_metadata_2025_split huggingface dataset:
A product theorem for ℱ_{𝓅} classes and an application
Authors: Pearce Kent 
Published: June 2010

For Re

p
>
0

p > 0

let

F

p

=
{
f

|

f
(
z
)
=

∫

|

x

|

=
1

(
1
−
x
z
)

−
p

d
μ
(
x
)

{\mathcal {F}_p} = \{ f|f(z) = \int _{|x| = 1} {{{(1 - xz)}^{ - p}}d\mu (x)}

,

|

z

|

>
1

|z| > 1

,

μ
\mu

a probability measure on

|

x

|

=
1

|x| = 1

and let

F

p

⋅

F

q

=
{
f
g

|

f
∈

F

p

,
g
∈

F

q

}

{\mathcal {F}_p} \cdot {\mathcal {F}_q} = \{ fg|f \in {\mathcal {F}_p},g \in {\mathcal {F}_q}\}

. Brickman, Hallenbeck, MacGregor and Wilken proved a product theorem for the

F

p

{\mathcal {F}_p}

classes; they showed that if

p
>
0

p > 0

,

q
>
0

q > 0

, then

F

p

⋅

F

q

⊂

F

p
+
q

{\mathcal {F}_p} \cdot {\mathcal {F}_q} \subset {\mathcal {F}_{p + q}}

. We give an (essentially complete) converse for the result of Brickman et al., i.e., we show that if

F

p

⋅

F

q

⊂

F

p
+
q

{\mathcal {F}_p}\cdot {\mathcal {F}_q} \subset {\mathcal {F}_{p + q}}

, then

p
>
0

p > 0

,

q
>
0

q > 0

or else

p
=
q
=
1
+
i
t

p = q = 1 + it

for some

t
t

real. As an immediate consequence we disprove a conjecture about the extreme points of the closed convex hulls of the classes

Sp(

γ

)

{\text {Sp(}}\gamma {\text {)}}

,

0
>

|

γ

|

>
π

/

2

0 > |\gamma | > \pi /2

, of

γ
\gamma

-spirallike univalent functions, i.e., writing

m
=
1
+

e

−
2
i
γ

m = 1 + {e^{ - 2i\gamma }}

, we show

{
z

/

(
1
−
x
z

)
m

|

|

x

|

=
1
}
⫋

E

K

Sp
⁡
(
γ
)

\{ z/(1 - xz)^m| |x| = 1\} \subsetneqq \mathcal {E}\mathcal {K} \operatorname {Sp}(\gamma )

,

0
>

|

γ

|

>
π

/

2

0 > |\gamma | > \pi /2

.
On certain comparison theorems for second order linear oscillation
Authors: Kwong Man Kam 
Published: June 2010

It is shown that if

(
p

y
′

)
′

+
q
y
=
0

(py’)’ + qy = 0

on

[
0
,
∞
)

[0,\infty )

is oscillatory then

(
p

z
′

)
′

+
a
q
z
=
0

(pz’)’ + aqz = 0

is also oscillatory for functions satisfying

a
(
t
)
⩾
1

a(t) \geqslant 1

and
\[

2
p
(
t
)

a
′

(
t
)
−
3

∫
0
t

p
(
s
)

a
′

2

(
s
)

a

−
1

(
s
)
d
s

2p(t)a’(t) - 3\int _0^t {p(s){{a’}^2}(s){a^{ - 1}}(s)ds}

\]
is nonincreasing.
Cauchy-type congruences for binomial coefficients
Authors: Hudson Richard H. , Williams Kenneth S. 
Published: June 2010

In 1840 Cauchy [2] showed that for a prime

p
=
e
f
+
1

p = ef + 1

,

e
=
20

e = 20

,
\[

(

10
f

f

)

≡
±

(

10
f

3
f

)

(
mod
p
)
,

\left ( {\begin {array}{* {20}{c}} {10f} \\ f \\ \end {array} } \right ) \equiv \pm \left ( {\begin {array}{* {20}{c}} {10f} \\ {3f} \\ \end {array} } \right )\quad (\bmod p),

\]
and it was not until 1965 that Whiteman [6] succeeded in removing the sign ambiguity in this congruence. In this paper we show how the Davenport-Hasse relation [3] in the form given by Yamamoto [8] can be used to resolve the sign ambiguity in other Cauchy-type congruences. Details are given just for

e
=
8
,
12
,

and

20

e = 8,12,{\text {and }}20

.
An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance
Authors: Ding Tung Ren 
Published: June 2010

In this paper, by using a generalized form of the Poincaré-Birkhoff Theorem, we demonstrate that the Duffing equation
\[

d
2

x

d

t
2

+
g
(
x
)
=
p
(
t
)

(
≡
p
(
t
+
2
π
)
)

\frac {{{d^2}x}} {{d{t^2}}} + g(x) = p(t)\quad ( \equiv p(t + 2\pi ))

\]
may also admit an infinite number of

2
π

2\pi

-periodic solutions even in a resonance case.
𝐿¹-convergence of Fourier series with complex quasimonotone coefficients
Authors: Stanojevic Vera B. 
Published: June 2010

A sequence of Fourier coefficients

{

f
^

(
n
)

}

\left \{ {\hat f(n)} \right \}

of a complex function in

L
1

(
T
)

{L^1}(T)

is said to be complex quasimonotone if there exists

θ
0

{\theta _0}

such that
\[

Δ

f
^

(
n
)
+

α
n

f
^

(
n
)
∈

{

z

|

|

arg
⁡
z

|

⩽

θ
0

>

π
2

}

\Delta \hat f(n) + \frac {\alpha } {n}\hat f(n) \in \left \{ {z|\left | {\arg z} \right | \leqslant {\theta _0} > \frac {\pi } {2}} \right \}

\]
for some

α
⩾
0

\alpha \geqslant 0

and for all

n
n

. It is proved that Fourier series with asymptotically even and complex quasimonotone coefficients, satisfying
\[

lim

n
→
∞

¯

n

1

/

q

max

n
⩽
j
⩽
[
λ
n
]

|

Δ

f
^

(
j
)

|

1

/

q

max

n
⩽
j
⩽
[
λ
n
]

|

f
^

(
j
)

|

1

/

p

=
o
(
1
)
,

λ
→
1
+
0
,

1
p

+

1
q

=
1
,

\overline {\lim \limits _{n \to \infty } } \;{n^{1/q}}\max \limits _{n \leqslant j \leqslant [\lambda n]} {\left | {\Delta \hat f(j)} \right |^{1/q}}\max \limits _{n \leqslant j \leqslant [\lambda n]} {\left | {\hat f(j)} \right |^{1/p}} = o(1),\; \lambda \to 1 + 0,\tfrac {1} {p} + \tfrac {1} {q} = 1,

\]
converges in

L
1

(
T
)

{L^1}(T)

-norm if and only if

f
^

(
n
)
lg
⁡

|
n
|

=
o
(
1
)

\hat f(n)\lg \left | n \right | = o(1)

,

n
→
∞

n \to \infty

. A recent result of Č V. Stanojević [3] is a special case of the corollary of the main theorem.
Strongly ergodic sequences of integers and the individual ergodic theorem
Authors: Blum J. R. , Reich J. I. 
Published: June 2010

Let

S
=
{

k
1

,

k
2

,
…
}

S = \{ {k_1},{k_2}, \ldots \}

be an increasing sequence of positive integers. We call

S
S

strongly ergodic if for every measure preserving transformation

T
T

on a probability space

(
Ω
,

F

,
P
)

(\Omega ,\mathcal {F},P)

and every

f
∈

L
1

(
Ω
)

f \in {L_1}(\Omega )

we have

lim

n
→
∞

(
1

/

n
)

∑

j
=
1

n

f
(

T

k
j

ω
)
=
P
f
(
ω
)

{\lim _{n \to \infty }}(1/n)\sum \nolimits _{j = 1}^n {f({T^{kj}}\omega ) = Pf(\omega )}

a.e. where

P
f

Pf

is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence

S
S

to be strongly ergodic and provide examples.
Rough and strongly rough norms on Banach spaces
Authors: Godini G. 
Published: June 2010

We give equivalent conditions for a norm on a Banach space

X
X

to be rough using the sets

A
(
x
)

A(x)

defined for each

x
∈
X

x \in X

by

A
(
x
)
=
{

f
∈

X
∗

|

f
(
x
)
=

|

|

x

|

|

,

|

|

f

|

|

=
1
}

A(x) = \{ {f \in {X^ * }|f(x) = || x ||,|| f || = 1\}}

. This enables us to obtain unitary characterizations for rough and strongly rough norms.
A formula for distribution trace characters on nilpotent Lie groups
Authors: Corwin L. , Greenleaf F. P. 
Published: June 2010

The distribution trace character

θ
π

{\theta _\pi }

of an irreducible representation

π
\pi

on a simply connected nilpotent Lie group

N
N

is described as a superposition of unitary characters on certain subgroups, in terms of the canonical objects introduced by R. Penney [8]. For

l
∈

n

∗

l \in {\mathfrak {n}^*}

, define

h

1

(
l
)
=

{\mathfrak {h}_1}(l) =

smallest ideal containing the radical

r

(
l
)

\mathfrak {r}(l)

, and

h

k
+
1

(
l
)
=

h

1

(
l

|

h

k

(
l
)
)

{\mathfrak {h}_{k + 1}}(l) = {\mathfrak {h}_1}(l|{\mathfrak {h}_k}(l))

. These subalgebras terminate in a subordinate subalgebra

h

∞

(
l
)

{\mathfrak {h}_\infty }(l)

after finitely many steps. If

H
∞

=
exp
⁡
(

h

∞

)

{H_\infty } = \exp ({\mathfrak {h}_\infty })

,

X

∞

=
(

e

2
π
i
l

)
∘
log
⁡

|

H
∞

{\mathcal {X}_\infty } = ({e^{2\pi il}}) \circ \log |{H_\infty }

, and

(

X

∞

,

H
∞

)
⋅
n
=
(

X

∞

⋅
n
,

H
∞

⋅
n
)

({\mathcal {X}_\infty },{H_\infty }) \cdot n = ({\mathcal {X}_\infty } \cdot n,\;{H_\infty } \cdot n)

, where

H
∞

⋅
n
=

n

−
1

H
∞

n

{H_\infty } \cdot n = {n^{ - 1}}{H_\infty }n

,

X

∞

⋅
n
(

h
′

)
=

X

∞

(
n

h
′

n

−
1

)

on

H
∞

⋅
n

{\mathcal {X}_\infty }\cdot n(h’) = {\mathcal {X}_\infty }(nh’{n^{ - 1}}){\text { on }}{H_\infty } \cdot n

, then

⟨

θ
π

,
ϕ

⟩

=

∫

K
∞

∖
N

⟨

(

X

∞

,

H
∞

)
⋅
n
,
ϕ

⟩

d

n
˙

\left \langle {{\theta _\pi },\phi } \right \rangle = {\smallint _{{K_\infty }\backslash N}}\left \langle {({\mathcal {X}_\infty },{H_\infty }) \cdot n,\phi } \right \rangle d\dot n

, where the pair

(

X

∞

,

H
∞

)
⋅
n

({\mathcal {X}_\infty },{H_\infty }) \cdot n

is regarded as the tempered distribution

⟨

(

X

∞

,

H
∞

)
⋅
n
,
ϕ

⟩

=

∫

H
∞

X

∞

(
h
)
ϕ
(

n

−
1

h
n
)
d
h

\left \langle {({\mathcal {X}_\infty },{H_\infty }) \cdot n,\phi } \right \rangle = {\smallint _{{H_\infty }}}{\mathcal {X}_\infty }(h)\phi ({n^{ - 1}}hn)dh

, and where

k

∞

=
{
X
∈

n

:
l
[

h
∞

,
X
]
=
0
}

{\mathfrak {k}_\infty } = \{ X \in \mathfrak {n}:l[{h_\infty },X] = 0\}

gives the stabilizer of the pair

(

X

∞

,

H
∞

)

({\mathcal {X}_\infty },{H_\infty })

. The integral over

K
∞

∖
N

{K_\infty }\backslash N

is absolutely convergent for any Schwartz function

ϕ
\phi

on

N
N

.
Asymptotic behavior of systems of linear ordinary differential equations
Authors: Tomastik E. C. 
Published: June 2010

Conditions will be placed on the

m
×
m

m \times m

matrices

G
(
t
)

G(t)

and

G
i

(
t
)

{G_i}(t)

to assure that for any integer

k
=
1
,
…
,
n

k = 1, \ldots ,n

, the linear differential system
\[

x
i
′

=

G
i

(
t
)

x

i
+
1

,

i
=
1
,
…
,
n
−
1
,

x
n
′

=
G
(
t
)

x
1

,

{x’_i} = {G_i}(t){x_{i + 1}},\quad i = 1, \ldots ,n - 1,\quad {x’_n} = G(t){x_1},

\]
where the

x
i

{x_i}

are

m
×
m

m \times m

matrices, has a solution

(

x
1

,
…
,

x
n

)

({x_1}, \ldots ,{x_n})

with the property that

x
k

(
t
)
→
I

{x_k}(t) \to I

(the identity matrix) and if

k
>
n

k > n

,

x
i

(
t
)
→
0

{x_i}(t) \to 0

,

i
=
k
+
1
,
…
,
n

i = k + 1, \ldots ,n

, as

t
→
∞

t \to \infty

. Furthermore, important bounds on the

x
i

(
t
)

{x_i}(t)

will be given. Some of these conditions will require that

∫
a
∞

|
G
|

>
∞

\int _a^\infty {\left | G \right | > \infty }

while others will not. Corollaries will be given for special cases such as

(
R
(
t
)
x
)
=
G
(
t
)
x

(R(t)x) = G(t)x

. No selfadjointness conditions are assumed; however, the results are new even in the selfadjoint case.
The game-theoretic value and the spectral radius of a nonnegative matrix
Authors: Cohen Joel E. , Friedland Shmuel 
Published: June 2010

We relate some minimax functions of matrices to some spectral functions of matrices. If

A
A

is a nonnegative

n
×
n

n \times n

matrix,

υ
(
A
)

\upsilon (A)

is the game-theoretic value of

A
A

, and

ρ
(
A
)

\rho (A)

is the spectral radius of

A
A

, then

υ
(
A
)
≤
ρ
(
A
)

\upsilon (A) \leq \rho (A)

. Necessary and sufficient conditions for

υ
(
A
)
=
ρ
(
A
)

\upsilon (A) = \rho (A)

are given. It follows that if

A
A

is nonnegative and irreducible and

n
>
1

n > 1

, then

υ
(
A
)
>
ρ
(
A
)

\upsilon (A) > \rho (A)

. Also, if, for a real matrix

A
A

and a positive matrix

B
B

,

υ
(
A
,
B
)
=

sup
X

inf
Y

X
T

A
Y

/

X
T

B
Y

\upsilon (A,B) = {\sup _X}{\inf _Y}{X^T}AY/{X^T}BY

over probability vectors

X
X

and

Y
Y

, then for nonnegative, nonsingular

A
A

and positive

B
B

,

ρ
(
A
B
)
=

[
υ
(

A

−
1

,
B
)

]

−
1

\rho (AB) = {[\upsilon ({A^{ - 1}},B)]^{ - 1}}

.
SLLN and convergence rates for nearly orthogonal sequences of random variables
Authors: Móricz Ferenc 
Published: June 2010

Let

{

X
k

:
k
⩾
1
}

\{ {X_k}:k \geqslant 1\}

be a sequence of random variables with finite second moments

E

X
k
2

=

σ
k
2

>
∞

EX_k^2 = \sigma _k^2 > \infty

for which

|

E

X
k

X
l

|

⩽

σ
k

σ
l

ρ
(

|

k
−
l

|

)

|E{X_k}{X_l}| \leqslant {\sigma _k}{\sigma _l}\rho (|k - l|)

, where

{
ρ
(
j
)
:
j
⩾
0
}

\{ \rho (j):j \geqslant 0\}

is a sequence of nonnegative numbers with

∑

j
=
0

∞

ρ
(
j
)

>
∞

\sum \nolimits _{j = 0}^\infty {\rho (j)} > \infty

. In particular, in the case of orthogonality,

ρ
(
j
)
=
0

\rho (j) = 0

for

j
⩾
1

j \geqslant 1

. We prove strong laws for the first arithmetic means

ζ
n

=

n

−
1

∑

k
=
1

n

X
k

{\zeta _n} = {n^{ - 1}}\sum \nolimits _{k = 1}^n {{X_k}}

and the Cesàro means
\[

τ
n

=

n

−
1

∑

k
=
1

n

(
1
−
(
k
−
1
)

n

−
1

)

X
k

.

{\tau _n} = {n^{ - 1}}\sum \limits _{k = 1}^n {(1 - (k - 1){n^{ - 1}}){X_k}} .

\]
The convergence rates are studied in the form

P
{

sup

n
>

2
p

|

ζ
n

|

>
ε
}

P\{ {\sup _{n > {2^p}}}|{\zeta _n}| > \varepsilon \}

and

P
{

sup

n
>

2
p

|

τ
n

|

>
ε
}

P\{ {\sup _{n > {2^p}}}|{\tau _n}| > \varepsilon \}

, where

ε
>
0

\varepsilon > 0

is fixed and

p
p

tends to

∞
\infty

. At the end, the case where

Σ

j
=
0

∞

ρ
(
j
)
=
∞

\Sigma _{j = 0}^\infty \rho (j) = \infty

is also treated.
Boundary behavior of Green potentials
Authors: Luecking Daniel H. 
Published: June 2010

A Green potential on the unit disk

{

|
z
|

>
1
}

\{ \left | z \right | > 1\}

is a function

u
(
z
)

u(z)

of the form
\[

u
(
z
)
=
∫

log
⁡

|

1
−

w
¯

z

z
−
w

|

d
α
(
w
)
,

u(z) = \int {\log \left | {\frac {{1 - \bar wz}}{{z - w}}} \right |{\text { }}} d\alpha (w),

\]
where

α
\alpha

is a positive measure such that

∫
(
1
−

|
w
|

)
d
α
(
w
)

\smallint (1 - \left | w \right |)d\alpha (w)

is finite. In this note I give a necessary and sufficient condition on a relatively closed subset

F
F

of the unit disk in order that, for all such

u
(
z
)

u(z)

,
\[

lim inf

F
∋
z
→
1

(
1
−

|
z
|

)
u
(
z
)
=
0.

\liminf _{F \ni z \to 1} (1 - \left | z \right |) u(z) = 0.

\]
The condition is that the hyperbolic capacity of the portion of

F
F

in arbitrarily small neighborhoods of 1 is bounded away from zero.
Convergence of cardinal series
Authors: de Boor Carl , Höllig Klaus , Riemenschneider Sherman 
Published: June 2010

The result of this paper is a generalization of our characterization of the limits of multivariate cardinal splines. Let

M
n

{M_n}

denote the

n
n

-fold convolution of a compactly supported function

M
∈

L
2

(

R

d

)

M \in {L_2}({{\mathbf {R}}^d})

and denote by
\[

S
n

:=

{

∑

j
∈

Z

d

c
(
j
)

M
n

(
⋅
−
j
)
:
c
∈

l
2

(

Z

d

)

}

{S_n}: = \left \{ {\sum \limits _{j \in {{\mathbf {Z}}^d}} {c(j){M_n}( \cdot - j):c \in {l_2}({{\mathbf {Z}}^d})} } \right \}

\]
the span of the translates of

M
n

{M_n}

. We prove that there exists a set

Ω
\Omega

with

vol
d

(
Ω
)
=

(
2
π

)
d

{\operatorname {vol} _d}(\Omega ) = {(2\pi )^d}

such that for any

f
∈

L
2

(

R

d

)

f \in {L_2}({{\mathbf {R}}^d})

,
\[

dist
⁡
(
f
,

S
n

)
→
0

as

n
→
∞
,

\operatorname {dist} (f,{S_n}) \to 0\quad {\text {as }}n \to \infty ,

\]
if and only if the support of the Fourier transform of

f
f

is contained in

Ω
\Omega

.
Pointwise estimates for convex polynomial approximation
Authors: Leviatan D. 
Published: June 2010

For a convex function

f
∈
C
[
−
1
,
1
]

f \in C[ - 1,1]

we construct a sequence of convex polynomials

p
n

{p_n}

of degree not exceeding

n
n

such that

|

f
(
x
)
=

p
n

(
x
)

|

≤
C

ω
2

(
f
,

1
−

x
2

/

n
)
,
−
1
≤
x
≤
1

|f(x) = {p_n}(x)| \leq C{\omega _2}(f,\sqrt {1 - {x^2}} /n), - 1 \leq x \leq 1

. If in addition

f
f

is monotone it follows that the polynomials are also monotone thus providing simultaneous monotone and convex approximation.
The 𝐾-theory of triangular matrix rings. II
Authors: Keating M. E. 
Published: June 2010

Let

T
T

be the upper triangular matrix ring defined by a pair of rings

R
R

and

S
S

and an

R
−
S

R - S

-bimodule

M
M

. We use the QP definition of algebraic

K
K

-theory to give a quick proof that the homomorphism
\[

π
m

:

K
m

(
T
)
→

K
m

(
R
)
⊕

K
m

(
S
)
,

m
⩾
0
,

{\pi _m}:{K_m}(T) \to {K_m}(R) \oplus {K_m}(S),\quad m \geqslant 0,

\]
induced by the obvious ring epimorphism is an isomorphism.
A formula for 𝐸_{𝑊}𝑒𝑥𝑝(-2⁻¹𝑎²‖𝑥+𝑦‖²₂)
Authors: Chiang Tzuu-Shuh , Chow Yun Shyong , Lee Yuh-Jia 
Published: June 2010

We prove that for a complex number

a
a

with

Re
⁡

a
2

>
−

π
2

/

4

\operatorname {Re} {a^2} > - {\pi ^2}/4

and

x
(
⋅
)
∈

L
2

[
0
,
1
]

x( \cdot ) \in {L^2}[0,1]

,
\[

E
W

{
exp
⁡
(
−

2

−
1

a
2

|

|

x
+
y

|

|

2
2

)
}
=

(
cosh
⁡
a

)

−
1

/

2

exp
⁡

[

2

−
1

(

∫
0
1

∫
0
1

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

−

a
2

∫
0
1

x
2

(
t
)
d
t

)

]

,

{E_W}\{ \exp ( - {2^{ - 1}}{a^2}||x + y||_2^2)\} = {(\cosh a)^{ - 1/2}}\exp \left [ {{2^{ - 1}}\left ( {\int _0^1 {\int _0^1 {k(s,t)x(s)x(t)dsdt} - {a^2}\int _0^1 {{x^2}(t)dt} } } \right )} \right ],

\]
, where

W
W

, the standard Wiener measure on

C
[
0
,
1
]

C[0,1]

, is the distribution of

y
y

and
\[

k
(
s
,
t
)
=

a
3

(
2
cosh
⁡
a

)

−
1

[
sinh
⁡
(
a
(
1
−

|

s
−
t

|

)
)
−
sinh
⁡
(
a
(
1
−

|

s
+
t

|

)
)
]
.

k(s,t) = {a^3}{(2\cosh a)^{ - 1}}[\sinh (a(1 - |s - t|)) - \sinh (a(1 - |s + t|))].

\]
.
A unimodality result in the enumeration of subgroups of a finite abelian group
Authors: Butler Lynne M. 
Published: June 2010

The number of subgroups of orderpk{p^k}in an abelian groupGGof orderpn{p^n}is a polynomial inp,α⋋(k;p)p,{\alpha _ \leftthreetimes }(k;p), determined by the typeλ\lambdaofGG. It is well known thatα⋋(k;p)=α⋋(n−k;p){\alpha _ \leftthreetimes }(k;p) = {\alpha _ \leftthreetimes }(n - k;p). Using a recent result from the theory of Hall-Littlewood symmetric functions, we prove thatα⋋(k;p){\alpha _ \leftthreetimes }(k;p), is a unimodal sequence of polynomials. That is, for1≤k≤n/2,αλ(k;p)−αλ(k−1;p)1 \leq k \leq n/2,{\alpha _\lambda }(k;p) - {\alpha _\lambda }(k - 1;p)is a polynomial inppwith nonnegative coefficients.
Universal functions and generalized classes of functions
Authors: Cichoń J. , Morayne M. 
Published: June 2010

For a class

A

\mathcal {A}

of subsets of a set

Z
Z

which is closed under countable unions we consider the families of functions
\[

M
_

A

=
{
f
:
Z
→
[
0
,
1
]
:
(
∀
c
)
(

f

−
1

(
(
c
,
1
]
)
∈

A

)
}

\underline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}((c,1]) \in \mathcal {A})\}

\]
and
\[

M
¯

A

=
{
f
:
Z
→
[
0
,
1
]
:
(
∀
c
)
(

f

−
1

(
[
0
,
c
)
)
∈

A

)
}

\overline M \mathcal {A} = \{ f:Z \to [0,1]:(\forall c)({f^{ - 1}}([0,c)) \in \mathcal {A})\}

\]
(for instance, if

Z
Z

is a topological space and

A

\mathcal {A}

is the family of all open subsets of

Z
Z

, then

M
_

A

\underline M \mathcal {A}

and

M
¯

A

\overline M \mathcal {A}

are the families of lower and upper semicontinuous functions from

Z
Z

to

[
0
,
1
]

[0,1]

, respectively). Using universal functions we show that under certain natural assumptions about

A

\mathcal {A}

there exists a function

f
∈

M
_

A

f \in \underline M \mathcal {A}

such that there is no partition

{

X
n

:
n
∈

N

}

\{ {X_n}:n \in {\mathbf {N}}\}

of

Z
Z

and a family of functions

{

h
n

:
n
∈

N

}
⊆
M

A

\{ {h_n}:n \in {\mathbf {N}}\} \subseteq M\mathcal {A}

such that

f
=

∪
n

(

h
n

|

X
n

)

f = { \cup _n}({h_n}|{X_n})

. This is a generalization of some results of this type proved by Novikov and Adian, Keldyš, and Laczkovich for the Baire hierarchy of functions. The universal functions technique we use is different from the methods of these authors.
Complemented invariant subspaces of 𝐻^{𝑝},0<𝑝<1, and the Hahn-Banach extension property
Authors: Cohn William S. 
Published: June 2010

Let

0

>

p

>

1

0{\text { > }}p{\text { > }}1

and let

H
p

{H^p}

denote the usual Hardy class of functions analytic on the disc. In this note we show that an invariant subspace of

H
p

{H^p}

is complemented in

H
p

{H^p}

if and only if it has the form

B

H
p

B{H^p}

where

B
B

is a Blaschke product whose zero sequence is a Carleson sequence. We also prove that this occurs if and only if the invariant subspace has the Hahn-Banach extension property.
Topological triviality of a family of zero-sets
Authors: Buchner Michael A. , Kucharz Wojciech 
Published: June 2010

This paper gives conditions on a mapping

F
:
U
×

R

k

→

K

p

(

U
⊂

K

n

open,

K

=

R

or

C

)

F:U \times {{\mathbf {R}}^k} \to {{\mathbf {K}}^p}\left ( {U \subset {{\mathbf {K}}^n}{\text {open, }}{\mathbf {K}} = {\mathbf {R}}\;{\text {or }}{\mathbf {C}}} \right )

such that the family

F
t

−
1

(
0
)

F_t^{ - 1}\left ( 0 \right )

is a topologically trivial family (i.e., does not change topologically as

t
∈

R

k

t \in {{\mathbf {R}}^k}

varies). As an application an easy proof is given of a counterexample to a conjecture of Thom concerning the number of topologically different realizations of a given jet.
A convergence problem connected with continued fractions
Authors: Larcher Gerhard 
Published: June 2010

The set

Z
α

:=
{
β

|

lim

n
→
∞

|

|

q
n

β

|

|

=
0
}

{Z_\alpha }: = \{ \beta |{\lim _{n \to \infty }}||{q_n}\beta || = 0\}

is considered, where

(

q
n

)

n
∈

N

{\left ( {{q_n}} \right )_{n \in {\mathbf {N}}}}

is the sequence of best approximation denominators of

α
\alpha

, and it is explicitly determined for

α
\alpha

with bounded continued fraction coefficients.
Transience of a pair of local martingales
Authors: Mountford T. S. 
Published: June 2010

We consider the process of windings of complex Brownian motion about two points

a
a

and

b
b

in the complex plane,

{
(

θ
a

(
t
)
,

θ
b

(
t
)
)
:
t
≥
0
}

\{ ({\theta ^a}(t),{\theta ^b}(t)):t \geq 0\}

. We show that this process is transient in the sense that

lim
t

→
∞

|

(

θ
a

(
t
)
,

θ
b

(
t
)
)

|

=
∞

{\lim _t} \to \infty |({\theta ^a}(t),{\theta ^b}(t))| = \infty

. This extends a result found in both Lyons and McKean (1984) and McKean and Sullivan (1984). We will mostly use facts and ideas found in the former paper.
Starlikeness and convexity from integral means of the derivative
Authors: Yamashita Shinji 
Published: June 2010

If

f
f

is analytic in

|

z

|

>
1

|z| > 1

and normalized:

f
(
0
)
=

f
′

(
0
)
−
1
=
0

f(0) = f’(0) - 1 = 0

, then

f
f

is univalent and starlike in

|

z

|

>
1
(
f
)

|z| > 1(f)

, where
\[

I
(
f
)
=
sup

r

{

(
2
π
)

−
1

∫
0

2
π

|

f
′

(
r

e

i
t

)

|

2

d
t

}

−
1

/

2

,

0
≤
r
≤
1.

I(f) = \sup \,r{\left \{ {{{(2\pi )}^{ - 1}}\int _0^{2\pi } {|f’(r{e^{it}}){|^2}dt} } \right \}^{ - 1/2}},\quad 0 \leq r \leq 1.

\]
Furthermore, there exists a normalized

f
f

such that

I
(
f
)
>
1

I(f) > 1

and that

f
′

f’

vanishes at a point on

|

z

|

=
I
(
f
)

|z| = I(f)

. If

f
f

is analytic and normalized in

|

z

|

>
1

|z| > 1

, then

f
f

is univalent and convex in

|

z

|

>
I
(
f
)

/

2

|z| > I(f)/2

.
Weak amenability of Banach algebras generated by some analytic semigroups
Authors: Galé Jośe E. 
Published: June 2010

In this paper it is shown that if

A
A

is a Banach algebra generated by an analytic semigroup

(

a
t

)
Re
⁡
t
>
0

({a^t})\operatorname {Re} t > 0

such tnat

|

|

a

1
+
i
y

|

|

=
O
(

|

y

|

ρ

)

(
y
∈

R

)

||{a^{1 + iy}}|| = O(|y{|^\rho }){\text { }}(y \in {\mathbf {R}})

, where

0
≤
ρ
>
1

/

2

0 \leq \rho > 1/2

, then

A
A

is weakly amenable, that is, each continuous derivation from

A
A

to a commutative

A
A

-module is null.
Functions not vanishing on trivial Gleason parts of Douglas algebras
Authors: Gorkin Pamela 
Published: June 2010

Let

B
B

denote a closed subalgebra of

L
∞

{L^\infty }

containing the space of bounded analytic functions. Let

M
(
B
)

M(B)

denote the maximal ideal space of

B
B

. Let

f
f

be a function in

B
B

such that

f
f

does not vanish on any Gleason part consisting of a single point. We show that if

g
g

is a function in

B
B

such that

|
g
|

≤

|
f
|

on

M
(
B
)

\left | g \right | \leq \left | f \right |{\text { on }}M(B)

, then

g

/

f
∈
B

g/f \in B

.
Powers’s binary shifts on the hyperfinite factor of type 𝐼𝐼₁
Authors: Enomoto Masatoshi , Watatani Yasuo 
Published: June 2010

A unit preserving

∗
*

-endomorphism

σ
\sigma

on the hyperfinite

I

I

1

{\text {I}}{{\text {I}}_1}

factor

R
R

is called a shift if

⋂

n
=
0

∞

σ
n

(
R
)
=
{
λ
1
;
λ
∈

C

}

\bigcap \nolimits _{n = 0}^\infty {{\sigma ^n}(R) = \{ \lambda 1;\lambda \in \mathbb {C}} \}

. A shift

σ
\sigma

is called Powers’ binary shift if there is a self-adjoint unitary

u
u

such that

R
=
{

σ
n

(
u
)
;
n
∈

N

∪
{
0
}
}

R = \{ {\sigma ^n}(u);n \in \mathbb {N} \cup \{ 0\} \}

and

σ
k

(
u
)
u
=
±
u

σ
k

(
u
)

{\sigma ^k}(u)u = \pm u{\sigma ^k}(u)

for

k
∈

N

∪
{
0
}

k \in \mathbb {N} \cup \{ 0\}

. Let

q
(
σ
)

q(\sigma )

be the number

min
{
k
∈

N

;

σ
k

(
R

)
′

∩
R
≠

C

1
}

\min \{ k \in \mathbb {N};{\sigma ^k}(R)’ \cap R \ne \mathbb {C}1\}

. It is shown that the number

q
(
σ
)

q(\sigma )

is not the complete outer conjugacy invariant for Powers’ binary shifts.