bluuebunny/crossref_metadata_2025_split huggingface dataset:
Abelian and Tauberian theorems for 0-regularly varying functions
Authors: Geluk J. L. 
Published: June 2010

A general kernel Abelian and Tauberian theorem is proved for functions

f
f

satisfying

lim
¯

t
→
∞

f
(
t
x
)

/

f
(
t
)
>
∞

for

x
>
0

{\overline {\lim } _{t \to \infty }}f(tx)/f(t) > \infty {\text { for }}x > 0

.
An area theorem for holomorphic functions
Authors: Yamashita Shinji 
Published: June 2010

Let

D
⫋

C

D \subsetneqq \mathbf {C}

and

Δ
⫋

C

\Delta \subsetneqq \mathbf {C}

be open sets in the plane

C

\mathbf {C}

containing 0. Suppose that

D
D

is connected and

f
f

is holomorphic in

D
D

with

f
(
0
)
=
0

f(0) = 0

. Then, for

Q
=
f
(
D
)
∩
Δ

Q = f(D) \cap \Delta

and

P
=

f

−
1

(
Q
)

P = {f^{ - 1}}(Q)

, we have
\[

A
(
P
)
+
A
(
Q
)
⩾
π

[
min
{
dis
⁡
(
∂
D
,
0
)
,
dis
⁡
(
∂
Δ
,
0
)
}

]
2

,

A(P) + A(Q) \geqslant \pi {[\min \{ \operatorname {dis}(\partial D,0),\operatorname {dis}(\partial \Delta ,0)\} ]^2},

\]

A
(
⋅
)

A( \cdot )

denoting the area. The constant

π
\pi

is sharp and the equality holds if and only if

f
≡
0
,
dis
⁡
(
∂
D
,
0
)
⩽
dis
⁡
(
∂
Δ
,
0
)

f \equiv 0,\operatorname {dis}(\partial D,0) \leqslant \operatorname {dis}(\partial \Delta ,0)

, and

D
D

is a disk of center 0.
Exponential sums of Lerch’s zeta functions
Authors: Wang Kai 
Published: June 2010

For

x
x

not an an integer and

Re
⁡
(
s
)
>
0

\operatorname {Re} (s) > 0

, let
\[

F
(
x
,
s
)
=

∑

k
=
1

∞

e

2
π
i
k
x

k
s

F(x,s) = \sum \limits _{k = 1}^\infty {\frac {{{e^{2\pi ikx}}}}{{{k^s}}}}

\]
be the Lerch’s zeta function. In this note, we will show that
\[

∑

γ
=
1

m
−
1

e

−
2
π
i
γ
α

/

m

F

(

γ
m

,
1
−
n

)

=

1
n

(

B
n

−

m
n

B
n

(

α
m

−

[

α
m

]

)

)

\sum \limits _{\gamma = 1}^{m - 1} {{e^{ - 2\pi i\gamma \alpha /m}}} F\left ( {\frac {\gamma }{m},1 - n} \right ) = \frac {1}{n}\left ( {{B_n} - {m^n}{B_n}\left ( {\frac {\alpha }{m} - \left [ {\frac {\alpha }{m}} \right ]} \right )} \right )

\]
where

α
\alpha

is an integer and

α
≢
0
(

mod

m
)

\alpha \not \equiv 0(\mod m)

and

n
⩾
1

n \geqslant 1

. For

n
=
1

n = 1

, this formula is equivalent to the classical Eisenstein formula
\[

α
m

−

[

α
m

]

−

1
2

=
−

1

2
m

∑

γ
=
1

m
−
1

sin
⁡

2
π
γ
α

m

cot
⁡

π
γ

n

.

\frac {\alpha }{m} - \left [ {\frac {\alpha }{m}} \right ] - \frac {1}{2} = - \frac {1}{{2m}}\sum \limits _{\gamma = 1}^{m - 1} {\sin \frac {{2\pi \gamma \alpha }}{m}} \cot \frac {{\pi \gamma }}{n}.

\]
A 𝑞-polynomial identity
Authors: Yang Kung-Wei 
Published: June 2010

We show that the

q
q

-polynomial coefficients

[

r

n
⋅
m

]

[_{\;r}^{n \cdot m}]

(see the definition preceding Theorem 1), which are the generating functions of the number of inversions between multisets, satisfy the beautiful identity
\[

∏

i
=
1

n

(

∑

j
=
0

m

q

j
m
(
i
−
1
)
+

(
2
j

)

t
j

)

=

∑

r
=
0

n
m

[

r

n
⋅
m

]

q

(
2
r

)

t
r

.

\prod \limits _{i = 1}^n {\left ( {\sum \limits _{j = 0}^m {{q^{jm(i - 1) + (_2^j)}}{t^j}} } \right ) = \sum \limits _{r = 0}^{nm} {[_{\;r}^{n \cdot m}} ]{q^{(_2^r)}}{t^r}} .

\]
Some sharp weak-type inequalities for holomorphic functions on the unit ball of 𝐶ⁿ
Authors: Tomaszewski Bogusław 
Published: June 2010

Let

B
n

=
{
z
∈

C

n

:

|

z

|

>
1
}

{B^n} = \{ z \in {{\mathbf {C}}^n}:|z| > 1\}

,

S
n

=
∂

B
n

{S^n} = \partial {B^n}

and let

σ
n

{\sigma _n}

be the Haar measure on

S
n

{S^n}

. Then for all

f
∈

H
p

(
1
⩽
p
>
∞
)

f \in {H^p}(1 \leqslant p > \infty )

such that

Im
⁡
(
f
(
0
)
)
=
0

\operatorname {Im} (f(0)) = 0

and

t
>
0

t > 0

,
\[

σ
n

(
{
z
∈

S
n

:

|

f
(
z
)

|

⩾
t
}
)
⩽

C
p

⋅

|

|

Re

f

|

|

p
p

t
p

{\sigma _n}(\{ z \in {S^n}:|f(z)| \geqslant t\} ) \leqslant {C_p} \cdot \frac {{||\operatorname {Re} \,f||_p^p}} {{{t^p}}}

\]
for some constant

C
p

{C_p}

depending only on

p
p

. The best constant

C
p

{C_p}

is found for

1
⩽
p
⩽
2

1 \leqslant p \leqslant 2

.
Selfadjoint nonoscillatory second order linear 𝐵*-algebra differential equations
Authors: Lopez Manuel 
Published: June 2010

The main result in this paper states that the second order linear

B
∗

{B^*}

-algebra differential equation

(
p
(
t
)

y
′

)
+
q
(
t
)
y
=
0

(p(t)y’) + q(t)y = 0

, where

p
(
t
)

p(t)

is positive and

q
(
t
)

q(t)

is Hermitian for each

t
t

, is nonoscillatory on

[

t
0

,
∞
)

[{t_0},\infty )

if the scalar equation

(

‖

p

−
1

(
t
)

‖

−
1

W
′

)
′

+

‖

q
(
t
)

‖

W
=
0

({\left \| {{p^{ - 1}}(t)} \right \|^{ - 1}}W’)’ + \left \| {q(t)} \right \|W = 0

is nonoscillatory on

[

t
0

,
∞
)

[{t_0},\infty )

. Consequently, every criterion on nonoscillation in the scalar case automatically produces another one in the

B
∗

{B^*}

-algebra case.
Long periodic orbits of the triangle map
Authors: Scarowsky Manny , Boyarsky Abraham 
Published: June 2010

Let

τ
:
[
0
,
1
]
→
[
0
,
1
]

\tau :[0,1] \to [0,1]

be defined by

τ
(
x
)
=
2
x

\tau (x) = 2x

on

[
0
,
1

/

2
]

[0,1/2]

and

τ
(
x
)
=
2
(
1
−
x
)

\tau (x) = 2(1 - x)

on

[
1

/

2
,
1
]

[1/2,1]

. We consider

τ
\tau

restricted to the domain

D
N

=
{
2
a

/

p
N

,
N
⩾
1
,
0
⩽
2
a
⩽

p
N

,
(
a
,
p
)
=
1
}

{D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\}

where

p
p

is any odd prime. Let

k
⩾
1

k \geqslant 1

be the minimum integer such that

p
N

|

2
k

±
1

{p^N}|{2^k} \pm 1

. Then there are

(
(

p

−
1
)
⋅

p

N
−
1

)

/

2
k

(({\text {p}} - 1) \cdot {p^{N - 1}})/2k

periodic orbits of

τ

|

D
N

\tau {|_{{D_N}}}

, having equal length, and there are

k
k

points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval

(
c
,
d
)

(c,d)

approaches

d
−
c

d - c

as

p

N
−
1

→
∞

{p^{N - 1}} \to \infty

. An application to the irreducibility of certain nonnegative matrices is given.
A Wiener inversion-type theorem
Authors: Holub James R. 
Published: June 2010

Let

W
(
D
)
=
{
f
(
z
)
=

Σ

n
=
0

∞

a
n

z
n

|

|

|

f

|

|

1

=

Σ

n
=
0

∞

|

a
n

|

>
+
∞
}

W(D) = \{ f(z) = \Sigma _{n = 0}^\infty {a_n}{z^n}|\;||f|{|_1} = \Sigma _{n = 0}^\infty |{a_n}| > + \infty \}

,

f
(
z
)

f(z)

a function in

W
(
D
)

W(D)

for which

f
(
0
)
=
1

f(0) = 1

, and

M
f

{M_f}

the operator of multiplication by

f
(
z
)

f(z)

on

W
(
D
)

W(D)

. It is shown that if

k
k

and

m
m

are integers for which

0
≤
m
≤
k
−
1

0 \leq m \leq k - 1

and

X
k
m

X_k^m

is the closed subspace of

W
(
D
)

W(D)

spanned by

{

z

n
k
+
i

|

n
=
0
,
1
,
…
;
i
=
0
,
1
,
…
,
m
}

\{ {z^{nk + i}}|n = 0,1, \ldots ;i = 0,1, \ldots ,m\}

, then

M
f

{M_f}

is bounded below on

X
k
m

⇔
f
(
z
)

X_k^m \Leftrightarrow f(z)

does not have

k
−
m

k - m

distinct zeros in any set of the form

{

w
i

z
0

|

0
≤
i
≤
k
−
1
;

|

z
0

|

=
1
}

\{ {w^i}{z_0}|0 \leq i \leq k - 1;|{z_0}| = 1\}

, where

w
w

is a primitive

k
k

th root of unity.
A best constant and the Gaussian curvature
Authors: Hong Chong Wei 
Published: June 2010

For axisymmetric

f
∈

C
∞

(

S
2

)

f \in {C^\infty }({S^2})

we find conditions to make

f
f

the scalar curvature of a metric pointwise conformal to the standard metric of

S
2

{S^2}

. Closely related to these results, we prove that in the inequality (Moser [8])
\[

∫

S
2

e
u

≤
C

e

‖

∇
u

‖

2
2

/

16
π

∀
u
∈

H
1
2

(

S
2

)

with

∫

S
2

u
=
0

,

\int _{{S^2}} {{e^u} \leq C{e^{\left \| {\nabla u} \right \|_2^2/16\pi \quad }}\forall u \in H_1^2({S^2})} {\text { with }}\int _{{S^2}} {u = 0} ,

\]
, the best constant

C
=

Vol(

S

2

)

C = {\text {Vol(}}{{\text {S}}^2}{\text {)}}

.
A companion to the Oseledec multiplicative ergodic theorem
Authors: Watkins Joseph C. 
Published: June 2010

Let

F
1

,

F
2

,
…

{F_1},{F_2}, \ldots

be a stationary sequence of continuously differentiable mappings from

[
0
,
1
]

[0,1]

into the set of

d
×
d

d \times d

matrices. Assume

F
k

(
0
)
=
I

{F_k}(0) = I

for each

k
k

and

E
[

sup

0
≤
p
≤
1

|

|

F
k
′

(
p
)

|

|

]
>
∞

E[{\sup _{0 \leq p \leq 1}}||{F’_k}(p)||] > \infty

. Let

I

\mathcal {I}

denote the invariant sigma field for the sequence. Then
\[

lim

n
→
∞

F
n

(

1
n

)

⋯

F
2

(

1
n

)

F
1

(

1
n

)

=
exp
⁡
E
[

F
1
′

(
0
)

|

I

]

\lim \limits _{n \to \infty } {F_n}\left ( {\frac {1}{n}} \right ) \cdots {F_2}\left ( {\frac {1}{n}} \right ){F_1}\left ( {\frac {1}{n}} \right ) = \exp E[{F’_1}(0)|\mathcal {I}]

\]
with probability one.
Generalizations of the Sidon-Telyakovskiĭ theorem
Authors: Stanojević Časlav V. , Stanojevic Vera B. 
Published: June 2010

The well-known Sidon-Telyakovskii integrability condition is considerably lightened as follows:
\[

1
n

∑

k
=
1

n

|

Δ
c
(
k
)

|

p

A
k
p

=
O
(
1
)
,

n
→
∞

,

\frac {1}{n}\sum \limits _{k = 1}^n {\frac {{|\Delta c(k){|^p}}}{{A_k^p}} = O(1),\quad n \to \infty } ,

\]
where

{
c
(
n
)
}

\{ c(n)\}

is a certain null-sequence and

1
>
p
≤
2

1 > p \leq 2

. It is proved that

∑

n
=
1

∞

n

p
−
1

|

Δ
c
(
n
)

|

p

ρ
p

(
n
)
>
∞

\sum \nolimits _{n = 1}^\infty {{n^{p - 1}}|\Delta c(n){|^p}{\rho ^p}(n) > \infty }

is also a sufficient integrability condition provided

∑

n
=
1

∞

(
1

/

n
ρ
(
n
)
)
>
∞

\sum \nolimits _{n = 1}^\infty {(1/n\rho (n)) > \infty }

, where

{
ρ
(
n
)
}

\{ \rho (n)\}

is an increasing sequence of positive numbers.
A mixed Parseval equation and the generalized Hankel transformations
Authors: Méndez J. M. 
Published: June 2010

Let

T
1

{T_1}

and

T
2

{T_2}

be two classical integral transforms whose inverse formulas coincide with themselves, satisfying the mixed Parseval equation
\[

∫
0
∞

f
(
x
)
g
(
x
)
d
x
=

∫
0
∞

F
1

(
y
)

G
2

(
y
)

d
y
,

\int _0^\infty {f(x)g(x)dx = \int _0^\infty {{F_1}(y){G_2}(y)} \;dy,}

\]
where

F
1

(
y
)
=
(

T
1

f
)
(
y
)

{F_1}(y) = ({T_1}f)(y)

and

G
2

(
y
)
=
(

T
2

g
)
(
y
)

{G_2}(y) = ({T_2}g)(y)

. We propose to define the generalized transformation

T
1
′

{T’_1}

as the adjoint operator of

T
2

{T_2}

, and conversely. This procedure provides a new approach to extend the Hankel transform to certain spaces of distributions.
Rings with associators in the commutative center
Authors: Kleinfeld Erwin 
Published: June 2010

Thedy has introduced the subject of rings which satisfy the identity
\[

(I)

(
R
,
(
R
,
R
,
R
)
)
=
0
,

{\text {(I) }}(R,(R,R,R)) = 0,

\]
where the commutator is defined by

(
a
,
b
)
=
a
b
−
b
a

(a,b) = ab - ba

, and the associator is defined by

(
a
,
b
,
c
)
=
a
b
⋅
c
−
a
⋅
b
c

(a,b,c) = ab \cdot c - a \cdot bc

and which satisfy one additional identity such as

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

. Assuming characteristic

≠
2

\ne 2

and simplicity, Thedy’s result is that such a ring

R
R

must be either commutative or associative. Thedy also showed that simplicity could not be relaxed to prime by presenting some examples which are neither associative nor commutative. We show here that the additional identity assumed by Thedy is in fact unnecessary for we show that if

R
R

is simple, characteristic

≠
2
,
3

\ne 2,3

and satisfies (I) then

R
R

must be either commutative or associative.
Landau’s inequality for the difference operator
Authors: Kwong Man Kam , Zettl A. 
Published: June 2010

The best constants for Landau’s inequality with the classical

p
p

-norms are known explicitly only when

p
=
1
,
2

and

∞

p = 1,2{\text { and }}\infty

. This is true for both the discrete and the continuous versions of the inequality and for both the "whole line" and "half line" cases. In each of the six known cases the best constant in the discrete version is the same as the best constant for the continuous version. Here we show that for many other values of

p
p

the discrete constants are strictly greater than the corresponding continuous ones. In addition, we show that the "three norm version" of the inequality, established by Nirenberg and Gabushin in the continuous case is also valid in the discrete case.
Functions whose derivatives take values in a half-plane
Authors: Gray Fernando , Ruscheweyh Stephan 
Published: June 2010

We derive sharp upper and lower bounds for

|

z

f
′

(
z
)

/

f
(
z
)

|

\left | {zf’(z)/f(z)} \right |

where

f
∈
R

f \in R

, i.e.

f
f

analytic in

D

{\mathbf {D}}

with

f
(
0
)
=
0
,

f
′

(
0
)
=
1

f(0) = 0,f’(0) = 1

and

e

i
α

f
′

(
z
)

>

0

{e^{i\alpha }}f’(z){\text { > }}0

in

D

{\mathbf {D}}

for a certain

α
=
α
(
f
)
∈

R

\alpha = \alpha (f) \in {\mathbf {R}}

. The extremal function is

k
(
z
)
=
−
z
−
2
log
⁡
(
1
−
z
)

k(z) = - z - 2\log (1 - z)

. This result improves an earlier one of D. K. Thomas.
A weak Krasnosel′skiĭ theorem in 𝑅^{𝑑}
Authors: Breen Marilyn 
Published: June 2010

Let

S
S

be a compact, locally starshaped set in

R
d

{R^d}

, and let

k
k

be a fixed integer,

0
≤
k
≤
d

0 \leq k \leq d

. If every

d
−
k
+
1

d - k + 1

points of

S
S

are clearly visible via

S
S

from a common point, then for every

k
k

-flat

F
′

F’

there exists a translate

F
F

of

F
′

F’

such that the following holds: To each point

s
0

s_0

in

S
∼
F

S \sim F

there correspond a point

s
m

{s_m}

in

F
F

and a polygonal path

⋃

{

[

s

i
−
1

,

s
i

]
:
1
≤
i
≤
m

}

\bigcup \left \{ {[{s_{i - 1}},{s_i}]:1 \leq i \leq m} \right \}

in

S
∩
aff
⁡
(

s
0

∪
F
)

S \cap \operatorname {aff} ({s_0} \cup F)

with

dist
⁡
(

s
i

,
F
)
>
dist
⁡
(

s

i
−
1

,
F
)
,
1
≤
i
≤
m

\operatorname {dist} ({s_i},F) > \operatorname {dist} ({s_{i - 1}},F),1 \leq i \leq m

. If

k
=
0

k = 0

or

k
=
d
−
1

k = d - 1

, then each point of

S
S

sees via

S
S

some point of

F
F

. Moreover, if

k
=
1

k = 1

, then

F
F

can be chosen so that

F
∩
S

F \cap S

is convex.
A characterization of all loglinear inequalities for three quermassintegrals of convex bodies
Authors: Gritzmann Peter 
Published: June 2010

We give a complete characterization of all inequalities of the type

W
i
α

(
K
)

W
j
β

(
K
)

W
k
γ

(
K
)
≥
c

W_i^\alpha (K)W_j^\beta (K)W_k^\gamma (K) \geq c

, where

K
K

is an arbitrary convex body of Euclidean

d
d

-space,

W
l

(
K
)
,
l
=
i
,
j
,
k

{W_l}(K),l = i,j,k

, denotes the

l
l

th quermassintegral of

K
K

and

α
,
β
,
γ

\alpha ,\beta ,\gamma

and

c
c

are arbitrary reals. A special case of such inequalities is the classical isoperimetric inequality for the volume and surface area of convex bodies. It turns out that all nontrivial inequalities of this type can be generated by means of

d
−
1

d - 1

basic inequalities, the Fenchel-Alexandrov inequalities.
A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations
Authors: Lasiecka I. , Triggiani R. 
Published: June 2010

We consider the solution operator
\[

(
L
u
)
(
t
)
=
A

∫
0
t

G
(
t
−
τ
)

A

−
1

B
u
(
τ
)
d
τ

(Lu)(t) = A\int _0^t {G(t - \tau )} {A^{ - 1}}Bu(\tau )d\tau

\]
corresponding to the abstract equation

x
˙

=
A
x
+
B
u

\dot x = Ax + Bu

on a reflexive Banach space

X
X

, where the linear operator

A
:
X
⊃

D

(
A
)
→
X

A:X \supset \mathcal {D}(A) \to X

is the infinitesimal generator of a (strongly continuous) group

G
(
t
)

G(t)

of bounded operators on

X
X

, and

B
:
U
⊃

D

(
B
)
→
X

B:U \supset \mathcal {D}(B) \to X

is a generally unbounded linear operator with

A

−
1

B
∈

L

(
U
,
X
)
,
U

{A^{ - 1}}B \in \mathcal {L}(U,X),U

being another reflexive Banach space (without loss of generality we take

A
A

to be boundedly invertible). Let

0
>
T
>
∞

0 > T > \infty

be given. We prove the following theorem: if

L
L

is continuous

L
p

(
0
,
T
;
U
)
→

L
p

(
0
,
T
;
X
)
,
1
>
p
>
∞

{L^p}(0,T;U) \to {L^p}(0,T;X), 1 > p > \infty

, then in fact

L
:

L:

continuous

L
p

(
0
,
T
;
U
)
→
C
[
0
,
T
]
;
X

{L^p}(0,T;U) \to C[0,T];X

, a lifting regularity theorem in the time variable. Moreover, we show by a parabolic example with nonhomogeneous term in the Dirichlet boundary conditions that the theorem fails to be true, if

G
(
t
)

G(t)

is merely a s.c. semigroup even if holomorphic. Applications of the theorem include mixed hyperbolic problems, including second order scalar hyperbolic equations defined on an open bounded domain

Ω
⊂

R
n

,
∂
Ω
=
Γ

\Omega \subset {R^n},\partial \Omega = \Gamma

, with nonhomogeneous term of class

L
2

(
0
,
T
;

L
2

(
Γ
)
)

{L^2}(0,T;{L^2}(\Gamma ))

acting in the Dirichlet or in the Neumann boundary conditions. In the former case, the theorem recovers the authors’ original procedure which yielded optimal regularity results for this dynamics [L-T.2]; in the latter, the theorem improves upon results of Lions-Magenes [L-M.1, vol. II]. Extension to

T
=
∞

T = \infty

is also studied.
Point evaluations and polynomial approximation in the mean with respect to harmonic measure
Authors: Akeroyd John 
Published: June 2010

For

1
≤
s
>
∞

1 \leq s > \infty

and crescents

1

^{1}

G
G

, with harmonic measure

ω
\omega

, the author examines the collection of bounded point evaluations,

bpe
⁡

(

P
s

(
ω
)

)

\operatorname {bpe}\left ( {{P^s}\left ( \omega \right )} \right )

, (resp. analytic bounded point evaluations,

abpe
⁡

(

P
s

(
ω
)

)

\operatorname {abpe}\left ( {{P^s}\left ( \omega \right )} \right )

) for polynomials with respect to the

L
s

(
ω
)

{L^s}\left ( \omega \right )

norm. If the polynomials are dense in the generalized Hardy space

H
s

(
G
)

{H^s}\left ( G \right )

, then

bpe
⁡

(

P
s

(
ω
)

)

=
abpe
⁡

(

P
s

(
ω
)

)

=
G

\operatorname {bpe}\left ( {{P^s}\left ( \omega \right )} \right ) = \operatorname {abpe}\left ( {{P^s}\left ( \omega \right )} \right ) = G

(Theorem 4). If the polynomials are not dense in

H
s

(
G
)

{H^s}\left ( G \right )

, then (with a mild restriction on

∂
G

\partial G

)

bpe
⁡

(

P
s

(
ω
)

)

=
abpe
⁡

(

P
s

(
ω
)

)

=
int
⁡
(

G
¯

∧

)

\operatorname {bpe} \left ( {{P^s}\left ( \omega \right )} \right ) = \operatorname {abpe} \left ( {{P^s}\left ( \omega \right )} \right ) = \operatorname {int} ({\bar G^ \wedge })

(Theorem 7).
Dense barrelled subspaces of uncountable codimension
Authors: Saxon Stephen A. , Robertson Wendy J. 
Published: June 2010

Let

E
E

be a Hausdorff barrelled space. If there exists a dense barrelled subspace

M
M

such that

(
codim
⁡
(
M
)
≥
c
)
[
codim
⁡
(
M
)
=
dim
⁡
(
E
)
]

(\operatorname {codim} (M) \geq c)[\operatorname {codim} (M) = \operatorname {dim} (E)]

, we say that (

M
M

is a satisfactory subspace [11]) [

E
E

is barrelledly fit], respectively. Robertson, Tweddle and Yeomans [11] proved that

E
E

has a barrelled countable enlargement (BCE) if it has a satisfactory subspace. (Trivially)

E
E

has a satisfactory subspace if

dim
⁡
(
E
)
≥
c

\dim (E) \geq c

and

E
E

is barrelledly fit. We show that

E
E

is barrelledly fit (and

dim
⁡
(
E
)
≥
c

\dim (E) \geq c

) if

E
≆
φ

E \ncong \varphi

and either (i)

E
E

is an (LF)-space, or (ii)

E
E

is an infinite-dimensional separable space and the continuum hypothesis holds. Conclusion: barrelledly fit spaces and their permanence properties arise from and advance the study of BCE’s.
Nondentable solid subsets in Banach lattices failing RNP. Applications to renormings
Authors: Werner Elisabeth 
Published: June 2010

We show that for every Banach lattice

E
E

failing RNP and not containing

c
0

{c_0}

(resp. containing

c
0

{c_0}

) and for every

ε
>
0

\varepsilon > 0

there exists a solid convex closed subset

D
D

of the unit ball of

E
E

, such that
\[

dist
⁡
[
ext
⁡
(

D
¯

σ
(

E

∗
∗

,

E
∗

)

)
,
E
]
>
1
−
ε
(
resp
.
>

1
2

−
ε
)

\operatorname {dist} [\operatorname {ext} ({\overline D ^{\sigma ({E^{ * * }},{E^ * })}}),E] > 1 - \varepsilon (\operatorname {resp} . > \tfrac {1}{2} - \varepsilon )

\]
and such that every slice of

D
D

has diameter bigger than

2
−
ε

2 - \varepsilon

. We also prove that these results are optimal. We apply them to construct rough lattice norms with almost optimal constant on non-Asplund Banach lattices.
On hyperspaces of polyhedra
Authors: Sakai Katsuro 
Published: June 2010

Let

Q
=

[
−
1
,
1

]
ω

Q = {[ - 1,1]^\omega }

be the Hilbert cube and
\[

Q
f

=

{

(

x
i

)
∈
Q

|

x
i

=
0

except for finitely many

i

}

.

{Q_f} = \left \{ {({x_i}) \in Q|{x_i} = 0{\text {except for finitely many }}i} \right \}.

\]
For a compact connected polyhedron

X
X

with

dim
⁡
X
>
0

\dim X > 0

, the hyperspaces of (nonempty) subcompacta, subcontinua, and compact subpolyhedra of

X
X

are denoted by

2
X

,
C
(
X
)

{2^X},C(X)

, and

Pol(

X

)

{\text {Pol(}}X{\text {)}}

, respectively. And let

C

Pol

(
X
)
=
C
(
X
)
∩

Pol(

X

)

{C^{{\text {Pol}}}}(X) = C(X) \cap {\text {Pol(}}X{\text {)}}

. It is shown that the pair

(

2
X

,

Pol(

X

)

)

({2^X},{\text {Pol(}}X{\text {)}})

is homeomorphic to

(
Q
,

Q
f

)

(Q,{Q_f})

. In case

X
X

has no free arc, it is also proved that the pair

(
C
(
X
)
,

C

Pol

(
X
)
)

(C(X),{C^{{\text {Pol}}}}(X))

is homeomorphic to

(
Q
,

Q
f

)

(Q,{Q_f})

.
Two operator functions with monotone property
Authors: Furuta Takayuki 
Published: June 2010

This paper proves that

F
(
p
)
=

(

B
r

A
p

B
r

)

(
1
+
2
r
)

/

(
p
+
2
r
)

F(p) = {({B^r}{A^p}{B^r})^{(1 + 2r)/(p + 2r)}}

is an increasing function of

p
p

for

p
≧
1

p \geqq 1

and

r
≧
0

r \geqq 0

whenever

A
≧
B
≧
0

A \geqq B \geqq 0

. This result is more precise than our previous result that

A
≧
B
≧
0

A \geqq B \geqq 0

ensures

(

B
r

A
p

B
r

)

(
1
+
2
r
)

/

(
p
+
2
r
)

≧

B

1
+
2
r

{({B^r}{A^p}{B^r})^{(1 + 2r)/(p + 2r)}} \geqq {B^{1 + 2r}}

for each

p
≧
1

p \geqq 1

and

r
≧
0

r \geqq 0

. We also cite three counterexamples related to Theorem 1.
Continuous selections of solution sets to evolution equations
Authors: Staicu Vasile 
Published: June 2010

We prove the existence of a continuous selection of the multivalued map

ξ
→

T

(
ξ
)

\xi \to \mathcal {T}(\xi )

, where

T

(
ξ
)

\mathcal {T}(\xi )

is the set of all weak (resp. mild) solutions of the Cauchy problem
\[

x
˙

(
t
)
∈
A
x
(
t
)
+
F
(
t
,
x
(
t
)
)
,

x
(
0
)
=
ξ

\dot x(t) \in Ax(t) + F(t,x(t)),\quad x(0) = \xi

\]
, assuming that

F
F

is Lipschitzian with respect to

x
x

and

−
A

- A

is a maximal monotone map (resp.

A
A

is the infinitesimal generator of a

C
0

{C_0}

-semigroup). We also establish an analog of Michael’s theorem for the solution sets of the Cauchy problem

x
˙

(
t
)
∈
F
(
t
,
x
(
t
)
)
,

x
(
0
)
=
ξ

\dot x(t) \in F(t,x(t)),\;x(0) = \xi

.
A note on the 𝐿^{𝑝} analogue of the “zero-two” law
Authors: Berend Daniel 
Published: June 2010

It was proved by R. Wittmann [2] that, given a positive linear contraction of

L
p

(

1
≤
p
>
∞

)

,

sup

‖
f
‖

p

≤
1

lim

n
→
∞

‖

T
n

f
−

T

n
+
1

f

‖

{L^p}\left ( {1 \leq p > \infty } \right ),{\operatorname {sup} _{{{\left \| f \right \|}_p} \leq 1}}{\operatorname {lim} _{n \to \infty }}\left \| {{T^n}f - {T^{n + 1}}f} \right \|

is either

≥

α
p

or

0

\geq {\alpha _p}{\text {or}}0

; the (best possible) value of

α
p

{\alpha _p}

is the

l
p

{l_p}

-norm of a certain

3
×
3

3 \times 3

matrix. In this paper

α
p

{\alpha _p}

is explicitly expressed as a function of

p
p

.
Decidability of the existential theory of the set of natural numbers with order, divisibility, power functions, power predicates, and constants
Authors: Terrier Véronique 
Published: June 2010

We construct an algorithm to test if a system of conditions of the types

μ
>
η
,
μ

/

η
,
μ
=

η
a

,

P
a

(
μ
)
,
¬
(
μ
>
η
)
,
¬
(
μ

/

η
)
,
¬
(
μ
=

η
a

)

\mu > \eta ,\mu /\eta ,\mu = {\eta ^a},{P_a}(\mu ),\neg (\mu > \eta ),\neg (\mu /\eta ),\neg (\mu = {\eta ^a})

, and

¬
(

P
a

(
μ
)
)

\neg ({P_a}(\mu ))

has a solution in natural numbers. (

a
∈
N

a \in N

, and

P
a

{P_a}

denotes the set

{

n
a

:
n
∈
N
}

\{ {n^a}:n \in N\}

.)