bluuebunny/crossref_metadata_2025 huggingface dataset:
Thin interpolating sequences and three algebras of bounded functions
Authors: Hedenmalm Håkan
Published: June 2010

We consider the closed subalgebra

A

{\mathbf {A}}

of

H
∞

{H^\infty }

generated by the thin interpolating Blaschke products, the smallest

C
∗

{C^*}

subalgebra

B

{\mathbf {B}}

of

L
∞

{L^\infty }

containing

A

{\mathbf {A}}

, and the Douglas algebra

E

{\mathbf {E}}

generated by the complex conjugates of thin interpolating Blaschke products. Our main result is that every

E

{\mathbf {E}}

-invertible inner function is a finite product of thin interpolating Blaschke products, making

B

=

C

E

{\mathbf {B}} = {C_{\mathbf {E}}}

. We apply results of Chang and Marshall to prove that

A

=

B

∩

H
∞

{\mathbf {A}} = {\mathbf {B}} \cap {H^\infty }

, that finite convex combinations of finite products of thin interpolating Blaschke products are dense in the closed unit ball of

A

{\mathbf {A}}

, and that the corona theorem holds for

A

{\mathbf {A}}

.
Symmetrization and optimal control for elliptic equations
Authors: Voas Charles , Yaniro Daniel
Published: June 2010

We consider an optimal control problem where

u
(
x
)

u(x)

satisfies

−
div
⁡
(
H
(
x
)
∇
u
)
=
1

- \operatorname {div}(H(x)\nabla u) = 1

in

Ω
\Omega

and

H
(
x
)

H(x)

is a control. We introduce the functional
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript normal upper Omega Baseline left-parenthesis upper H right-parenthesis equals StartAbsoluteValue normal upper Omega EndAbsoluteValue Superscript negative 1 Baseline integral Underscript normal upper Omega Endscripts u left-parenthesis x right-parenthesis d x">

J
Ω

(
H
)
=

|

Ω

|

−
1

∫
Ω

u
(
x
)

d
x

{J_\Omega }(H) = {|\Omega |^{ - 1}}\int \limits _\Omega {u(x)} dx

and show using a symmetrization argument that if the distribution function of

H
H

is fixed, then

J
Ω

(
H
)

{J_\Omega }(H)

is largest when

Ω
\Omega

is a ball and

H
H

is radial and decreasing on radii.
An alternating procedure for operators on 𝐿_{𝑝} spaces
Authors: Akcoglu M. A. , Sucheston L.
Published: June 2010

Let

L
p

{L_p}

be the usual Banach spaces over a

σ
\sigma

-finite measure space. If

1

 > 

p

 > 

∞

1{\text { > }}p{\text { > }}\infty

and

q
=
p

(
p
−
1

)

−
1

q = p{(p - 1)^{ - 1}}

, then

ψ
p

:

L
p

→

L
q

{\psi _p}:{L_p} \to {L_q}

denotes the duality mapping defined by the requirements that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis f comma psi Subscript p Baseline f right-parenthesis equals double-vertical-bar f double-vertical-bar Subscript p Superscript p Baseline equals double-vertical-bar f double-vertical-bar Subscript p Baseline double-vertical-bar psi Subscript p Baseline f double-vertical-bar q comma f element-of upper L Subscript p Baseline">

(
f
,

ψ
p

f
)
=

‖
f
‖

p
p

=

‖
f
‖

p

‖

ψ
p

f

‖

q
,
f
∈

L
p

(f,{\psi _p}f) = \left \| f \right \|_p^p = {\left \| f \right \|_p}\left \| {{\psi _p}f} \right \|q,f \in {L_p}

. If

T
:

L
p

→

L
p

T:{L_p} \to {L_p}

is a bounded linear operator, then

M
(
T
)
:

L
p

→

L
p

M(T):{L_p} \to {L_p}

is the mapping defined by

M
(
T
)
=

ψ
q

T
∗

ψ
p

T

M(T) = {\psi _q}{T^ * }{\psi _p}T

, where

T
∗

:

L
q

→

L
q

{T^ * }:{L_q} \to {L_q}

is the adjoint of

T
T

. It is proved that if

T
n

{T_n}

is a sequence of operators on

L
p

{L_p}

such that

‖

T
n

‖

≤
1

\left \| {{T_n}} \right \| \leq 1

for all

n
n

, then

M
(

T
n

⋯

T
2

T
1

)
f

M({T_n} \cdots {T_2}{T_1})f

converges in

L
p

{L_p}

for all

f
∈

L
p

f \in {L_p}

.
Free actions on products of even-dimensional spheres
Authors: Cusick Larry W.
Published: June 2010

We show that if

G
G

is a finite group acting freely on

∏

j
=
1

k

S

2

n
j

\prod _{j = 1}^k{S^{2{n_j}}}

and if the induced action on

(
mod
2
)

(\bmod 2)

homology is trivial, then 2 for some

l
≤
k

l \leq k

. We also show that if

G
G

acts freely on

G
G

and

G
G

is cyclic of order

2
l

{2^l}

, then

2

l
−
1

≤
k

{2^{l - 1}} \leq k

.
The 𝑆𝑈𝑃=𝑀𝐴𝑋 problem for 𝛿
Authors: Berner Andrew J. , Juhász István
Published: June 2010

Let

δ

(
X
)

=
sup
⁡
{
d
(
D
)
:
D

\delta \left ( X \right ) = \operatorname {sup}\{ d(D):D

is a dense subspace of

X
}

X\}

. It is shown that if

κ
\kappa

is a limit cardinal, but not a strong limit, and

cf

(
κ
)

>
ω

{\text {cf}}\left ( \kappa \right ) > \omega

, then there is a

0
0

-dimensional Hausdorff space

X
X

such that

δ

(
X
)

=
κ

\delta \left ( X \right ) = \kappa

, but for all dense

D
⊂
X
,
d
(
D
)
>
κ

D \subset X,d(D) > \kappa

. For all other values of

κ
\kappa

, if

X
X

is Hausdorff and

δ

(
X
)

=
κ

\delta \left ( X \right ) = \kappa

, then there is a dense

D
⊂
X

D \subset X

such that

d

(
D
)

=
κ

d\left ( D \right ) = \kappa

.
Spectral asymptotics for spinor Laplacians and multiplicities
Authors: Moscovici Henri
Published: June 2010

We use Getzler’s formula for the heat kernel of the spinor Laplacian to derive information about the asymptotic distribution of multiplicities in the quasi-regular representation of a semisimple Lie group

G
G

modulo a co-compact discrete subgroup

Γ
\Gamma

.
𝑁𝐾₁ of finite groups
Authors: Harmon Dennis R.
Published: June 2010

For

R
R

any ring with unity, let

N

K
1

(
R
)

N{K_1}(R)

denote the kernel of the homomorphism

ε
∗

:

K
1

(
R
[
t
]
)
→

K
1

(
R
)

{\varepsilon _*}:{K_1}(R[t]) \to {K_1}(R)

induced by the augmentation

ε
:
t
→
0

\varepsilon :t \to 0

. We show that if

π
\pi

is a finite group of square-free order, then

N

K
1

(
Z
π
)
=
0

N{K_1}(Z\pi ) = 0

.
Almost exactness in normed spaces
Authors: Harte Robin
Published: June 2010

"Almost exactness" is defined for pairs of bounded operators between normed spaces.
The complement of the stable manifold for one-dimensional endomorphisms
Authors: Arteaga Carlos
Published: June 2010

Let

N
N

denote either the circle

S
1

{S^1}

or the closed interval

I
=
[
0
,
1
]

I = [0,1]

and let

f
f

be a

C
1

{C^1}

endomorphism of

N
N

. Let

Σ
(
f
)

\Sigma (f)

be the complement of the union of the stable manifolds of the sinks of

f
f

. In this paper we give necessary and sufficient conditions for

Σ
(
f
)

\Sigma (f)

to consist of eventually periodic points.
How many sets are porous?
Authors: Zamfirescu Tudor
Published: June 2010

The notion of a

σ
\sigma

-porous set is often used to sharpen results on sets of first Baire category or of measure zero. It essentially uses the related notion of porosity. We find out in this note that there are quite a few porous sets: In complete convex metric spaces, most totally bounded closed sets are porous! Then we strengthen this result for the case of a Banach space.
The roots of a polynomial vary continuously as a function of the coefficients
Authors: Harris Gary , Martin Clyde
Published: June 2010

We present an elementary topological proof that the roots of a polynomial vary continuously as a function of the coefficients.
For every continuous 𝑓 there is an absolutely continuous 𝑔 such that [𝑓=𝑔] is not bilaterally strongly porous
Authors: Buczolich Z.
Published: June 2010

For any Darboux function

f
:

[

0
,
1

]

→

R

f:\left [ {0,1} \right ] \to {\mathbf {R}}

and any

0
>
δ
>
1

0 > \delta > 1

there is a point

x
∈

[

0
,
1
−
δ

]

x \in \left [ {0,1 - \delta } \right ]

and a sequence

x
n

{x_n}

such that (a)

x
n

∈

[

x
+

δ

n
+
1

,
x
+

δ
n

]

(

n
=
1
,
2
,
…

)

{x_n} \in \left [ {x + {\delta ^{n + 1}},x + {\delta ^n}} \right ]\left ( {n = 1,2, \ldots } \right )

and (b)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 2 Overscript normal infinity Endscripts StartAbsoluteValue f left-parenthesis x Subscript n Baseline right-parenthesis minus f left-parenthesis x Subscript n minus 1 Baseline right-parenthesis EndAbsoluteValue greater-than plus normal infinity">

∑

n
=
2

∞

|

f

(

x
n

)

−
f

(

x

n
−
1

)

|

>
+
∞

\sum \nolimits _{n = 2}^\infty {\left | {f\left ( {{x_n}} \right ) - f\left ( {{x_{n - 1}}} \right )} \right |} > + \infty

. Consequently, for every

f
∈
C

[

0
,
1

]

f \in C\left [ {0,1} \right ]

there is an absolutely continuous function

g
g

such that

{

x
:
f

(
x
)

=
g

(
x
)

}

\left \{ {x:f\left ( x \right ) = g\left ( x \right )} \right \}

is not bilaterally strongly porous.
Remarks on linear differential systems with measurable coefficients
Authors: Johnson Russell A.
Published: June 2010

We discuss several topics having to do with linear differential equations with locally integrable coefficients. These include gap labelling and Cantor spectrum for the random Schrodinger operator, and Coppel’s perturbation theorem.
On a characterization of 𝑊-sets and the dimension of hyperspaces
Authors: Grispolakis J. , Tymchatyn E. D.
Published: June 2010

A subcontinuum

A
A

of a continuum

X
X

is a

W
W

-set if for each mapping

f
:
Y
↠
X

f:Y \twoheadrightarrow X

of an arbitrary continuum

Y
Y

onto

X
X

there is a continuum in

Y
Y

which is mapped by

f
f

onto

A
A

. We characterize

W
W

-sets in terms of accessibility by small continua. We localize several known results on continua all of whose subcontinua are

W
W

-sets. Finally, we extend a result of J. T. Rogers by proving that if

X
X

is an atriodic continuum whose first Čech cohomology group is finitely generated then the hyperspace

C
(
X
)

C(X)

of subcontinua of

X
X

is two dimensional.
External equivalence classes in decompositions of spaces
Authors: Dębski W. , Tymchatyn E. D.
Published: June 2010

Let

X
X

be an indecomposable continuum in the plane. Krasinkiewicz has shown that the union of the external composants of

X
X

is a first category

F
σ

{F_\sigma }

set in

X
X

. We give extensions of this theorem to quite general decompositions of a space

X
X

embedded in a locally connected ambient space

Y
Y

.
Pointwise convergence approximate identities of dilated radially decreasing kernels
Authors: Kerman R. A.
Published: June 2010

Let

ϕ
\phi

be integrable on

R
n

{R^n}

with

∫

R
n

ϕ
(
y
)
d
y
=
1

\int _{{R^n}} {\phi (y)dy = 1}

. It is shown that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript epsilon right-arrow 0 plus Endscripts left-parenthesis phi Subscript upper F Superscript asterisk Baseline f right-parenthesis left-parenthesis x right-parenthesis equals limit Underscript epsilon right-arrow 0 plus Endscripts epsilon Superscript negative n Baseline integral Underscript upper R Superscript n Baseline Endscripts left-parenthesis StartFraction x minus y Over epsilon EndFraction right-parenthesis f left-parenthesis y right-parenthesis d y equals f left-parenthesis x right-parenthesis">

lim

ε
→
0
+

(

ϕ
F
∗

f
)
(
x
)
=

lim

ε
→
0
+

ε

−
n

∫

R
n

(

x
−
y

ε

)
f
(
y
)
d
y
=
f
(
x
)

{\lim _{\varepsilon \to 0 + }}(\phi _F^*f)(x) = {\lim _{\varepsilon \to 0 + }}{\varepsilon ^{ - n}}\int _{{R^n}} {(\frac {{x - y}} {\varepsilon })f(y)dy = f(x)}

\]
a.e. on

R
n

{R^n}

, whenever the least radially decreasing majorant of

ϕ
\phi

, defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="psi left-parenthesis x right-parenthesis equals sup Underscript StartAbsoluteValue y EndAbsoluteValue greater-than-or-slanted-equals StartAbsoluteValue x EndAbsoluteValue Endscripts StartAbsoluteValue phi left-parenthesis y right-parenthesis EndAbsoluteValue">

ψ
(
x
)
=

sup

|

y

|

⩾

|

x

|

|

ϕ
(
y
)

|

\psi (x) = {\sup _{|y| \geqslant |x|}}|\phi (y)|

, is such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue x EndAbsoluteValue Superscript n Baseline psi left-parenthesis x right-parenthesis equals StartAbsoluteValue x EndAbsoluteValue Superscript n Baseline psi left-parenthesis StartAbsoluteValue x EndAbsoluteValue right-parenthesis">

|

x

|

n

ψ
(
x
)
=

|

x

|

n

ψ
(

|

x

|

)

|x{|^n}\psi (x) = |x{|^n}\psi (|x|)

is nonincreasing in

|

x

|

|x|

when

|

x

|

|x|

is large and

(

ψ

ε
0

∗

|

f

|

)
(

x
0

)
>
∞

(\psi _{{\varepsilon _0}}^*|f|)({x_0}) > \infty

for some

x
0

∈

R
n

{x_0} \in {R^n}

and

ε
0

>
0

{\varepsilon _0} > 0

.
𝐾ⁿ-positive maps in 𝐶*-algebras
Authors: Itoh Takashi
Published: June 2010

Let

K
n

{K^n}

be the set of

n
n

-positive maps of

B
(
H
)

B(H)

to

B
(
H
)

B(H)

. A

K
n

{K^n}

-positive map of a

C
∗

{C^ * }

-algebra

A
A

to

B
(
H
)

B(H)

is a positive linear map

ϕ
\phi

such that

∑

Tr(

ϕ

(

a
i

)

b
i
t

)
≥
0

\sum {{\text {Tr(}}\phi {\text {(}}} {a_i})b_i^t) \geq 0

for any
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation a Subscript i Baseline circled-times b Subscript i Baseline element-of left-brace x element-of upper A circled-times Subscript gamma Baseline upper T left-parenthesis upper H right-parenthesis vertical-bar upper K Superscript n Baseline contains-as-member Superscript for-all Baseline alpha comma left-parenthesis id circled-times alpha right-parenthesis left-parenthesis x right-parenthesis greater-than-or-equal-to 0 right-brace">

∑

a
i

⊗

b
i

∈
{
x
∈
A

⊗
γ

T
(
H
)

|

K
n

∋
∀

α
,
(

id

⊗
α

)(

x

)

≥

0

}

\sum {{a_i} \otimes {b_i} \in \{ x \in A{ \otimes _\gamma }T(H)|{K^n}} { \ni ^\forall }\alpha ,({\text {id}} \otimes \alpha {\text {)(}}x{\text {)}} \geq {\text {0}}\}

. It is shown that the following three statements are equivalent. (1) Every

K
n

{K^n}

-positive map of

A
A

to

B
(
H
)

B(H)

is

K

n
+
1

{K^{n + 1}}

-positive. (2) Every

K
n

{K^n}

-positive map of

A
A

to

B
(
H
)

B(H)

is completely positive. (3)

A
A

is an

n
n

-subhomogeneous

C
∗

{C^ * }

-algebra.
On the Calkin representations of 𝐵(ℋ)
Authors: Behncke H.
Published: June 2010

Irreducible representations of

B
(

H

)

B(\mathcal {H})

, with

H

\mathcal {H}

a not necessarily separable Hilbert space, are constructed and analyzed along the lines of a similar study of Reid for separable Hilbert spaces.
The convergence of moments in the central limit theorem for 𝜌-mixing sequences of random variables
Authors: Peligrad Magda
Published: June 2010

In this paper we establish maximal inequalities for

ρ
\rho

-mixing sequences and, as a consequence, we obtain the convergence of the expected value of functions of partial sums to the corresponding ones of the normal distribution.
Two notes on imbedded prime divisors
Authors: Ratliff L. J.
Published: June 2010

The first note shows that if

R
>
T

R > T

are any two Noetherian rings, then there exists a Noetherian ring

B
B

between

R
R

and

T
T

which has a maximal ideal

N
N

such that

grade

(
N
)
≤
1

{\text {grade}}(N) \leq 1

and

N
∩
R

N \cap R

is a maximal ideal. The second note shows that if

R
R

is a Noetherian ring, then there exists a free quadratic integral extension ring

B
B

of

R
R

such that

Spec
⁡
(
B
)
≅
Spec
⁡
(
R
)

\operatorname {Spec}(B) \cong \operatorname {Spec}(R)

and such that if

I
I

is any regular ideal in

R
R

and

P
1

∩
⋯
∩

P
g

{P_1} \cap \cdots \cap {P_g}

are prime ideals in

R
R

containing

I
I

, then there exists an ideal

J
J

in

B
B

integrally dependent on

I
B

IB

such that the prime ideals corresponding to the

P
i

{P_i}

are prime divisors of

J
n

{J^n}

for all

n
≥
1

n \geq 1

.
𝑝-adic continued fractions of Liouville type
Authors: Laohakosol Vichian , Ubolsri Patchara
Published: June 2010

A

p
p

-adic analogue of a recent algebraic independence criterion of Adams is proved. It is then applied to construct Liouville type

p
p

-adic continued fractions of the kinds first considered by Ruban, and by Schneider.
Finitely many primitive positive clones
Authors: Burris S. , Willard R.
Published: June 2010

Given a finite set

A
A

there are only finitely many sequences of the form

⟨

Con
⁡
(

A

n

)

⟩

n
≥
1

{\left \langle {\operatorname {Con}({{\mathbf {A}}^n})} \right \rangle _{n \geq 1}}

or

⟨

Hom
⁡
(

A

n

,

A

)

⟩

n
≥
1

{\left \langle {\operatorname {Hom}({{\mathbf {A}}^n},{\mathbf {A}})} \right \rangle _{n \geq 1}}

, where

A

{\mathbf {A}}

is any algebra on

A
A

. From this we derive the fact that there are only finitely many primitive positive clones on

A
A

, which solves a problem posed by A. F. Danil’čenko in the 1970s. Consequently there are only finitely many model companions for universal Horn classes generated by an algebra of a given finite size.
Perturbations of the unilateral shift and transitive operator algebras
Authors: Ansari Mohamad A.
Published: June 2010

An operator

T
T

on a Hilbert space

H

\mathcal {H}

is said to have the transitive algebra property if

L

(

H

)

\mathcal {L}(\mathcal {H})

is the only transitive operator algebra which contains

T
T

. It was shown by Arveson that the unilateral shift has this property. It is the purpose of the present paper to show that perturbations of the unilateral shift by a large class of finite rank operators have the transitive algebra property. Our results are partial solutions of the well-known transitive algebra problem.
The simplicial bundle of a CW fibration
Authors: Barnes Donald W.
Published: June 2010

Suppose we are given a fibration

p
:
E
→
B

p:E \to B

over a connected base, with both base and fibre having the homotopy type of CW complexes. We construct a fibre bundle over

B
B

fibre homotopy equivalent to the given fibration and with fibre a simplicial complex. Further, the transformations of the fibre arising from the transition functions of this bundle are simplicial maps. From this, we deduce that the weak spectral sequence constructor axioms are sufficient to determine the Serre spectral sequence of a CW fibration.
A compact semilattice on the Hilbert cube with no interval homomorphism
Authors: Gierz Gerhard
Published: June 2010

An example for a compact semilattice with no interval homomorphism is given. The underlying space of the semilattice is homeomorphic to the Hilbert cube.
A short proof of a theorem on compositions of d.c. mappings
Authors: Veselý Libor
Published: June 2010

In this note, there is presented an essentially shorter proof of the theorem that a composition of d.c. mappings is locally d.c., proved for finite dimensions by P. Hartman in 1959.
A finiteness problem for one-dimensional maps
Authors: de Melo W.
Published: June 2010

We discuss the connection between the density of structurally stable maps in the space of unimodal maps of the interval, the finiteness of attractors and the nonexistence of wandering intervals. We show that in the space of unimodal maps having an eventually periodic flat critical point, there is a residual subset whose maps have infinitely many sinks. In this space there are also maps having a wandering interval.
A further refinement of the Bruhat decomposition
Authors: Curtis Charles W.
Published: June 2010

Kawanaka obtained explicit formulas for the structure constants in the Hecke algebra

H

(

G

(
q
)

,
B

(
q
)

)

H\left ( {G\left ( q \right ),B\left ( q \right )} \right )

of a finite Chevalley group

G

(
q
)

G\left ( q \right )

. This note contains a geometric interpretation of these formulas, involving decompositions of Bruhat cells, in connected reductive algebraic groups.
Unique solvability of an extended Stieltjes moment problem
Authors: Njåstad Olav
Published: June 2010

Let

a
1

,

a
2

,
…
,

a
p

{a_1},{a_2}, \ldots ,{a_p}

be given real numbers ordered by size, and let

[
α
,
β
]

[\alpha ,\beta ]

be a real interval disjoint from the set

{

a
1

,

a
2

,
…
,

a
p

}

\{ {a_1},{a_2}, \ldots ,{a_p}\}

. Let

{

c
j

(
i
)

:
j
=
1
,
2
,
…
}

\{ c_j^{(i)}:j = 1,2, \ldots \}

, be sequences of real numbers and

c
0

{c_0}

be a real number. The extended Stieltjes moment problem is to find a distribution function

ψ
\psi

with all its points of increase in

[
α
,
β
]

[\alpha ,\beta ]

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript alpha Superscript beta Baseline d psi left-parenthesis t right-parenthesis equals c 0 comma integral Subscript alpha Superscript beta Baseline StartFraction d psi left-parenthesis t right-parenthesis Over left-parenthesis t minus a Subscript i Baseline right-parenthesis Superscript j Baseline EndFraction equals c Subscript j Superscript left-parenthesis i right-parenthesis Baseline comma i equals 1 comma ellipsis comma p comma j equals 1 comma 2 comma ellipsis period">

∫
α
β

d
ψ
(
t
)
=

c
0

,

∫
α
β

d
ψ
(
t
)

(
t
−

a
i

)

j

=

c
j

(
i
)

,

i
=
1
,
…
,
p
,

j
=
1
,
2
,
…
.

\int _\alpha ^\beta {d\psi (t) = {c_0},\quad \int _\alpha ^\beta {\frac {{d\psi (t)}}{{{{(t - {a_i})}^j}}} = c_j^{(i)},\quad i = 1, \ldots ,p,\;j = 1,2, \ldots .} }

\]
Necessary and sufficient conditions for the existence of a unique solution of the problem are given. Orthogonal

R
R

-functions and Gaussian quadrature formulas play important roles in the proof.
A remark on the existence of positive periodic solutions of superlinear parabolic problems
Authors: Esteban Maria J.
Published: June 2010

We prove the existence of a solution for the following problem:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential Subscript t Baseline u minus normal upper Delta u equals f left-parenthesis t comma x comma u right-parenthesis in left-parenthesis 0 comma upper T right-parenthesis times normal upper Omega comma u greater-than 0 in left-parenthesis 0 comma upper T right-parenthesis times normal upper Omega comma u left-parenthesis upper T right-parenthesis equals u left-parenthesis 0 right-parenthesis in normal upper Omega comma u equals 0 on left-parenthesis 0 comma upper T right-parenthesis times partial-differential normal upper Omega comma">

∂
t

u
−
Δ
u
=
f
(
t
,
x
,
u
)

 in (

0
,
T
)
×
Ω
,

u

 > 

0

 in (

0
,
T
)
×
Ω
,
u
(
T
)
=
u
(
0
)

 in 

Ω
,

u
=
0

 on (

0
,
T
)
×
∂
Ω
,

{\partial _t}u - \Delta u = f(t,x,u){\text { in (}}0,T) \times \Omega ,\;\quad u{\text { > }}0{\text { in (}}0,T) \times \Omega ,u(T) = u(0){\text { in }}\Omega ,\;\quad u = 0{\text { on (}}0,T) \times \partial \Omega ,

\]
where

Ω
\Omega

is a bounded domain of

R
N

{R^N}

and the function

f
(
t
,
x
,
⋅
)

f(t,x, \cdot )

grows more slowly than

u
α

{u^\alpha }

at

+
∞

+ \infty

, with

α

 > 

N

/

(
N
−
2
)

\alpha {\text { > }}N/(N - 2)

. On démontre ici l’existence d’une solution positive pour le problème parabolique périodique suivant
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row partial-differential Subscript t Baseline u minus normal upper Delta u equals f left-parenthesis t comma x comma u right-parenthesis in left-parenthesis 0 comma upper T right-parenthesis times normal upper Omega comma u greater-than 0 in left-parenthesis 0 comma upper T right-parenthesis times normal upper Omega comma u left-parenthesis upper T right-parenthesis equals u left-parenthesis 0 right-parenthesis in normal upper Omega comma u equals 0 on left-parenthesis 0 comma upper T right-parenthesis times partial-differential normal upper Omega comma EndLayout">

∂
t

u
−
Δ
u
=
f
(
t
,
x
,
u
)

 in (0,

T

)

×
Ω

,

u

 > 

0

 in (0,

T
)
×
Ω
,
u
(
T
)
=
u
(
0
)

 in 

Ω
,

u
=
0

 on (

0
,
T
)
×
∂
Ω
,

\begin {gathered} {\partial _t}u - \Delta u = f(t,x,u){\text { in (0,}}T{\text {)}} \times \Omega {\text {,}}\quad \;u{\text { > }}0{\text { in (0,}}T) \times \Omega , u(T) = u(0){\text { in }}\Omega ,\quad \;u = 0{\text { on (}}0,T) \times \partial \Omega ,\end {gathered}

\]
où

Ω
\Omega

est un domaine borné de

R
N

{R^N}

et la fonction

f
(
t
,
x
,
⋅
)

f(t,x, \cdot )

croit plus lentement que

u
α

{u^\alpha }

à l’infini, avec

α

 > 

N

/

(
N
−
2
)

\alpha {\text { > }}N/(N - 2)

.
A note on hyper-Hermitian four-manifolds
Authors: Boyer Charles P.
Published: June 2010

It is shown that the only hyperhermitian four-manifolds are, up to conformal equivalence, tori and

K
3

{K^3}

surfaces with their standard hyperKähler structures and certain conformally flat Hopf surfaces.
A characterization of 𝐿^{𝑝}-improving measures
Authors: Hare Kathryn E.
Published: June 2010

A Borel measure

μ
\mu

on a compact abelian group

G
G

is

L
p

{L^p}

-improving if

μ
\mu

convolves

L
p

(
G
)

{L^p}(G)

to

L

p
+
ε

(
G
)

{L^{p + \varepsilon }}(G)

for some

ε
>
0

\varepsilon > 0

. We characterize

L
p

{L^p}

-improving measures by means of their Fourier transforms.
𝑐₀-subspaces and fourth dual types
Authors: Farmaki Vasiliki A.
Published: June 2010

For a separable Banach space

X
X

, we define the notion of a

c

0
+

{c_{0 + }}

-type on

X
X

and show that the existence of such a type is equivalent to the embeddability of

c
0

{c_0}

in

X
X

. All these types are weakly null and fourth dual (i.e. of the form

τ
(
x
)
=

‖

x
+
g

‖

\tau (x) = \left \| {x + g} \right \|

for

g
∈

X

∗
∗
∗
∗

g \in {X^{****}}

). We define the

l

l
+

{l^{{l^ + }}}

-dual types on

X
X

(these are generated by sequences in

X

∗
∗

{X^{**}}

) and prove that they coincide with the fourth dual types on

X
X

. We also prove that

c

0
+

{c_{0 + }}

-types are fourth dual types.
On the relation between 𝐶*-algebras of foliations and those of their coverings
Authors: Wang Xiaolu
Published: June 2010

By using the theory of groupoid equivalence of P. S. Muhly, J. N. Renault and D. P. Williams (cf. [5, 7]), we identify the relation between the

C
∗

{C^*}

-algebra of a foliated manifold and those of its regular covering foliations.
A construction of finite and 𝜎-finite invariant measures in measure spaces
Authors: Kubokawa Yoshihiro
Published: June 2010

Let

T
T

be a bijective nonsingular transformation on a finite measure space. We shall first construct a

σ
\sigma

-finite and finite invariant measure by a unified method which is valid for both cases. Secondly we shall give another construction of a finite invariant measure. We shall also give a new necessary and sufficient condition of a unified form for the existence of

σ
\sigma

-finite and finite invariant measures. Further, we shall discuss in detail ergodic transformations.
A note on the 𝑔-function
Authors: Wilson J. Michael
Published: June 2010

We prove a pointwise inequality relating real-variable Littlewood-Paley

g
g

-functions and real-variable Lusin area functions.
Subdirectly irreducible members of products of lattice varieties
Authors: Grätzer George , Kelly David
Published: June 2010

In this paper we prove: Theorem.LetV\mathbf {V}andW\mathbf {W}be nontrivial lattice varieties. IfL∈V∘WL \in \mathbf {V} \circ \mathbf {W},then there is a subdirectly irreducibleS∈V∘WS \in \mathbf {V} \circ \mathbf {W}containingLLas a sublattice. Moreover, ifLLis finite,SScan also be chosen to be finite.
Analytic and differentiable functions vanishing on an algebraic set
Authors: Kucharz Wojciech
Published: June 2010

Let

U
U

be an open semi-algebraic subset of

R

n

{{\mathbf {R}}^n}

and let

X
X

be a closed analytic subset of

U
U

which is also a semi-algebraic set (e.g.,

U
=

R

n

U = {{\mathbf {R}}^n}

and

X
X

is an algebraic subset of

R

n

{{\mathbf {R}}^n}

). It is proved that the ideal of analytic functions on

U
U

vanishing on

X
X

is finitely generated provided that the set

X
X

is coherent. The ideal of infinitely differentiable functions on

U
U

vanishing on

X
X

is finitely generated if and only if the set

X
X

is coherent.
The structure of bounded bilinear forms on products of 𝐶*-algebras
Authors: Ylinen Kari
Published: June 2010

Let

A
1

{A_1}

and

A
2

{A_2}

be

C
∗

{C^ * }

-algebras and

B
:

A
1

×

A
2

→

C

B:{A_1} \times {A_2} \to {\mathbf {C}}

be a bounded bilinear form. It is proved that there exist a Hilbert space

H
H

, two Jordan morphisms

μ
i

:

A
i

→
L
(
H
)
,
i
=
1
,
2

{\mu _i}:{A_i} \to L(H),i = 1,2

, and two vectors

ξ
1

,

ξ
2

∈
H

{\xi _1},{\xi _2} \in H

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B left-parenthesis x comma y right-parenthesis equals left-parenthesis mu 1 left-parenthesis x right-parenthesis xi 1 vertical-bar mu 2 left-parenthesis y Superscript asterisk Baseline right-parenthesis xi 2 right-parenthesis for all x element-of upper A 1 comma y element-of upper A 2 period">

B
(
x
,
y
)
=
(

μ
1

(
x
)

ξ
1

|

μ
2

(

y
∗

)

ξ
2

)

 for all 

x
∈

A
1

,
y
∈

A
2

.

B(x,y) = ({\mu _1}(x){\xi _1}\left | {{\mu _2}({y^ * }} \right .){\xi _2}){\text { for all }}x \in {A_1},y \in {A_2}.

\]
The proof depends on the Grothendieck-Pisier-Haagerup inequality and Halmos’s unitary dilation theorem. An extemely elementary proof of the latter is given.
Measurable Darboux functions
Authors: Brown J. B. , Humke P. , Laczkovich M.
Published: June 2010

We investigate how certain Darboux-like properties of real functions (including connectivity, almost continuity, and peripheral continuity) are related to each other within certain measurability classes (including the classes of Lebesgue measureable, Borel, and Baire-

1
1

functions).
𝑘-partitions and a characterization for compact unions of 𝑘 starshaped sets
Authors: Breen Marilyn
Published: June 2010

For natural numbers

k
k

and

j
j

, define

σ

(

1
,
j

)

=
j
+
2
,

σ

(

2
,
1

)

=
6

\sigma \left ( {1,j} \right ) = j + 2,\;\sigma \left ( {2,1} \right ) = 6

, and

σ

(

k
,
j

)

=

k
2

j
+
1

\sigma \left ( {k,j} \right ) = {k^2}j + 1

otherwise. The set

S
S

in

R
d

{R^d}

has property

A

(
k
)

A\left ( k \right )

if and only if

S
S

is a finite union of one-dimensional convex sets and for every

σ

(

k
,
1

)

\sigma \left ( {k,1} \right )

-member subset

F
F

of

S
S

there correspond points

c
1

,
…
,

c
k

{c_1}, \ldots ,{c_k}

(depending on

F
F

) such that each point of

F
F

sees via

S
S

some

c
i

,
1
⩽
i
⩽
k

{c_i},1 \leqslant i \leqslant k

. The following results are established. (1) Let

k
k

and

j
j

be fixed natural numbers, and let

L

\mathcal {L}

be a collection of sets such that every

j
+
1

j + 1

members meet in at most one point. Then

L

\mathcal {L}

has a

k
k

-partition

L

1

,
…
,

L

k

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

with

∩

{

L
:
L

in

L

i

}

≠
∅

\cap \left \{ {L:L\;{\text {in}}\;{\mathcal {L}_i}} \right \} \ne \emptyset

if and only if every

σ

(

k
,
j

)

\sigma \left ( {k,j} \right )

or fewer members of

L

\mathcal {L}

have such a

k
k

-partition. (2) Let

S
S

be compact in

R
d

{R^d}

. The set

S
S

is a union of

k
k

starshaped sets if and only if there is a sequence of compact sets

{

S
i

}

\left \{ {{S_i}} \right \}

converging to

S
S

(relative to the Hausdorff metric) such that each set

S
i

{S_i}

satisfies property

A

(
k
)

A\left ( k \right )

. The first result yields a piercing number for certain families of sets, while the second provides a characterization for compact unions of

k
k

starshaped sets in

R
d

{R^d}

.
Spaces which admit AR-resolutions
Authors: Koyama A. , Mardešić S. , Watanabe T.
Published: June 2010

It is proved that a topological space

X
X

admits an AR-resolution (in the sense of [6]) if and only if

X
X

has trivial (strong) shape.
Endomorphisms of an extremal algebra
Authors: Kamowitz Herbert , Wortman Dennis
Published: June 2010

Let

E
a
[
−
1
,
1
]

Ea[ - 1,1]

denote the extremal algebra on

[
−
1
,
1
]

[ - 1,1]

as defined in Bonsall and Duncan, Numerical ranges. II. We show that every nonzero endomorphism

T
T

on

E
a
[
−
1
,
1
]

Ea[ - 1,1]

has the form

T
f
(
x
)
→
f
(
A
x
+
B
)

Tf(x) \to f(Ax + B)

where

A
A

and

B
B

are real and

|

A

|

+

|

B

|

≤
1

|A| + |B| \leq 1

. Further, the endomorphism

T
T

is an automorphism if, and only if,

B
=
0

B = 0

and

A
=
1

A = 1

or

−
1

-1

, while

T
T

is a nonzero compact endomorphism if, and only if,

T
:
f
(
x
)
→
f
(
B
)

T:f(x) \to f(B)

for some

B
B

in

[
−
1
,
1
]

[ - 1,1]

. Also included in this note are several results related to compact endomorphisms of regular commutative semisimple Banach algebras.
Sets of disjoint elements in free products of lattice-ordered groups
Authors: Powell Wayne B. , Tsinakis Constantine
Published: June 2010

We show that for every infinite cardinal number

m
m

there exist two totally ordered abelian groups whose free product in any nontrivial variety of lattice-ordered groups has a disjoint set of cardinality

m
m

. This answers problem 10.7 of [13] and extends the results in [12]. We further prove that for the variety of abelian lattice-ordered groups or the variety of all lattice-ordered groups, the free product of two nontrivial members of the variety will always contain an infinite disjoint set.
The 𝐻-deviation of a lift
Authors: Williams Frank
Published: June 2010

Let

X
X

be an

H
H

-space,

x
∈
X

x \in X

a primitive element, and

α
\alpha

a stable primary operation that vanishes on

x
x

. Let

y
y

represent

x
x

in

H
∗

(

P
2

X
)

{H^*}({P_2}X)

. (

P
2

X

{P_2}X

is the projective plane of

X
X

.) Let

D
~

\tilde D

be the

H
H

-deviation of the lift of

x
x

to the two-stage Postnikov system with

k
k

-invariant

α
\alpha

. We obtain a formula that relates

D
~

\tilde D

to the representation of

α
(
y
)

\alpha (y)

as a product in

H
∗

(

P
2

X
)

{H^*}({P_2}X)

.
On the modulus of weakly compact operators and strongly additive vector measures
Authors: Schmidt Klaus D.
Published: June 2010

Aliprantis and Burkinshaw proved that each weakly compact operator from an AL-space into a KB-space has a weakly compact modulus. In the present paper it is shown that this is also true for weakly compact operators from a Banach lattice having an order continuous dual norm into an order complete AM-space with unit. A corresponding result is obtained for strongly additive vector measures.
Weighted capacity and the Choquet integral
Authors: Adams David R.
Published: June 2010

The capacity set function that is naturally associated with a linear second-order elliptic partial differential operator in divergence form is related to the concept of the Choquet integral of a weight function with respect to Newtonian capacity. The weight function comes from the coefficients of the differential operator. This idea is reminiscent of the Radon-Nikodym Theorem, but now for capacities instead of measures.
A remark on the global embedding problem for three-dimensional CR manifolds
Authors: Barrett David E.
Published: June 2010

Examples are constructed of compact strictly pseudoconvex real analytic three-dimensional CR manifolds which admit no nonconstant CR functions. Also, the embedding problem for certain CR manifolds is shown to be related to a result of Hadamard on immersed convex hypersurfaces in

R

n

{{\mathbf {R}}^n}

.
Short proofs of three theorems on harmonic functions
Authors: Boas H. P. , Boas R. P.
Published: June 2010

We present elementary proofs—shorter than any others that we know—for three related theorems.
On uniformly amenable groups
Authors: Wysoczański Janusz
Published: June 2010

We consider the conditions of the uniform amenability of discrete groups. The equivalence of the uniform Følner and Kesten conditions is proved. Also an example of the group which is amenable but not uniformly amenable is given.
Pure state extensions of the trace on the Choi algebra
Authors: Lazar Aldo , Tsui Sze-Kai , Wright Stephen
Published: June 2010

Pure state extensions of a nonhyperfinite

I

I

1

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

-factor state on the Choi algebra to the Cuntz algebra

O
2

{O_2}

is constructed.
A short proof of the equivalence of KMP and RNP in Banach lattices and preduals of von Neumann algebras
Authors: Caselles V.
Published: June 2010

In this note we give a unified approach to the equivalence between the Krein-Milman property and the Radon-Nikodym property in Banach lattices and preduals of von Neumann algebras.
On quasiminimal e-degrees and total e-degrees
Authors: Sorbi Andrea
Published: June 2010

We show that there exists a set

A
A

such that the

e
e

-degree of

A
A

is quasi-minimal and the

e
e

-degree of the complement of

A
A

is total. This provides also a counterexample to a conjecture in [1].
Bifurcation to badly ordered orbits in one-parameter families of circle maps, or angels fallen from the devil’s staircase
Authors: Hockett Kevin , Holmes Philip
Published: June 2010

We discuss the structure of the bifurcation set of a one-parameter family of endomorphisms of

S
1

{S^1}

having two critical points and negative Schwarzian derivative. We concentrate on the case in which one of the endpoints of the rotation set is rational, providing a partial characterization of components of the nonwandering set having specified rotation number and the bifurcations in which they are created. In particular we find, for each rational rotation number

p
′

/

q
′

p’/q’

less than the upper boundary of the rotation set

p

/

q

p/q

, infinitely many saddle-node bifurcations to badly ordered periodic orbits of rotation number

p
′

/

q
′

p’/q’

.
A generalized converse measurability theorem
Authors: Chang K. S. , Ryu K. S.
Published: June 2010

It has long been known that measurability questions in Wiener space and Yeh-Wiener space are often rather delicate. Some converse measurability theorems in Wiener and Yeh-Wiener spaces were proved by Köehler, Skoug, and the first author. In this paper, we establish a generalized converse measurability theorem by which the above measurability theorems are proved as corollaries.
Cyclic homology and the Beĭlinson-Manin-Schechtman central extension
Authors: Getzler Ezra
Published: June 2010

We construct central extensions of the Lie algebra of differential operators on a one-dimensional affine variety over a field of characteristic zero, generalizing the Virasoro extension. The construction is an application of recent calculations of the Hochschild and cyclic homology of algebras of differential operators.
Alternating procedures in uniformly smooth Banach spaces
Authors: Assani I.
Published: June 2010

Let

E
E

be a uniformly smooth Banach space and

C
C

the set of real continuous strictly increasing functions

μ
\mu

on

R

+

{{\mathbf {R}}_ + }

such that

μ
(
0
)
=
0

\mu (0) = 0

. At each

μ
\mu

we can associate a unique duality map

J
μ

:
E
→

E
∗

{J_\mu }:E \to {E^ * }

such that

(

J
μ

x
,
x
)
=

‖

J
μ

x

‖

⋅

‖
x
‖

({J_\mu }x,x) = \left \| {{J_\mu }x} \right \| \cdot \left \| x \right \|

and

‖

J
μ

x

‖

=
μ

(

‖
x
‖

)

\left \| {{J_\mu }x} \right \| = \mu \left ( {\left \| x \right \|} \right )

. We prove in this note that if

T
n

{T_n}

is a sequence of linear contractions on

E
E

the sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T 1 Superscript asterisk Baseline upper T 2 Superscript asterisk Baseline midline-horizontal-ellipsis upper T Subscript n Superscript asterisk Baseline upper J Subscript mu Baseline upper T Subscript n Baseline midline-horizontal-ellipsis upper T 2 upper T 1 x">

T
1
∗

T
2
∗

⋯

T
n
∗

J
μ

T
n

⋯

T
2

T
1

x

T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_2}{T_1}x

converges strongly in

E
∗

{E^ * }

norm for all

x
x

in

E
E

. In particular if

E
∗

{E^ * }

is also uniformly smooth then for any

μ
\mu

and

ν
\nu

in

C
C

the sequence
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript nu Superscript asterisk Baseline upper T 1 Superscript asterisk Baseline upper T 2 Superscript asterisk Baseline midline-horizontal-ellipsis upper T Subscript n Superscript asterisk Baseline upper J Subscript mu Baseline upper T Subscript n Baseline midline-horizontal-ellipsis upper T 1 x">

J
ν
∗

T
1
∗

T
2
∗

⋯

T
n
∗

J
μ

T
n

⋯

T
1

x

J_\nu ^ * T_1^ * T_2^ * \cdots T_n^ * {J_\mu }{T_n} \cdots {T_1}x

converges in

E
E

norm. This generalizes a result of M. Akcoglu and L. Sucheston [1].
Plane curves whose singular points are cusps
Authors: Yoshihara Hisao
Published: June 2010

Let

C
C

be an irreducible curve of degree

d
d

in the complex projective plane. We assume that each singular point is a one place point with multiplicity 2 or 3. Let

σ
\sigma

be the sum of "the Milnor numbers" of the singularities. Then we shall show that

7
σ
>
6

d
2

−
9
d

7\sigma > 6{d^2} - 9d

. This gives a necessary condition for the existence of such a curve, for example, if

C
C

is rational, then

d
≤
10

d \leq 10

.
Construction of Hamilton sequences for certain Teichmüller mappings
Authors: Reich Edgar
Published: June 2010

Suppose
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace vertical-bar double-integral Underscript StartAbsoluteValue z EndAbsoluteValue greater-than 1 Endscripts kappa left-parenthesis z right-parenthesis phi left-parenthesis z right-parenthesis d x d y vertical-bar colon phi left-parenthesis z right-parenthesis">

sup
{

|

∬

|
z
|

>
1

κ
(
z
)
φ
(
z
)
d
x
d
y

|

:
φ
(
z
)

\sup \{ |\iint _{\left | z \right | > 1} {\kappa (z)\varphi (z)dxdy|:\varphi (z)}

analytic for
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue z EndAbsoluteValue greater-than 1 comma double-integral Underscript StartAbsoluteValue z EndAbsoluteValue greater-than 1 Endscripts StartAbsoluteValue phi left-parenthesis z right-parenthesis EndAbsoluteValue d x d y equals 1 right-brace equals ess sup StartSet StartAbsoluteValue kappa left-parenthesis z right-parenthesis EndAbsoluteValue colon StartAbsoluteValue z EndAbsoluteValue greater-than 1 EndSet">

|
z
|

>
1
,

∬

|
z
|

>
1

|

φ
(
z
)

|

d
x
d
y
=
1
}
=

ess

sup

{

|

κ
(
z
)

|

:

|
z
|

>
1
}

\left | z \right | > 1,\iint _{\left | z \right | > 1} {|\varphi (z)|dxdy = 1\} = {\text {ess}}\sup }\{ |\kappa (z)|:\left | z \right | > 1\}

. The question of constructive determination of extremal sequences

{

φ
n

}

\{ \varphi _n \}

is considered for some classes of functions

κ
(
z
)

\kappa (z)

that arise in connection with plane quasiconformal mappings. For example, such a sequence

{

φ
n

}

\{ \varphi _n \}

is constructed explicitly for the

κ
(
z
)

\kappa (z)

that arises in connection with the affine stretch of Strebel’s chimney domain.
Smoothness of the billiard ball map for strictly convex domains near the boundary
Authors: Kovachev Valery
Published: June 2010

The billiard ball map in bounded strictly convex domains in

R

n

{{\mathbf {R}}^n}

with boundaries of class

C
k

,
k
≥
2

{C^k},k \geq 2

, is considered and its smoothness of class

C

k
−
1

{C^{k - 1}}

up to the boundary is proved.
A nonorientable complete minimal surface in 𝑅³ between two parallel planes
Authors: López Francisco J.
Published: June 2010

In this paper, we show how to produce nonorientable, regular complete, minimal surfaces in

R
3

{R^3}

between two parallel planes.
Fixed point theorems for compact acyclic metric spaces
Authors: Park Sehie
Published: June 2010

Multifunctions

F
:
X
→

2
M

F:X \to {2^M}

are studied where

X
⊂
M

X \subset M

is either a compact acyclic finite-dimensional ANR or a compact acyclic lc subspace. Conditions are found for the existence of

v
v

such that

d
(
v
,
F
v
)
=
d
(
X
,
F
v
)

d(v,Fv) = d(X,Fv)

and, also, for the existence of fixed points.
Compactly generated subgroups and open subgroups of locally compact groups
Authors: Bagley R. W. , Wu T. S. , Yang J. S.
Published: June 2010

This paper contains results of the following sort: If

G
G

is a locally compact group and

H
H

is a closed subgroup such that the coset space

G

/

H

G/H

is locally connected, then

H

G
0

H{G_0}

is open in

G
G

. If

G
G

is a locally compact group such that

G

/

G
0

G/{G_0}

is compact, then every closed subgroup of

G
G

is compactly generated if and only if

G
0

{G_0}

has no noncompact simple factor.
Nontangential maximal functions over compact Riemannian manifolds
Authors: Blank B. E.
Published: June 2010

The nontangential maximal function associated to the Poisson semigroup for a compact Riemannian manifold is shown to be weak type (1,1) and

L
p

{L^p}

bounded

(
p
>
1
)

(p > 1)

.
An irreducible not admissible Banach representation of 𝑆𝐿(2,𝑅)
Authors: Soergel Wolfgang
Published: June 2010

This weird representation can be constructed by inducing from a solution of the invariant subspace problem.
The number of generators of modules over polynomial rings
Authors: Lyubeznik Gennady
Published: June 2010

Let

k
k

be an infinite field and

B
=
k
[

X
1

,
…
,

X
n

]

B = k[{X_1}, \ldots ,{X_n}]

a polynomial ring over

k
k

. Let

M
M

be a finitely generated module over

B
B

. For every prime ideal

P
⊂
B

P \subset B

let

μ
(

M
P

)

\mu ({M_P})

be the minimum number of generators of

M
P

{M_P}

, i.e.,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu left-parenthesis upper M Subscript upper P Baseline right-parenthesis equals dimension upper B Subscript upper P Baseline slash upper P Subscript upper P Baseline left-parenthesis upper M Subscript upper P Baseline circled-times Subscript upper B Sub Subscript upper P Subscript Baseline left-parenthesis upper B Subscript upper P Baseline slash upper P Subscript upper P Baseline right-parenthesis right-parenthesis">

μ
(

M
P

)
=
dim
⁡

B
P

/

P
P

(

M
P

⊗

B
P

(

B
P

/

P
P

)
)

\mu ({M_P}) = \dim {B_P}/{P_P}({M_P}{ \otimes _{{B_P}}}({B_P}/{P_P}))

. Set
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta left-parenthesis upper M right-parenthesis equals max left-brace mu left-parenthesis upper M Subscript upper P Baseline right-parenthesis plus dimension left-parenthesis upper B slash upper P right-parenthesis vertical-bar upper P element-of upper S p e c upper B such that upper M Subscript upper P Baseline is not free right-brace">

η
(
M
)
=
max
{
μ
(

M
P

)
+
dim
⁡
(
B

/

P
)

|

P
∈
Spec

B

such

that

M

P

is

not

free

}

\eta (M) = \max \{ \mu ({M_P}) + \dim (B/P)\left | {P \in \operatorname {Spec} } \right .B\;{\text {such}}\;{\text {that}}\;{M_{P\;}}{\text {is}}\;{\text {not}}\;{\text {free}}\}

. Then

M
M

can be generated by

η
(
M
)

\eta (M)

elements. This improves earlier results of A. Sathaye and N. Mohan Kumar on a conjecture of Eisenbud-Evans.
Cesàro and Borel-type summability
Authors: Borwein David , Markovich Tom
Published: June 2010

Though summability of a series by the Cesàro method

C
p

{C_p}

does not in general imply its summability by the Borel-type method

(
B
,
α
,
β
)

(B,\alpha ,\beta )

, it is shown that the implication holds under an additional condition.
A note on the generalized Riemann integral
Authors: Pfeffer Washek F.
Published: June 2010

We show that the generalized Riemann integral can be defined by means of gage functions which are upper semicontinuous when restricted to a suitable subset whose complement has measure zero.
An extension theorem for normal functions
Authors: Järvi Pentti
Published: June 2010

Given a domain

Ω
⊂

C

n

\Omega \subset {{\mathbf {C}}^n}

, an analytic subvariety

V
V

of

Ω
\Omega

and a normal function

f
:
Ω
∖
V
→

C

^

f:\Omega \backslash V \to \widehat {\mathbf {C}}

, we show that

f
f

can be extended to a holomorphic mapping

f
∗

:
Ω
→

C

^

{f^*}:\Omega \to \widehat {\mathbf {C}}

provided the singularities of

V
V

are normal crossings.
Intersection of Lebesgue spaces 𝐿₁ and 𝐿₂
Authors: Dilworth S. J.
Published: June 2010

The linear topological properties of the intersection of Lebesgue spaces

L
1

{L_1}

and

L
2

{L_2}

are investigated.
Approximating PA
Authors: Wiid F. G. J.
Published: June 2010

It is shown that the ring of continuous real-valued functions on a compact Hausdorff space of dimension

≤
1

\leq 1

is a BCS-ring. This result follows from a study of the preservation of the PA and BCS properties under certain ring-theoretic constructions.
Automatic continuity of concave functions
Authors: Howe Roger
Published: June 2010

A necessary and sufficient condition is given that a semicontinuous, nonnegative, concave function on a finite dimensional closed convex set

X
X

necessarily be continuous at a point

x
0

∈
X

{x_0} \in X

. Application of this criterion at all points of

X
X

yields a characterization, due to Gale, Klee and Rockafellar, of convex polyhedra in terms of continuity of their convex functions.
Compact parallelizable four-dimensional symplectic and complex manifolds
Authors: Fernández Marisa , Gotay Mark J. , Gray Alfred
Published: June 2010

Examples of compact symplectic manifolds with no complex and/or Kähler structures are presented.
Clear visibility and 𝐿₂ sets
Authors: Stavrakas N.
Published: June 2010

Let

S
⊂

R
2

S \subset {R^2}

be a closed connected set whose points of local nonconvexity are compact. Suppose any two points of local nonconvexity are clearly visible from a common point of

S
S

. Then

S
S

is almost starshaped and

S
S

is

2
2

-polygonally connected. This generalizes a result of Breen.