bluuebunny/crossref_metadata_2025 huggingface dataset:
Fay’s trisecant formula and cross-ratios
Authors: Poor Cris
Published: June 2010

This note considers Fay’s trisecant formula as a relation between cross-ratio functions and thereby gives a simple proof of the trisecant formula. In this proof the expression for the cross-ratio function is lifted from the theta locus to the entire Jacobian. Variations of the formula as used by different authors are also given.
The Fekete-Szegő problem for strongly close-to-convex functions
Authors: Abdel-Gawad H. R. , Thomas D. K.
Published: June 2010

Let

K
(
β
)

K(\beta )

denote the class of normalized analytic strongly close-to-convex functions of order

β
≥
0

\beta \geq 0

, defined in the unit disc

D
D

and let

f
∈
K
(
β
)

f \in K(\beta )

, with

f
(
z
)
=
z
+

a
2

z
2

+

a
3

z
3

+
⋯

f(z) = z + {a_2}{z^2} + {a_3}{z^3} + \cdots

, for

z
∈
D

z \in D

. Sharp bounds are obtained for

|

a
3

−
μ

a
2
2

|

|{a_3} - \mu a_2^2|

when

μ
\mu

is real.
Extremal compressions of closed operators
Authors: Förster K.-H. , Jahn K.
Published: June 2010

Let

X
X

be a Banach space,

A
A

a closed linear operator on

X
X

, and

λ
1

,
…
,

λ
n

{\lambda _1}, \ldots ,{\lambda _n}

isolated eigenvalues of

A
A

of finite multiplicity. If

P
P

is a projection on

X
X

such that

λ
1

,
…
,

λ
n

{\lambda _1}, \ldots ,{\lambda _n}

belong to the resolvent of the compression of

A
A

on the range of

P
P

it is easy to see that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension upper N left-parenthesis upper P right-parenthesis greater-than-or-equal-to max left-brace right-brace colon of dimdim times times upper N left-parenthesis right-parenthesis minus minus lamda iA colon less-than-or-equal-to less-than-or-equal-to less-than-or-equal-to 1 in period">

dim
⁡
N

(
P
)

≥
max

{

dim
⁡
N

(

λ
i

−
A

)

:
1
≤
i
≤
n

}

.

\dim N\left ( P \right ) \geq \max \left \{ {\dim N\left ( {{\lambda _i} - A} \right ):1 \leq i \leq n} \right \}.

\]
It is shown that there exist such projections where we have equality in this inequality.
A family of type 𝐼𝐼𝐼 extensions of the trace on the Choi algebra
Authors: Stacey P. J.
Published: June 2010

An infinite family of nonequivalent type III factor states of

0

2

0_{2}

is constructed, each one of which extends the trace on the Choi algebra.
Convergence and integrability of trigonometric series with coefficients of bounded variation of order (𝑚,𝑝)
Authors: Stanojevic Vera B.
Published: June 2010

Let

{
c
(
n
)
}

\{ c(n)\}

be a complex null sequence such that for some integer

m
≥
1

m \geq 1

and some

p
∈
(
1
,
2
]

p \in (1,2]

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript StartAbsoluteValue n EndAbsoluteValue greater-than normal infinity Endscripts StartAbsoluteValue normal upper Delta Superscript m Baseline c left-parenthesis n right-parenthesis EndAbsoluteValue Superscript p Baseline greater-than normal infinity and sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartAbsoluteValue normal upper Delta left-parenthesis c left-parenthesis n right-parenthesis minus c left-parenthesis negative n right-parenthesis right-parenthesis EndAbsoluteValue log base 10 n greater-than normal infinity period">

∑

|

n

|

>
∞

|

Δ
m

c
(
n
)

|

p

>
∞

and

∑

n
=
1

∞

|

Δ
(
c
(
n
)
−
c
(
−
n
)
)

|

lg
⁡
n
>
∞
.

\sum \limits _{|n| > \infty } {|{\Delta ^m}c(n){|^p} > \infty \quad {\text {and}}\quad \sum \limits _{n = 1}^\infty {|\Delta (c(n) - c( - n))|\lg n > \infty .} }

\]
It is shown that the series
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis sigma-summation Underscript StartAbsoluteValue n EndAbsoluteValue greater-than normal infinity Endscripts c left-parenthesis n right-parenthesis e Superscript i n t Baseline comma t element-of upper T equals StartFraction double-struck upper R Over 2 pi double-struck upper Z EndFraction">

(
∗
)

∑

|

n

|

>
∞

c
(
n
)

e

int

,

t
∈
T
=

R

2
π

Z

( * )\quad \sum \limits _{|n| > \infty } {c(n)} {e^{\operatorname {int} }},\quad t \in T = \frac {\mathbb {R}}{{2\pi \mathbb {Z}}}

\]
converges a.e. and that the well-known condition

C
w

{C_w}

of J. W. Garrett and C. V. Stanojevic [4, 3] implies that the series (*) is the Fourier series of its sum. This generalizes results of W. O. Bray and C. V. Stanojevic [1]. An important consequence of the main result is that

n
Δ
c
(
n
)
=
0
(
1
)
,

|

n

|

→
∞

n\Delta c(n) = 0(1),\quad |n| \to \infty

, implies that the condition

C
w

{C_w}

is equivalent to the de la Vallee Poussin summability of partial sums

{

S
n

(
c
)
}

\{ {S_n}(c)\}

as conjectured in [8].
On 𝑝-radical blocks of finite groups
Authors: Hida Akihiko
Published: June 2010

We give a sufficient condition on a

p
p

-block of a finite group under which the block is

p
p

-radical.
Orthosymmetric ortholattices
Authors: Mayet R.
Published: June 2010

Orthosymmetric ortholattices, which form an equational class of algebras, approximate ortholattices of closed subspaces of Hilbert spaces, and more generally projection lattices of von Neumann algebras, more closely than orthomodular lattices
Regular PI metric flows are equicontinuous
Authors: Glasner Eli
Published: June 2010

Let

(
X
,
T
)

(X,T)

be a metrizable minimal flow. We show that a homomorphism

X

→

π

Y

X \stackrel {\pi }{\to } Y

, which is regular, and PI can be decomposed as

X

→

σ

Z

→

ρ

Y

X \stackrel {\sigma }{\to } Z \stackrel {\rho }{\to } Y

,

π
=
ρ
∘
σ

\pi = \rho \circ \sigma

, where

ρ
\rho

is proximal and

σ
\sigma

is a compact group extension. In particular, assuming further that

T
T

is abelian and taking

Y
Y

to be the trivial one point flow, we find that a metric regular PI flow is a compact group rotation.
The field of definition of a real representation of a quiver 𝑄
Authors: Schofield Aidan
Published: June 2010

In answer to a question of Kac, we show that if

α
\alpha

is a real root for the quiver

Q
Q

, then there is a unique indecomposable of dimension vector

α
\alpha

defined over the rational numbers.
Compositions of continuous functions and connected functions
Authors: Kellum Kenneth R. , Rosen Harvey
Published: June 2010

Suppose

f
:
X
→
Y

f:X \to Y

is continuous and onto and

g
:
Y
→
Z

g:Y \to Z

is such that

g
∘
f
:
X
→
Z

g \circ f:X \to Z

has a property we are interested in. For which properties of functions can we infer that

g
g

has the same property? Properties for which we can infer this include continuity and the Darboux property. Properties for which we cannot include almost continuity.
A study of almost-everywhere singleton-valued Filippovs
Authors: Biles Daniel C. , Spraker John S.
Published: June 2010

Filippov solutions of

x
′

(
t
)
=
f
(
t
,
x
(
t
)
)

x’(t) = f(t,x(t))

are defined in terms of a certain set-valued function called the Filippov of

f
f

. Under some fairly general assumptions, we answer the following question: The Filippovs of

f
f

and

g
g

are equal if and only if

f
f

and

g
g

relate in what way?
Orbits and characters associated to highest weight representations
Authors: Collingwood David H.
Published: June 2010

We relate two different orbit decompositions of the flag variety. This allows us to pass from the closed formulas of Boe, Enright, and Shelton for the formal character of an irreducible highest weight representation to closed formulas for the distributional character written as a sum of characters of generalized principal series representations. Otherwise put, we give a dictionary between certain Lusztig-Vogan polynomials arising in Harish-Chandra module theory and the Kazhdan-Lusztig polynomials associated to a relative category

O

\mathcal {O}

of Hermitian symmetric type.
A proof of the Popov conjecture for tori
Authors: Wehlau David L.
Published: June 2010

We prove a lemma which reduces much of the invariant theory of torus representations to the theory of faithful stable torus representations (Lemma 2). Using this reduction we obtain a structure theorem (Theorem 1) for equidimensional representations of tori. This theorem shows that the weights of an equidimensional torus representation are arranged in a very special manner within the lattice of characters. Understanding this arrangement allows us to prove that equidimensional representations of tori must be cofree (the Popov conjecture for tori).
The Hilbert transform with exponential weights
Authors: Colzani Leonardo , Vignati Marco
Published: June 2010

We study the operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper H f left-parenthesis x right-parenthesis equals 2 Superscript negative x Baseline integral Subscript 0 Superscript plus normal infinity Baseline StartFraction 2 Superscript y Baseline f left-parenthesis y right-parenthesis Over x minus y EndFraction d y">

H

f
(
x
)
=

2

−
x

∫
0

+
∞

2
y

f
(
y
)

x
−
y

d
y

\mathcal {H}f(x) = {2^{ - x}}\int _0^{ + \infty } {\frac {{{2^y}f(y)}}{{x - y}}dy}

\]
on Lorentz spaces on

R

+

{\mathbb {R}_ + }

with respect to the measure

4
x

d
x

{4^x}dx

. This is related to the harmonic analysis of radial functions on hyperbolic spaces. We prove that this operator is bounded on the Lorentz spaces

L

2
,
9

(

R

+

,

4
x

d
x
)
,
1
>
q
>
+
∞

{L^{2,9}}({\mathbb {R}_ + },{4^x}dx),1 > q > + \infty

, and it maps the Lorentz space

L

2
,
1

(

R

+

,

4
x

d
x
)

{L^{2,1}}({\mathbb {R}_ + },{4^x}dx)

into a space that we call WEAK-

L

2
,
1

(

R

+

,

4
x

d
x
)

{L^{2,1}}({\mathbb {R}_ + },{4^x}dx)

. We also prove that

H

\mathcal {H}

maps

L
1

(

R

+

,

4
x

d
x
)

{L^1}({\mathbb {R}_ + },{4^x}dx)

into WEAK-
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1 Baseline left-parenthesis double-struck upper R Subscript plus Baseline comma 4 Superscript x Baseline d x right-parenthesis plus upper L squared left-parenthesis double-struck upper R Subscript plus Baseline comma 4 Superscript x Baseline d x right-parenthesis">

L
1

(

R

+

,

4
x

d
x
)
+

L
2

(

R

+

,

4
x

d
x
)

{L^1}({\mathbb {R}_ + },{4^x}dx) + {L^2}({\mathbb {R}_ + },{4^x}dx)

.
On the period-energy relation
Authors: Bates Larry , Śniatycki Jędrzej
Published: June 2010

We give a new proof of the period-energy relation for Hamiltonian systems.
Compact perturbations of certain CSL algebras
Authors: Katsoulis E. G. , Power S. C.
Published: June 2010

In this note we show that several CSL algebras, including
\[

Alg
⁡

L

(

2
∞

,
≤
,

m
p

)
,
0
>
p
>
1
,

\operatorname {Alg}\mathcal {L}({2^\infty }, \leq ,{m_p}),0 > p > 1,

\]
have compact perturbations that are norm closed.
Group completions and orbifolds of variable negative curvature
Authors: Stark Christopher W.
Published: June 2010

W. J. Floyd’s comparison of the Furstenberg maximal boundary of a noncompact,

R

{\mathbf {R}}

-rank one, connected semisimple Lie group

G
G

with finite center and the group completion of a discrete, cocompact subgroup

Γ
\Gamma

of

G
G

is extended to a homeomorphism between the group completion of the fundamental group

Γ
\Gamma

of a closed Riemannian orbifold

M
=
Γ
∖
X

M = \Gamma \backslash X

of strictly negative sectional curvatures and the sphere at infinity in the Eberlein-O’Neill compactification

X
¯

\overline X

of the universal cover

X
X

of

M
M

.
Compact measures have Loeb preimages
Authors: Ross David
Published: June 2010

A compact measure is a (possibly nontopological) measure that is inner-regular with respect to a compact family of measurable sets. The main result of this paper is that every compact probability measure is the image, under a measure-preserving transformation, of a Loeb probability space. This generalizes a well-known result about Radon topological probability measures. It is also proved that a compact probability space can be topologized in such a way that the measure is essentially Radon.
On the abstract Cauchy problem in Fréchet spaces
Authors: Henríquez Hernán R. , Hernández Eduardo A.
Published: June 2010

Let

A
A

be a linear operator in a Fréchet space

X
X

with the resolvent defined for all

λ
>
0

\lambda > 0

. In this note we prove that both the first and the second order abstract Cauchy problems associated to

A
A

are well posed on certain maximal subspaces of

X
X

. Our results extend those of Kantorovitz [5] and Cioranescu [2].
A sharp estimate in an operator inequality
Authors: McEachin R.
Published: June 2010

Let

H

\mathcal {H}

and

K

\mathcal {K}

be Hilbert spaces, and suppose

A
∈

B

(

H

)

A \in \mathcal {B}(\mathcal {H})

and

B
∈

B

(

K

)

B \in \mathcal {B}(\mathcal {K})

are selfadjoint operators with

dist
⁡
(
σ
(
A
)
,
σ
(
B
)
)
≥
δ
>
0

\operatorname {dist} (\sigma (A),\sigma (B)) \geq \delta > 0

. It is known that for any

Q
∈

B

(

K

,

H

)

Q \in \mathcal {B}(\mathcal {K},\mathcal {H})

we must have
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction pi Over 2 EndFraction StartAbsoluteValue EndAbsoluteValue upper A upper Q minus upper Q upper B StartAbsoluteValue EndAbsoluteValue greater-than-or-equal-to delta StartAbsoluteValue EndAbsoluteValue upper Q StartAbsoluteValue EndAbsoluteValue">

π
2

|

|

A
Q
−
Q
B

|

|

≥
δ

|

|

Q

|

|

\tfrac {\pi }{2}||AQ - QB|| \geq \delta ||Q||

. In this paper we give examples proving that

π

/

2

\pi /2

is sharp in this inequality.
Subharmonic solutions of conservative systems with nonconvex potentials
Authors: Fonda A. , Lazer A. C.
Published: June 2010

We consider the system of second order differential equations
\[

u
+
∇
G
(
u
)
=
e
(
t
)
≡
e
(
t
+
T
)
,

u + \nabla G(u) = e(t) \equiv e(t + T),

\]
, where the potential

G
:

R

n

→

R

G:{\mathbb {R}^n} \to \mathbb {R}

is not necessarily convex. Using critical point theory, we give conditions under which the system has infinitely many subharmonic solutions.
Illumination for unions of boxes in 𝑅^{𝑑}
Authors: Breen Marilyn
Published: June 2010

Let

S
S

be a finite union of boxes (polytopes whose edges are parallel to the coordinate axes) in

R
d

{R^d}

. If every two vertices of

S
S

are clearly illumined by some common translate of the box

T
T

, then there is a translate of

T
T

that clearly illumines every point of

S
S

. A similar result holds when appropriate boundary points of

S
S

are illumined (rather than clearly illumined) by translates of box

T
T

.
A note on the largest regular subalgebra of a Banach algebra
Authors: Inoue Jyunji , Takahasi Sin-ei
Published: June 2010

A simple proof of Albrecht’s result on the existence of the largest closed regular subalgebra of a semisimple commutative Banach algebra is given.
An arc of finite 2-measure that is not rationally convex
Authors: Bagby Thomas , Gauthier P. M.
Published: June 2010

A subset of

C

2

{{\mathbf {C}}^2}

having the properties in the title is constructed.
Uniform limits of sequences of polynomials and their derivatives
Authors: Ball Joseph A. , Fanney Thomas R.
Published: June 2010

Let

E
E

be a compact subset of the unit interval

[
0
,
1
]

[0,1]

, and let

C
(
E
)

C(E)

denote the space of functions continuous on

E
E

with the uniform norm. Consider the densely defined operator

D
:
C
(
E
)
→
C
(
E
)

D:C(E) \to C(E)

given by

D
p
=

p
′

Dp = p’

for all polynomials

p
p

. Let

G
G

represent the graph of

D
D

, that is

G
=
{
(
p
,

p
′

)
:
p

G = \{ (p,p’):p

polynomials} considered as a submanifold of

C
(
E
)
×
C
(
E
)

C(E) \times C(E)

. Write the interior of the set

E
,

int
⁡
E

E,\;\operatorname {int} E

as a countable union of disjoint open intervals and let

E
^

\widehat {E}

be the union of the closure of these intervals. The main result is that the closure of

G
G

is equal to the set of all functions

(
h
,
k
)
∈
C
(
E
)
×
C
(
E
)

(h,k) \in C(E) \times C(E)

such that

h
h

is absolutely continuous on

E
^

\widehat {E}

and

k

|

E
^

=

h
′

|

E
^

k|\widehat {E} = h’|\widehat {E}

. As a consequence, the operator

D
D

is closable if and only if the set

E
E

is the closure of its interior. On the other extreme,

G
G

is dense in

C
(
E
)
×
C
(
E
)

C(E) \times C(E)

i.e. for any pair

(
f
,
g
)
∈
C
(
E
)
×
C
(
E
)

(f,g) \in C(E) \times C(E)

, there exists a sequence of polynomials

{

p
n

}

\{ {p_n}\}

so that

p
n

→
f

{p_n} \to f

and

p
n
′

→
g

{p’_n} \to g

uniformly on

E
E

, if and only if the interior

int
⁡
E

\operatorname {int} E

of

E
E

is empty.
The Dunford-Pettis property in the predual of a von Neumann algebra
Authors: Bunce L. J.
Published: June 2010

The von-Neumann algebras whose predual has the Dunford-Pettis property are characterised as being Type I finite. This answers the question asked by Chu and Iochum in The Dunford Pettis property in

C
∗

{C^*}

-algebras, Studia Math. 97 (1990), 59-64.
A Stone-Weierstrass theorem without closure under suprema
Authors: McAfee R. Preston , Reny Philip J.
Published: June 2010

For a compact metric space

X
X

, consider a linear subspace

A
A

of

C

(
X
)

C\left ( X \right )

containing the constant functions. One version of the Stone-Weierstrass Theorem states that, if

A
A

separates points, then the closure of

A
A

under both minima and maxima is dense in

C

(
X
)

C\left ( X \right )

. By the Hahn-Banach Theorem, if

A
A

separates probability measures,

A
A

is dense in

C

(
X
)

C\left ( X \right )

. It is shown that if

A
A

separates points from probability measures, then the closure of

A
A

under minima is dense in

C

(
X
)

C\left ( X \right )

. This theorem has applications in economic theory.
Continuity of multidimensional Brownian local times
Authors: Sheu Shey Shiung
Published: June 2010

The local time of a multidimensional semimartingle at a hypersurface will be defined via Tanaka’s formula. One can define a certain distance between hypersurfaces so that the continuity properties of local time can be discussed when the underlying process is Brownian motion.
Le calcul fonctionnel dans l’espace de Besov critique
Authors: Bourdaud G.
Published: June 2010

Si

F
F

une fonction de la variable complexe qui opère, par composition à gauche, sur l’espace de Besov

B
p

s
,
q

(

R

n

)

B_p^{s,q}({\mathbb {R}^n})

ou sur l’espace de Triebel-Lizorkin

F
p

s
,
q

(

R

n

)

F_p^{s,q}({\mathbb {R}^n})

, où

0
>
s
>
1
,
s
=
n

/

p
,
1
≤
q
≤
+
∞
(
q
>
1

0 > s > 1,s = n/p,1 \leq q \leq + \infty (q > 1

, dans le cas de l’espace de Besov), alors

F
F

est globalement lipschitzienne. Ce résultat achève la description du calcul fonctionnel dans

B
p

s
,
q

(

R

n

)

B_p^{s,q}({\mathbb {R}^n})

et dans

F
p

s
,
q

(

R

n

)

F_p^{s,q}({\mathbb {R}^n})

, pour

0
>
s
>
1

0 > s > 1

. Let

F
F

be a complex variable function which acts, via left composition, on the Besov space

B
p

s
,
q

(

R

n

)

B_p^{s,q}({\mathbb {R}^n})

or the Triebel-Lizorkin space

F
p

s
,
q

(

R

n

)

F_p^{s,q}({\mathbb {R}^n})

, where

0
>
s
>
1
,
s
=
n

/

p
,
1
≤
q
≤
+
∞
(
q
>
1

0 > s > 1,s = n/p,1 \leq q \leq + \infty (q > 1

, in the Besov case); then

F
F

is globally lipschitz. This theorem—added to previous results on the noncritical case—provides a complete characterization of the functional calculus on

B
p

s
,
q

(

R

n

)

B_p^{s,q}({\mathbb {R}^n})

and

F
p

s
,
q

(

R

n

)

F_p^{s,q}({\mathbb {R}^n})

, for

0
>
s
>
1

0 > s > 1

.
A remark on the abstract Cauchy problem on spaces of Hölder continuous functions
Authors: Hieber Matthias , Räbiger Frank
Published: June 2010

We prove that the generator of a

C
0

{C_0}

-semigroup on

C
α

(

R

n

)

{C^\alpha }({\mathbb {R}^n})

is a bounded operator. Nevertheless, certain elliptic differential operators generate

β
\beta

-times integrated semigroups on

C
α

(

R

n

)

{C^\alpha }({\mathbb {R}^n})

whenever

β
>
n

/

2
+
1

\beta > n/2 + 1

.
𝐾₀ of certain subdiagonal subalgebras of von Neumann algebras
Authors: Baker Richard
Published: June 2010

We show that

K
0

{K_0}

of any finite maximal subdiagonal subalgebra of a separably acting finite von Neumann algebra is isomorphic to

K
0

{K_0}

of the diagonal of the subalgebra. It results that

K
0

{K_0}

of any finite,

σ
\sigma

-weakly closed, maximal triangular subalgebra of a separably acting finite von Neumann algebra is isomorphic to

K
0

{K_0}

of the diagonal of the subalgebra, provided that the diagonal of the subalgebra is a Cartan subalgebra of the von Neumann algebra. In addition, given any separably acting type

II
1

{\text {II}_1}

factor

M

\mathcal {M}

, we explicitly compute

K
0

{K_0}

of those triangular subalgebras

T

\mathcal {T}

of

M

\mathcal {M}

that have the property that there exists a UHF subalgebra

A

\mathcal {A}

of

M

\mathcal {M}

and a standard triangular UHF algebra

S

\mathcal {S}

in

A

\mathcal {A}

such that

A

\mathcal {A}

is

σ
\sigma

-weakly dense in

M

\mathcal {M}

and

T

\mathcal {T}

is the

σ
\sigma

-weak closure of

S

\mathcal {S}

.
Complex cycles on real algebraic models of a smooth manifold
Authors: Bochnak J. , Kucharz W.
Published: June 2010

Let

M
M

be a compact connected orientable

C
∞

{C^\infty }

submanifold of

R

n

{\mathbb {R}^n}

with

2
dim
⁡
M
+
1
≤
n

2\dim M + 1 \leq n

. Let

G
G

be a subgroup of

H
2

(
M
,

Z

)

{H^2}(M,\mathbb {Z})

such that the quotient group

H
2

(
M
,

Z

)

{H^2}(M,\mathbb {Z})

has no torsion. Then

M
M

can be approximated in

R

n

{\mathbb {R}^n}

by a nonsingular algebraic subset

X
X

such that

H

C

-
a
l
g

2

(
X
,

Z

)

H_{\mathbb {C} \operatorname {- alg}}^{2}(X,\mathbb {Z})

is isomorphic to

G
G

. Here

H

C

-
a
l
g

2

(
X
,

Z

)

H_{\mathbb {C}\operatorname { - alg}}^2(X,\mathbb {Z})

denotes the subgroup of

H
2

(
X
,

Z

)

{H^2}(X,\mathbb {Z})

generated by the cohomology classes determined by the complex algebraic hypersurfaces in a complexification of

X
X

.
Interpolation between weighted Hardy spaces
Authors: Cwikel Michael , McCarthy John E. , Wolff Thomas H.
Published: June 2010

We prove that

H
p

(

w
0

1
−
s

w
1
s

)

{H^p}(w_0^{1 - s}w_1^s)

is an interpolation space of exponent

s
s

between

H
p

(

w
0

)

{H^p}({w_0})

and

H
p

(

w
1

)

{H^p}({w_1})

if and only if

log
⁡
(

w
1

/

w
0

)

\log ({w_1}/{w_0})

is in BMO. If

log
⁡
(

w
1

/

w
0

)

\log ({w_1}/{w_0})

fails to be in BMO,

H
p

(

w
0

1
−
s

w
1
s

)

{H^p}(w_0^{1 - s}w_1^s)

can still be an interpolation space, provided the range of

w
w

has sufficiently large gaps.
Regular matrices and 𝑃-sets in 𝛽𝑁\𝑁. II
Authors: Atalla Robert E.
Published: June 2010

It was discovered by Henriksen and Isbell that the support in

β
N
∖
N

\beta N\backslash N

of a regular matrix is a

P

{\text {P}}

-set. We study conditions under which a

P

{\text {P}}

-subset of a matrix support set contains another matrix support set.
Some applications of convolution of operators on Banach spaces
Authors: Dowling Patrick N.
Published: June 2010

Using convolution between functions and cone absolutely summing operators, we obtain characterizations of Banach spaces with the Radon-Nikodym property, the analytic Radon-Nikodym property and Banach spaces not containing a copy of

c
0

{c_0}

.
Graph-theoretic bound on number of A.C.I.M. for random transformation
Authors: Góra Paweł
Published: June 2010

Upper and lower bounds are given for the number of ergodic absolutely continuous invariant measures for a general random map. They are based on a communication graph describing the interaction of deterministic transformations used to define the random one. In particular, the case of piecewise expanding transformations is discussed.
Uniqueness of unbounded Loeb measure using Choquet’s theorem
Authors: Živaljević Boško
Published: June 2010

Uniqueness of the Carathéodory extension of the standard part map of an internal unbounded measure

μ
\mu

defined on an internal algebra

A

\mathcal {A}

of subsets of an internal set

Ω
\Omega

has been proved by Henson using the notion of a countably determined set. Here we show how Choquet’s capacitability theorem can be used in the proof of the same result.
Cylinder functions in the Fresnel class of functions on abstract Wiener spaces
Authors: Chung Dong Myung , Hwang Hong Taek
Published: June 2010

In this paper we consider the class of cylinder functions on abstract Wiener space

B
B

and give necessary and sufficient conditions of cylinder functions on

B
B

to be in the Banach algebra

F

(
B
)

\mathfrak {F}\left ( B \right )

(resp.

F

∗

(
B
)

{\mathfrak {F}^*}\left ( B \right )

) of analytic (resp. sequential) Feynman integrable functions on

B
B

. The results here subsume similar known results obtained by Chang, Johnson, and Skoug in the setting of Hilbert and Wiener spaces.
Defeat of the 𝐹𝑃²𝐹 conjecture: Huckaba’s example
Authors: Faith Carl
Published: June 2010

A commutative ring

R
R

is

F

P
2

F

F{P^2}F

(resp. FPF) provided that all finitely presented (resp. finitely generated) faithful modules generate the category

mod
−
R

\bmod - R

of all

R
R

-modules. A conjecture of the author dating to the middle 1970s states that any

F

P
2

F

F{P^2}F

ring

R
R

has FP-injective classical quotient ring

Q
=

Q

c
l

(
R
)

Q = {Q_{cl}}(R)

. It was shown by the author (Injective quotient rings. II, Lecture Notes in Pure and Appl. Math., vol. 72, Dekker, New York, 1982, pp. 71-105) that FPF rings

R
R

have injective

Q
Q

and by the author and P. Pillay (Classification of commutative FPF rings, Notas Math., vol. 4, Univ. de Murcia, Murcia, Spain, 1990) that

C
F

P
2

F

CF{P^2}F

local rings (defined below) have FP-injective

Q
Q

. The counterexample is a difficult example due to Huckaba of a strongly Prüfer ring without "Property A." (A ring with Property A was labelled a McCoy ring by the author.) This counterexample is

C
F

P
2

F

CF{P^2}F

in the sense that every factor ring of

R
R

is

F

P
2

F

F{P^2}F

.
A note on extensions of semigroups of *-endomorphisms
Authors: Arveson William , Kishimoto Akitaka
Published: June 2010

We present a relatively elementary proof that every

E
0

{E_0}

-semigroup acting on a type

I
∞

{I_\infty }

factor can be extended to a one-parameter group of

∗
*

-automorphisms acting on a larger type

I
∞

{I_\infty }

factor.
A compact regular operator without modulus
Authors: Abramovich Y. A. , Wickstead A. W.
Published: June 2010

We construct an operator with the properties mentioned in the title. Without the regularity assumption such an example was first discovered by U. Krengel.
Self-similar sets. VII. A characterization of self-similar fractals with positive Hausdorff measure
Authors: Bandt Christoph , Graf Siegfried
Published: June 2010

For self-similar sets with nonoverlapping pieces, Hausdorff dimension and measure are easily determined. We express "absence of overlap" in terms of discontinuous action of a family of similitudes, thus improving the usual "open set condition".
Special 𝛼-limit points for maps of the interval
Authors: Hero Michael W.
Published: June 2010

The notion of a special

α
\alpha

-limit point is defined. For maps of the interval, it is shown that a point is a special

α
\alpha

-limit point if and only if it is an element of the attracting center.
Varieties attached to an 𝑆𝐿₂(2^{𝑘})-module
Authors: Mason Geoffrey
Published: June 2010

Let

G
k

=

SL
2

⁡
(

2
k

)

{G_k} = \operatorname {SL}_2({2^k})

and

V
V

be an

F

G
k

F{G_k}

-module with

F
F

a field containing

GF
⁡
(

2
k

)

\operatorname {GF}(2^k)

. We show that

V
V

is irreducible if and only if there is a subgroup

U
0

{U_0}

contained in a

2
2

-Sylow of

G
k

{G_k}

such that

V
V

affords the regular representation of

U
0

{U_0}

. We further show how to construct a variety, defined over an algebraic closure of

GF
⁡
(
2
)

\operatorname {GF}(2)

, whose

GF
⁡
(

2
k

)

\operatorname {GF}(2^k)

-rational points parameterize those conjugacy classes of subgroups of

G
k

{G_k}

, isomorphic to

U
0

{U_0}

, that are not represented regularly on

V
V

.
On the Aleksandrov problem of conservative distances
Authors: Mielnik Bogdan , Rassias Themistocles M.
Published: June 2010

A case of the Aleksandrov problem for unit distance preserving mappings between metric spaces is solved. The relevance of methods used in mathematical foundations of quantum mechanics is shown for another case of Aleksandrov problem involving angular distances

π

/

2

\pi /2

on the unit sphere.
Sample path-valued conditional Yeh-Wiener integrals and a Wiener integral equation
Authors: Park Chull , Skoug David
Published: June 2010

In this paper we evaluate the conditional Yeh-Wiener integral

E
(
F
(
x
)

|

x
(
s
,
t
)
=
ξ
)

E(F(x)|x(s,t) = \xi )

for functions

F
F

of the form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x right-parenthesis equals exp left-brace integral Subscript 0 Superscript t Baseline integral Subscript 0 Superscript s Baseline phi left-parenthesis sigma comma tau comma x left-parenthesis sigma comma tau right-parenthesis right-parenthesis d sigma d tau right-brace period">

F
(
x
)
=
exp
⁡
{

∫
0
t

∫
0
s

ϕ

(
σ
,
τ
,
x
(
σ
,
τ
)
)
d
σ
d
τ
}
.

F(x) = \exp \{ \int _0^t {\int _0^s \phi } (\sigma ,\tau ,x(\sigma ,\tau ))d\sigma d\tau \} .

\]
The method we use to evaluate this conditional integral is to first define a sample path-valued conditional Yeh-Wiener integral of the type

E
(
F
(
x
)

|

x
(
s
,
)
=
ψ
(
)
)

E(F(x)|x(s,) = \psi ())

and show that it satisfies a Wiener integral equation. We next obtain a series solution for

E
(
F
(
x
)

|

x
(
s
,
)
=
ψ
(
)
)

E(F(x)|x(s,) = \psi ())

by solving this Wiener integral equation. Finally, we integrate this series solution appropriately in order to evaluate

E
(
F
(
x
)

|

x
(
s
,
t
)
=
ξ
)

E(F(x)|x(s,t) = \xi )

.
On a conjecture of Nitsche
Authors: Crow Gregory D.
Published: June 2010

We show that under the hypothesis of bounded Gaussian curvature, certain minimal surfaces are in fact of finite total curvature. We can then answer the following version of a conjecture of Nitsche (J. Math. Mech. 11 (1962), 295) under the hypothesis of bounded Gaussian curvature: Conjecture. Let

M
2

⊂

R

3

{M^2} \subset {{\mathbf {R}}^3}

be a complete minimal surface such that for some height function

H
H

, the level sets are (compact) Jordan curves. Then

M
M

is a catenoid.
Parametrized Borsuk-Ulam theorems for multivalued maps
Authors: Izydorek Marek , Jaworowski Jan
Published: June 2010

By combining parametrized Borsuk-Ulam theorems proved by Dold with methods using the Vietoris mapping theorem we show that Dold’s results can be extended to multivalued maps. Such methods were invented by Eilenberg and Montgomery who applied them to multivalued fixed-point theorems, and they were used by Jaworowski to prove a multivalued version of the Borsuk-Ulam theorem. Subsequently they were extended and refined in various ways by Górniewicz and others. We also indicate how our results can be proved in the relative case, for pairs of spaces rather than for single spaces only. This allows us to obtain positive results for bundles over manifolds with boundary; for instance, over a closed interval.
One radius theorem for the eigenfunctions of the invariant Laplacian
Authors: Kwon E. G.
Published: June 2010

Let

B
B

be the open unit ball in

C

n

{{\mathbf {C}}^n}

with its boundary

S
S

. Suppose that

α
≥

1
2

\alpha \geq \tfrac {1}{2}

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis z right-parenthesis equals left-parenthesis 1 minus StartAbsoluteValue z EndAbsoluteValue squared right-parenthesis Superscript n left-parenthesis 1 minus alpha right-parenthesis Baseline upper F left-parenthesis z right-parenthesis">

u
(
z
)
=

(
1
−

|
z
|

2

)

n
(
1
−
α
)

F
(
z
)

u(z) = {(1 - {\left | z \right |^2})^{n(1 - \alpha )}}F(z)

for some

F
(
z
)
∈
C
(

B
¯

)

F(z) \in C(\overline B )

. If for every

z
∈
B

z \in B

there corresponds an

r
(
z
)
:
0
>
r
(
z
)
>
1

r(z):0 > r(z) > 1

and an automorphism

ψ
z

{\psi _z}

with

ψ
z

(
0
)
=
z

{\psi _z}(0) = z

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis z right-parenthesis equals StartFraction 1 Over g Subscript alpha Baseline left-parenthesis r left-parenthesis z right-parenthesis right-parenthesis EndFraction integral Underscript upper S Endscripts u ring psi Subscript z Baseline left-parenthesis r left-parenthesis z right-parenthesis zeta right-parenthesis d sigma left-parenthesis zeta right-parenthesis comma">

u
(
z
)
=

1

g
α

(
r
(
z
)
)

∫
S

u
∘

ψ
z

(
r
(
z
)
ζ
)
d
σ
(
ζ
)

,

u(z) = \frac {1}{{{g_\alpha }(r(z))}}\int _S {u \circ {\psi _z}(r(z)\zeta )d\sigma (\zeta )} ,

\]
then

Δ
~

u
(
z
)
=
−
4

n
2

α
(
1
−
α
)
u
(
z
)
,
z
∈
B

\tilde \Delta u(z) = - 4{n^2}\alpha (1 - \alpha )u(z),z \in B

. Here

Δ
~

\tilde \Delta

is the invariant Laplacian and

g
α

(
r
)

{g_\alpha }(r)

is the hypergeometric function

F
(
n
−
n
α
,
n
−
n
α
,
n
;

r
2

)

F(n - n\alpha ,n - n\alpha ,n;{r^2})

.
Existence of maximal element and equilibrium for a nonparacompact 𝑁-person game
Authors: Kim Won Kyu
Published: June 2010

In this paper, we will introduce the concept of

L
S

{L_S}

-majorized correspondence and prove a new maximal element existence theorem on nonparacompact sets. As applications, we prove a new existence theorem of equilibrium for a nonparacompact

1
1

-person game with

L
S

{L_S}

-majorized preference correspondences, and then we prove that a nonparacompact N-person game with preference correspondences of class

L
L

can be reduced to a

1
1

-person game with

L
S

{L_S}

-majorized preference correspondences.
A group determinant determines its group
Authors: Mansfield Richard
Published: June 2010

The determinant of a group’s multiplication table determines the group.
On the mapping class group of spherical 3-orbifolds
Authors: Cuccagna Scipio , Zimmermann Bruno
Published: June 2010

We prove that the mapping class group of a spherical

3
3

-orbifold with nonempty singular set is finite and can be realized by a finite group of diffeomorphisms; we also indicate how to compute this group.
On a singular value inequality of Ky Fan and Hoffman
Authors: Dodds Peter G. , Dodds Theresa K.-Y.
Published: June 2010

It is shown that the identity operator is a best unitary approximant to any positive measurable operator affiliated with a semifinite von Neumann algebra equipped with a distinguished faithful normal semifinite trace.
On nilpotency of the separating ideal of a derivation
Authors: Garimella Ramesh V.
Published: June 2010

We prove that the separating ideal

S
(
D
)

S(D)

of any derivation

D
D

on a commutative unital algebra

B
B

is nilpotent if and only if

S
(
D
)
∩
(
⋂

R
n

)

S(D) \cap (\bigcap {{R^n})}

is a nil ideal, where

R
R

is the Jacobson radical of

B
B

. Also we show that any derivation

D
D

on a commutative unital semiprime Banach algebra

B
B

is continuous if and only if

⋂

(
S
(
D
)
)

n

=
{
0
}

\bigcap {{{(S(D))}^n} = \{ 0\} }

. Further we show that the set of all nilpotent elements of

S
(
D
)

S(D)

is equal to

⋂

(
S
(
D
)
∩
P
)

\bigcap {(S(D) \cap P)}

, where the intersection runs over all nonclosed prime ideals of

B
B

not containing

S
(
D
)

S(D)

. As a consequence, we show that if a commutative unital Banach algebra has only countably many nonclosed prime ideals then the separating ideal of a derivation is nilpotent.
Mapping spaces of compact Lie groups and 𝑝-adic completion
Authors: Blanc David , Notbohm Dietrich
Published: June 2010

If

B
G

,

B
H

{\mathbf {BG}},\;{\mathbf {BH}}

are the classifying spaces of compact Lie groups, with

H

{\mathbf {H}}

connected, then the mapping space functor

m
a
p

(

B
G

,
−
)

{\mathbf {map}}({\mathbf {BG}}, - )

commutes with

p
p

-completion on

B
H

{\mathbf {BH}}

: i.e., for each

f
:

B
G

→

B
H

f:{\mathbf {BG}} \to {\mathbf {BH}}

the component

(

m
a
p

(

B
G

,

B
H

)
f

)
p
∧

({\mathbf {map}}{({\mathbf {BG}},{\mathbf {BH}})_f})_p^ \wedge

is

p
p

-complete, and is homotopy equivalent to

m
a
p

(

B
G

,

B
H

p
∧

)

i
∘
f

{\mathbf {map}}{({\mathbf {BG}},{\mathbf {BH}}_p^ \wedge )_{i \circ f}}

.
A generalization of the punctured neighborhood theorem
Authors: Lee Woo Young
Published: June 2010

If

T
∈

L

(
X
)

T \in \mathcal {L}(X)

is regular on a Banach space

X
X

, with finite- dimensional intersection

T

−
1

(
0
)
∩
T
(
X
)

{T^{ - 1}}(0) \cap T(X)

, and if

S
,

S
′

S,\;S’

are invertible, commute with

T
T

and have sufficiently small norm, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension left-parenthesis upper T minus upper S prime right-parenthesis Superscript negative 1 Baseline left-parenthesis 0 right-parenthesis equals dimension left-parenthesis upper T minus upper S right-parenthesis Superscript negative 1 Baseline left-parenthesis 0 right-parenthesis">

dim
⁡

(
T
−

S
′

)

−
1

(
0
)
=
dim
⁡

(
T
−
S

)

−
1

(
0
)

\dim {(T - S’)^{ - 1}}(0) = \dim {(T - S)^{ - 1}}(0)

and

dim
⁡
X

/

(
T
−

S
′

)
X
=
dim
⁡
X

/

(
T
−
S
)
X

\dim X/(T - S’)X = \dim X/(T - S)X

.
The mountain climbers’ problem
Authors: Keleti Tamás
Published: June 2010

We show that two climbers can climb a mountain in such a way that at each moment they are at the same height above the sea level, supposing that the mountain has no plateau. That is, if

f
f

and

g
g

are continuous functions mapping

[
0
,
1
]

[0,1]

to

[
0
,
1
]

[0,1]

with

f
(
0
)
=
g
(
0
)
=
0

f(0) = g(0) = 0

and

f
(
1
)
=
g
(
1
)
=
1

f(1) = g(1) = 1

, and if neither

f
f

nor

g
g

has an interval of constancy then there exist continuous functions

k
k

and

h
:
[
0
,
1
]
→
[
0
,
1
]

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

satisfying

k
(
0
)
=
h
(
0
)
=
0
,

k
(
1
)
=
h
(
1
)
=
1

k(0) = h(0) = 0,\;k(1) = h(1) = 1

, and

f
∘
k
=
g
∘
h

f \circ k = g \circ h

.
A quantitative Dirichlet-Jordan test for Walsh-Fourier series
Authors: Móricz Ferenc
Published: June 2010

We consider the Walsh-Fourier series

∑

a
k

w
k

(
x
)

\sum {{a_k}{w_k}(x)}

of a function

f
f

assumed to be of bounded fluctuation on the interval

[
0
,
1
)

[0,1)

. Every function of bounded variation is also of bounded fluctuation on the same interval, but not conversely. We present an estimate for the difference of

f
(
x
)

f(x)

at a point

x
∈
[
0
,
1
)

x \in [0,1)

and the partial sum of its Walsh-Fourier series in terms of the bounded fluctuation operator. This gives rise to a local convergence result. As special cases, we obtain a Walsh analogue of the Dirichlet-Jordan test and a global convergence result due to Onneweer.
On two absolute Riesz summability factors of infinite series
Authors: Sarıgöl Mehmet Ali
Published: June 2010

This paper gives a necessary and sufficient condition in order that a series

∑

a
n

ε
n

\sum {{a_n}} {\varepsilon _n}

should be summable

|

R
,

q
n

|

|R,{q_n}|

whenever

∑

a
n

\sum {{a_n}}

is summable

|

R
,

p
n

|

k

,

k
⩾
1

|R,{p_n}{|_k},\;k \geqslant 1

, and so extends the known result of Bosanquet to the case

k
>
1

k > 1

.
Distinct 3-manifolds with all 𝑆𝑈(2)_{𝑞} invariants the same
Authors: Lickorish W. B. R.
Published: June 2010

Witten’s

SU
⁡

(
2

)
q

\operatorname {SU} {(2)_q}

invariants do not classify

3
3

-manifolds.
A remark on curves covered by coverings
Authors: Coppens Marc
Published: June 2010

Let

f
:

C
′

→
C

f:C’ \to C

be a covering between two smooth irreducible projective curves. Let

p
p

be a prime number. If

C
′

C’

is a covering of degree

p
p

of a curve of genus

h
h

and if

g
(
C
)
⩾

(
2
p

)
(
h
+
3
)
+
h
+
3
−
2
p

g(C) \geqslant (_2^p)(h + 3) + h + 3 - 2p

, then

C
C

is a covering of degree

p
p

of a curve of genus at most

h
h

.
Identities of the natural representation of the infinitely based semigroup
Authors: Al’shanskii Leonid , Kushkuley Alexander
Published: June 2010

An equational theory of a very small semigroup may fail to be finitely presented. A well-known example of such a semigroup was studied in detail by Peter Perkins some twenty years ago. We prove that the natural representation of his semigroup has a finite basis of identical relations and discuss this fact in a general context of universal algebra.
There are many Ostaszewski spaces
Authors: LaBerge Timothy J.
Published: June 2010

We show that

♢

ω
1

{\diamondsuit _{{\omega _1}}}

implies there is a family of

2

ω
1

{2^{{\omega _1}}}

many pairwise nonhomeomorphic Ostaszewski spaces, with the additional property that any continuous image of one in another is compact. We also present diamond versions of Fleissner’s forcing results concerning the normality of products of Ostaszewski spaces and their compactifications.
A heat kernel characterization of asymptotic harmonicity
Authors: Ledrappier François
Published: June 2010

A compact negatively curved manifold is asymptotically harmonic if and only if the relation

4

λ
1

=
β

4{\lambda _1} = \beta

holds, where

λ
1

{\lambda _1}

is the spectral gap of the Laplacian on the universal cover, and

β
\beta

is the Kaimanovich entropy.
A central limit theorem on Heisenberg type groups. II
Authors: Ohring Peter
Published: June 2010

We present a Liapounov type central limit theorem for random variables associated to a commutative Banach algebra of "radial" measures on Heisenberg type groups. This theorem improves on a result presented by the author in Proc. Amer. Math. Soc. 113 (1991), 529-536.
Hermite expansions on 𝑅ⁿ for radial functions
Authors: Thangavelu S.
Published: June 2010

It is proved that the Riesz means

S
R
δ

f
,

δ
>
0

S_R^\delta f,\,\delta > 0

, for the Hermite expansions on

R

n

,

n
⩾
2

{\mathbb {R}^n},\,n \geqslant 2

, satisfy the uniform estimates

‖

S
R
δ

f

‖

p

⩽
C

‖
f
‖

p

{\left \| {S_R^\delta f} \right \|_p} \leqslant C{\left \| f \right \|_p}

for all radial functions if and only if

p
p

lies in the interval

2
n

/

(
n
+
1
+
2
δ
)
>
p
>
2
n

/

(
n
−
1
−
2
δ
)

2n/(n + 1 + 2\delta ) > p > 2n/(n - 1 - 2\delta )

.
The Hasse norm principle for elementary abelian extensions
Authors: Horie Mitsuko
Published: June 2010

Let

K

/

k

K/k

be an elementary abelian extension of finite algebraic number fields. The Hasse norm principle for

K

/

k

K/k

and its relation to the Hasse norm principles for all proper subextensions of

K

/

k

K/k

will be discussed. The central class field of

K

/

k

K/k

with

k
=

Q

k = \mathbb {Q}

will also be studied.
Divergence of averages obtained by sampling a flow
Authors: Akcoglu Mustafa , Bellow Alexandra , del Junco Andrés , Jones Roger L.
Published: June 2010

In this paper we consider ergodic averages obtained by sampling at discrete times along a measure preserving ergodic flow. We show, in particular, that if

U
t

{U_t}

is an aperiodic flow, then averages obtained by sampling at times

n
+

t
n

n + {t_n}

satisfy the strong sweeping out property for any sequence

t
n

→
0

{t_n} \to 0

. We also show that there is a flow (which is periodic) and a sequence

t
n

→
0

{t_n} \to 0

such that the Cesaro averages of samples at time

n
+

t
n

n + {t_n}

do converge a.e. In fact, we show that every uniformly distributed sequence admits a perturbation that makes it a good Lebesgue sequence.
Infinite-ods in arcwise connected continua
Authors: Vought Eldon J.
Published: June 2010

The theorem is proved that a continuum that has only finitely many arc components (hence an arcwise connected continuum) and contains an

n
n

-od for every integer

n
n

must contain an infinite-od.
Strongly extreme points and the Radon-Nikodým property
Authors: Hu Zhibao
Published: June 2010

We prove that if

K
K

is a bounded and convex subset of a Banach space

X
X

and

x
x

is a point in

K
K

, then

x
x

is a strongly extreme point of

K
K

if and only if

x
x

is a strongly extreme point of

K
¯

∗

{\overline K ^{\ast }}

which is the weak

∗

^{{\ast }}

closure of

K
K

in

X

∗

∗

{X^{{\ast }{\ast }}}

. We also prove that a Banach space

X
X

has the Radon-Nikodým property if and only if for any equivalent norm on

X
X

, the unit ball has a strongly extreme point.
On groups related to the Hecke groups
Authors: Knopp Marvin I. , Newman Morris
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column a m p semicolon lamda 1 2nd Row 1st Column 0 2nd Column a m p semicolon 1 EndMatrix comma Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column a m p semicolon 0 2nd Row 1st Column lamda 2 2nd Column a m p semicolon 1 EndMatrix">

[

1

a
m
p
;

λ
1

0

a
m
p
;
1

]

,

[

1

a
m
p
;
0

λ
2

a
m
p
;
1

]

\left [ {\begin {array}{*{20}{c}} 1 & {{\lambda _1}} \\ 0 & 1 \\ \end {array} } \right ],\left [ {\begin {array}{*{20}{c}} 1 & 0 \\ {{\lambda _2}} & 1 \\ \end {array} } \right ]

be parabolic elements of

PSL
⁡
(
2
,
R
)

\operatorname {PSL} (2,R)

, where

λ
1

,

λ
2

>
0

{\lambda _1},{\lambda _2} > 0

. The principal result shown here is that

K
(

λ
1

,

λ
2

)

K({\lambda _1},{\lambda _2})

, the group generated by these elements, is discrete if and only if

λ
1

λ
2

⩾
4

{\lambda _1}{\lambda _2} \geqslant 4

, or

λ
1

λ
2

=
4

cos
2

(
π

/

p
)

{\lambda _1}{\lambda _2} = 4{\cos ^2}(\pi /p)

, where

p
p

is an integer

⩾
3

\geqslant 3

. When

λ
1

λ
2

=
4

cos
2

(
π

/

p
)
,

K
(

λ

1
,

λ
2

)

{\lambda _1}{\lambda _2} = 4{\cos ^2}(\pi /p),\;K({\lambda _{1,}}{\lambda _2})

is conjugate to the classical Hecke group

H
(
2
cos
⁡
(
π

/

p
)
)

H(2\cos (\pi /p))

if

p
p

is odd; while if

p
p

is even,

K
(

λ
1

,

λ
2

)

K({\lambda _1},{\lambda _2})

is conjugate to a subgroup of

H
(
2
cos
⁡
(
π

/

p
)
)

H(2\cos (\pi /p))

of index

2
2

. When

λ
1

λ
2

⩾
4
,

K
(

λ
1

,

λ
2

)

{\lambda _1}{\lambda _2} \geqslant 4,\;K({\lambda _1},{\lambda _2})

is conjugate to a subgroup of

H
(

(

λ
1

λ
2

)

)

H(\sqrt {({\lambda _1}{\lambda _2})} )

of index

2
2

. In all of these cases

K
(

λ
1

,

λ
2

)

K({\lambda _1},{\lambda _2})

is the free product of two cyclic groups.
Restrictions to continuous functions and Boolean algebras
Authors: Recław Ireneusz
Published: June 2010

We show that every Borel function

f
:

R

→

R

f:\mathbb {R} \to \mathbb {R}

is continuous on a set

A
∉

J

A \notin \mathcal {J}

if

B
(

R

)

/

J

B(\mathbb {R})/\mathcal {J}

is weakly distributive. We also show that CCC is not sufficient. We investigate some other conditions considering the problem of restrictions to continuous functions.
Equations 𝑎𝑢^{𝑙}_{𝑛}=𝑏𝑢^{𝑘}_{𝑚} satisfied by members of recurrence sequences
Authors: Schlickewei H. P. , Schmidt W. M.
Published: June 2010

Let

{

u
n

}

n
∈

Z

{\{ {u_n}\} _{n \in \mathbb {Z}}}

be a linear recurrence sequence. Given

a
≠
0
,

b
≠
0

a \ne 0,\,b \ne 0

, and natural

k
≠
l

k \ne l

, we study equations as indicated in the title in unknowns

n
,
m

n,m

. It turns out that under natural conditions on the sequence

{

u
n

}

\{ {u_n}\}

, there are only finitely many solutions.
A counterexample to the infinity version of the Hyers and Ulam stability theorem
Authors: Casini Emanuele , Papini Pier Luigi
Published: June 2010

Hyers and Ulam proved a stability result for convex functions, defined in a subset of

ℜ
n

{\Re ^n}

. Here we give an example showing that their result cannot be extended to those functions defined in infinite-dimensional normed spaces. Also, we give a positive result for a particular class of approximately convex functions, defined in a Banach space, whose norm satisfies the so-called convex approximation property.
Measurability properties of sets of Vitali’s type
Authors: Solecki Sławomir
Published: June 2010

Assume a group

G
G

acts on a set. Given a subgroup

H
H

of

G
G

, by an

H
H

-selector we mean a selector of the set of all orbits of

H
H

. We investigate measurability properties of

H
H

-selectors with respect to

G
G

-invariant measures.