bluuebunny/crossref_metadata_2025 huggingface dataset:
Generalizations of the Sidon-Telyakovskiĭ theorem
Authors: Stanojević Časlav V. , Stanojevic Vera B.
Published: June 2010

The well-known Sidon-Telyakovskii integrability condition is considerably lightened as follows:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction 1 Over n EndFraction sigma-summation Underscript k equals 1 Overscript n Endscripts StartFraction StartAbsoluteValue normal upper Delta c left-parenthesis k right-parenthesis EndAbsoluteValue Superscript p Baseline Over upper A Subscript k Superscript p Baseline EndFraction equals upper O left-parenthesis 1 right-parenthesis comma n right-arrow normal infinity comma">

1
n

∑

k
=
1

n

|

Δ
c
(
k
)

|

p

A
k
p

=
O
(
1
)
,

n
→
∞

,

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

\]
where

{
c
(
n
)
}

\{ c(n)\}

is a certain null-sequence and

1
>
p
≤
2

1 > p \leq 2

. It is proved that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts n Superscript p minus 1 Baseline StartAbsoluteValue normal upper Delta c left-parenthesis n right-parenthesis EndAbsoluteValue Superscript p Baseline rho Superscript p Baseline left-parenthesis n right-parenthesis greater-than normal infinity">

∑

n
=
1

∞

n

p
−
1

|

Δ
c
(
n
)

|

p

ρ
p

(
n
)
>
∞

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

is also a sufficient integrability condition provided

∑

n
=
1

∞

(
1

/

n
ρ
(
n
)
)
>
∞

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

, where

{
ρ
(
n
)
}

\{ \rho (n)\}

is an increasing sequence of positive numbers.
Function spaces and local characters of topological spaces
Authors: Tereda Toshiji
Published: June 2010

We write

V
≃
W

V \simeq W

to mean that the two linear topological spaces

V
V

and

W
W

are linearly homeomorphic. In this paper we prove: (1) There are compact spaces

X
,
Y

X,Y

for which

C
p

(
X
)

≃

C
p

(
Y
)

{C_p}\left ( X \right ) \simeq {C_p}\left ( Y \right )

and

χ

(
X
)

≠
χ

(
Y
)

\chi \left ( X \right ) \ne \chi \left ( Y \right )

are satisfied. (2) For each infinite cardinal

κ
\kappa

, there are spaces

X
,
Y

X,Y

for which

C
p

(
X
)

≃

C
p

(
Y
)

,
χ

(
X
)

=
ω

{C_p}\left ( X \right ) \simeq {C_p}\left ( Y \right ),\chi \left ( X \right ) = \omega

and

ψ

(
Y
)

=
κ

\psi \left ( Y \right ) = \kappa

. (3) For each infinite cardinal

κ
\kappa

, there are spaces

X
,
Y

X,Y

for which

C
p

(
X
)

≃

C
p

(
Y
)

,
π
χ

(
X
)

=
ω

{C_p}\left ( X \right ) \simeq {C_p}\left ( Y \right ),\pi \chi \left ( X \right ) = \omega

and

π
χ

(
Y
)

=
κ

\pi \chi \left ( Y \right ) = \kappa

.
One-relator quotients and free products of cyclics
Authors: Fine Benjamin , Howie James , Rosenberger Gerhard
Published: June 2010

It is proven that the Freiheitssatz holds for all one-relator products of cyclic groups if the relator is cyclically reduced and a proper power. The method of proof involves representing such groups in

PS

L

2

(

C

)

{\text {PS}}{{\text {L}}_2}({\mathbf {C}})

and is a refinement of a technique of Baumslag, Morgan and Shalen. The technique allows the extension of the Freiheitssatz result to many additional one-relator products.
Spans of simple triods
Authors: West Thelma
Published: June 2010

We calculate and estimate the spans of objects in certain classes of continua; our main interest is in those objects with positive span.
An equivalence relation that is not freely generated
Authors: Adams Scot
Published: June 2010

In this paper it is shown that there exists a Borel equivalence relation with countable equivalence classes that is not generated by a free Borel action of a countable discrete group.
Convergence rates for Tikhonov regularization in finite-dimensional subspaces of Hilbert scales
Authors: Engl Heinz W. , Neubauer Andreas
Published: June 2010

The main result of this paper states how the discretization parameter and regularization parameter should be chosen in relation to the noise level in order to yield the optimal convergence rate for the Tikhonov-regularized solution of an ill-posed linear operator equation in a finite-dimensional subspace in the framework of Hilbert scales. The results apply to a wide class of spline and finite-element subspaces of Sobolev scales.
A note on measurability and almost continuity
Authors: Burke Maxim R. , Fremlin David H.
Published: June 2010

We prove that it is consistent with ZFC that there exist a measurable function

f
:

[

0
,
1

]

→

ω
1

f:\left [ {0,1} \right ] \to {\omega _1}

which is not almost continuous.
A mixed Parseval equation and the generalized Hankel transformations
Authors: Méndez J. M.
Published: June 2010

Let

T
1

{T_1}

and

T
2

{T_2}

be two classical integral transforms whose inverse formulas coincide with themselves, satisfying the mixed Parseval equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Subscript 0 Superscript normal infinity Baseline f left-parenthesis x right-parenthesis g left-parenthesis x right-parenthesis d x equals integral Subscript 0 Superscript normal infinity Baseline upper F 1 left-parenthesis y right-parenthesis upper G 2 left-parenthesis y right-parenthesis d y comma">

∫
0
∞

f
(
x
)
g
(
x
)
d
x
=

∫
0
∞

F
1

(
y
)

G
2

(
y
)

d
y
,

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

\]
where

F
1

(
y
)
=
(

T
1

f
)
(
y
)

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

and

G
2

(
y
)
=
(

T
2

g
)
(
y
)

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

. We propose to define the generalized transformation

T
1
′

{T’_1}

as the adjoint operator of

T
2

{T_2}

, and conversely. This procedure provides a new approach to extend the Hankel transform to certain spaces of distributions.
Homotopy type finiteness theorems for certain precompact families of Riemannian manifolds
Authors: Yamaguchi Takao
Published: June 2010

In this paper, we consider a precompact family of Riemannian manifolds with respect to the Hausdorff distance, and prove the homotopy type finiteness of elements in the family. This is an extension in the homotopy type version of the Cheeger and Weinstein finiteness theorems.
On dendroids with Kelley’s property
Authors: Czuba Stanisław T.
Published: June 2010

It is proved that if a dendroid has Kelley’s property, then it is smooth. This is a correction of an error from [4].
Rigid Stone spaces within 𝑍𝐹𝐶
Authors: Dow A. , Gubbi A. V. , Szymański A.
Published: June 2010

We construct within ZFC a family of pairwise nonhomeomorphic dense-in-itself rigid extremally disconnected compact separable spaces. Any such space is determined by a sequence of weak

P
P

-points of

N

∗

{{\mathbf {N}}^ * }

having different types.
A new approach to the multipliers of Pedersen’s ideal
Authors: Phillips N. Christopher
Published: June 2010

We obtain a simple description of the multiplier algebra

Γ

(

K
A

)

\Gamma \left ( {{K_A}} \right )

of Pedersen’s ideal

K
A

{K_A}

of a

C
∗

{C^ * }

-algebra

A
A

as a pro-

C
∗

{C^ * }

-algebra (inverse limit of

C
∗

{C^ * }

-algebras). This description enables us to derive a number of the results obtained by Lazar and Taylor in their study of

Γ

(

K
A

)

\Gamma \left ( {{K_A}} \right )

directly from corresponding facts about pro-

C
∗

{C^ * }

-algebras, and to give simplified proofs of several other results.
A note on generalized inverse functions
Authors: Harte Robin
Published: June 2010

We offer the following generalization of an observation of Graham Allen about left inverses: if the elements of a semigroup of Banach algebra elements can all be given mutually commuting generalized inverses, then they can all be given invertible generalized inverses.
Cut locus contained in a hypersurface
Authors: Gómez F. , Muñoz M. C.
Published: June 2010

We prove that if the cut locus

C
(
p
)

C(p)

of a point

p
p

in a compact connected Riemannian manifold

M
M

is contained in a connected hypersurface

N
N

, then

M
M

is homeomorphic to

S
m

{S^m}

if

C
(
p
)
≠
N

C(p) \ne N

and

M
M

is homotopically equivalent to

R

P
m

{\mathbf {R}}{P^m}

if

C
(
p
)
=
N

C(p) = N

.
Some remarks on the Pompeiu problem for groups
Authors: Scott David , Sitaram Alladi
Published: June 2010

A Borel set

E
E

in a topological group

G
G

is said to be a

P
P

-set for the space of integrable functions on

G
G

if the zero function is the only integrable function whose integral over all left and right translates of

E
E

by elements of

G
G

is zero. For a "sufficiently nice" group

G
G

and a Borel set

E
E

of finite Haar measure a certain condition on the Fourier transform of a function related to

E
E

is shown to be a sufficient condition for

E
E

to be a

P
P

-set. This condition is then applied to several classes of groups including certain compact groups, certain semisimple Lie groups, the Heisenberg groups and the Euclidean motion group of the plane.
Smooth extensions of Lipschitzian real functions
Authors: Ricceri Biagio
Published: June 2010

In this short note we point out that any Lipschitzian real function

f
f

defined in a subset

K
K

of a Banach space

E
E

, with

span

¯

(K)

≠

E

\overline {{\text {span}}} {\text {(K)}} \ne {\text {E}}

, can be extended to a surjective, open and Lipschitzian real function

g
g

on

E
E

in such a way that, for every

r
∈

R

r \in {\mathbf {R}}

, the set

g

−
1

(
r
)

{g^{ - 1}}(r)

is arcwise connected. In fact, a more refined result is proved.
On exponents of homology and cohomology of finite groups
Authors: Carlson Jon F.
Published: June 2010

Let

G
G

be a finite group and let

r
r

be the maximum of the

p
p

-ranks of

G
G

for all primes

p
p

dividing the order to

G
G

. There exist positive integers

m
m

and

n
n

such that the exponents of

H
n

(
G
,

Z

)

{H^n}(G,{\mathbf {Z}})

and

H
m

(
G
,

Z

)

{H_m}(G,{\mathbf {Z}})

are greater than

|

G

|

1

/

r

|G{|^{1/r}}

.
𝐿_{𝑝}-continuity of positive semigroups on finite von Neumann algebras
Authors: Watanabe Seiji
Published: June 2010

Let

M
M

be a

σ
\sigma

-finite, finite von Neumann algebra with a faithful normal tracial state

τ
\tau

. Let

α
\alpha

be a one-parameter semigroup of normal positive contractions of

M
M

. Then it is shown that

α
\alpha

is continuous with respect to the

L
p

{L_p}

-norm

(
1
≤
p
>
∞
)

(1 \leq p > \infty )

induced by

τ
\tau

if and only if it is

σ
\sigma

-weakly continuous.
A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations
Authors: Jensen R. , Lions P.-L. , Souganidis P. E.
Published: June 2010

In this note we extend some recent results of R. Jensen concerning the uniqueness of viscosity solutions of scalar, second order, fully nonlinear, elliptic, possibly degenerate, partial differential equations.
On the holomorphicity of harmonic maps from compact Kähler manifolds to hyperbolic Riemann surfaces
Authors: Ono Kaoru
Published: June 2010

We give a sufficient condition in order that a harmonic map from a compact Kähler manifold with negative first Chern class to a compact hyperbolic Riemann surface be

±
\pm

holomorphic. The above condition generalizes that of Eells and Wood concerning harmonic maps between Riemann surfaces. As a corollary we get a generalization of Kneser’s theorem.
A nonmovable space with movable components
Authors: Dydak J. , Segal J. , Spież S.
Published: June 2010

In this paper we construct a nonmovable complete metric space of which every component is movable. This construction is based on P. Roy’s famous example of a complete metric space

X
X

which has inductive dimension ind

X
=
0

X = 0

and covering dimension

dim
⁡
X
=
1

\dim X = 1

.
The free metabelian group of rank two contains continuously many nonisomorphic subgroups
Authors: Baumslag G. , Stammbach U. , Strebel R.
Published: June 2010

It is proved that the free metabelian group of rank two contains continuously many nonisomorphic subgroups.
Super-rigid families of strongly Blackwell spaces
Authors: Droste Manfred
Published: June 2010

We construct a complete subfield

F
F

of

P
(

R

)

P({\mathbf {R}})

, isomorphic to

P
(

R

)

P({\mathbf {R}})

, of pairwise non-Borel-isomorphic rigid strong Blackwell subsets of

R

{\mathbf {R}}

such that there are only ’very few’ measurable functions between any two members of

F
F

. As a consequence, we obtain large chains and antichains of non-isomorphic rigid strong Blackwell subsets of

R

{\mathbf {R}}

. Also, there is a collection of continuously many dense subsets of

R

{\mathbf {R}}

such that any two of them differ only by two elements, but none of them is a continuous image of any other.
On two problems concerning Baire sets in normal spaces
Authors: Balogh Zoltán T.
Published: June 2010

Two problems will be dealt with. The first problem, due to Katetov, asks whether there is a normal, nonperfect

T
2

{T_2}

space

X
X

such that the Baire and Borel algebras in

X
X

coincide. The second problem, due to Ross and Stromberg, asks whether each closed Baire set has to be zero set in a normal, locally compact

T
2

{T_2}

space. Several consistent examples of spaces satisfying the requirements of the first problem will be constructed. A counterexample to the second problem is given in ZFC.
Mixed-norm generalizations of Bergman spaces and duality
Authors: Gadbois Steve
Published: June 2010

Conditions sufficient for boundedness of the Bergman projection on certain "mixed-norm" spaces of functions on the unit ball of

C

N

{{\mathbf {C}}^N}

are given, and this is used to identify the dual space of such mixed-norm spaces. Several representation theorems that follow from the duality are also given.
A simple proof of Gabber’s theorem on projective modules over a localized local ring
Authors: Swan Richard G.
Published: June 2010

Let

A
A

be a regular local ring of dimension 3 and let

u
u

be an element of

A
A

not in the square of the maximal ideal. Gabber has shown that all projective modules over

A
[

u

−
1

]

A[{u^{ - 1}}]

are free. An elementary proof of this fact is given here.
Banach space properties of Ciesielski-Pol’s 𝐶(𝐾) space
Authors: Godefroy G. , Pelant J. , Whitfield J. H. M. , Zizler V.
Published: June 2010

A

C
(
K
)

C(K)

space

X
0

{X_0}

which Ciesielski and Pol show does not continuously linearly inject into any

c
0

(
Γ
)

{c_0}(\Gamma )

has an equivalent

C
∞

{C^\infty }

-norm, is Lipschitz equivalent to a

c
0

(
Γ
)

{c_0}(\Gamma )

, and the density character of

X
0

{X_0}

is equal to the

w
∗

{w^*}

-density character of

X
0
∗

X_0^*

.
Linear independence of iterates and entire solutions of functional equations
Authors: Christensen Jens Peter Reus , Fischer Pal
Published: June 2010

A classical result of Pólya concerning the growth of Nevanlinna’s characteristics of composite functions is used to prove linear independence of some iterates. The same result of Pólya is also used to show the nonexistence of entire solutions of the Feigenbaum functional equation.
The convolution of Radon measures
Authors: Kharaghani H.
Published: June 2010

The convolution of a pair of bounded Radon measures is extended to Čech-complete topological semigroups with separately continuous multiplication.
An extension of the closed unbounded filter
Authors: Mignone Robert J.
Published: June 2010

A natural extension of the closed unbounded filter is introduced. This extension coincides with the closed unbounded filter on uncountable, regular cardinals

κ
\kappa

, but in general does not for

P
k

λ

{P_k}\lambda

and

[
λ
]

κ

{\left [ \lambda \right ]^\kappa }

.
Paraproducts and commutators of martingale transforms
Authors: Bañuelos Rodrigo , Bennett Andrew G.
Published: June 2010

We define paraproducts in a probabilistic setting and study their boundedness properties. As an application of paraproduct techniques, we prove a generalization of the commutator result of Coifman-Rochberg-Weiss [6].
Rings with associators in the commutative center
Authors: Kleinfeld Erwin
Published: June 2010

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

(I) 

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

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

\]
where the commutator is defined by

(
a
,
b
)
=
a
b
−
b
a

(a,b) = ab - ba

, and the associator is defined by

(
a
,
b
,
c
)
=
a
b
⋅
c
−
a
⋅
b
c

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

and which satisfy one additional identity such as

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

. Assuming characteristic

≠
2

\ne 2

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

R
R

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

R
R

is simple, characteristic

≠
2
,
3

\ne 2,3

and satisfies (I) then

R
R

must be either commutative or associative.
𝐿^{𝑝} continuity of pseudo-differential operators with discontinuous symbol
Authors: Alvarez J. , Durán R.
Published: June 2010

A set

Γ
⊂

R

n

×

R

n

\Gamma \subset {{\mathbf {R}}^n} \times {{\mathbf {R}}^n}

is called conic if

(

x
,
ξ

)

∈
Γ

\left ( {x,\xi } \right ) \in \Gamma

implies

(

x
,
t
ξ

)

∈
Γ

\left ( {x,t\xi } \right ) \in \Gamma

, for all

t
>
0

t > 0

. In localizing a partial differential equation, the following question arose: Let us consider a pseudo-differential operator

L
L

whose symbol is the characteristic function of a conic subset of

R

n

×

R

n

{{\mathbf {R}}^n} \times {{\mathbf {R}}^n}

. Will

L
L

be a bounded operator in

L
2

(

R

n

)

{L^2}\left ( {{{\mathbf {R}}^n}} \right )

? We present as a counterexample a pseudo-differential operator

L
L

whose symbol is the characteristic function of a conic subset of

R

2

×

R

2

{{\mathbf {R}}^2} \times {{\mathbf {R}}^2}

, which is unbounded on

L
p

(

R

2

)

{L^p}\left ( {{{\mathbf {R}}^2}} \right )

for

1
≤
p
≤
2

1 \leq p \leq 2

.
Landau’s inequality for the difference operator
Authors: Kwong Man Kam , Zettl A.
Published: June 2010

The best constants for Landau’s inequality with the classical

p
p

-norms are known explicitly only when

p
=
1
,
2

 and 

∞

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

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

p
p

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

We derive sharp upper and lower bounds for

|

z

f
′

(
z
)

/

f
(
z
)

|

\left | {zf’(z)/f(z)} \right |

where

f
∈
R

f \in R

, i.e.

f
f

analytic in

D

{\mathbf {D}}

with

f
(
0
)
=
0
,

f
′

(
0
)
=
1

f(0) = 0,f’(0) = 1

and

e

i
α

f
′

(
z
)

 > 

0

{e^{i\alpha }}f’(z){\text { > }}0

in

D

{\mathbf {D}}

for a certain

α
=
α
(
f
)
∈

R

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

. The extremal function is

k
(
z
)
=
−
z
−
2
log
⁡
(
1
−
z
)

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

. This result improves an earlier one of D. K. Thomas.
𝑀_{𝑛} as a 0,1-sublattice of 𝐶𝑜𝑛𝐴 does not force the term condition
Authors: Willard Ross
Published: June 2010

For every

n
≥
3

n \geq 3

there exists a finite nonabelian algebra whose congruence lattice has

M
n

{M_n}

as a

0
,
1

0, 1

-sublattice. This answers a question of R. McKenzie and D. Hobby.
Polynomials and numerical ranges
Authors: Li Chi-Kwong
Published: June 2010

Let

A
A

be an

n
×
n

n \times n

complex matrix. For

1
≤
k
≤
n

1 \leq k \leq n

we study the inclusion relation for the following polynomial sets related to the matrix

A
A

. (a) The classical numerical range of the

k
k

th compound of the matrix

λ
I
−
A

\lambda I - A

. (b) The

k
k

th decomposable numerical range of the matrix

λ
I
−
A

\lambda I - A

. (c) The convex hull of the set of all monic polynomials of degree

k
k

that divide the characteristic polynomial of

A
A

. Moreover, we give an example showing that the set described in (a) is not convex in general. This settles a question raised by C. Johnson.
An unsolvable Cousin problem
Authors: Rubel Lee A.
Published: June 2010

We construct a Laurent series

∑

n
=
−
∞

∞

A
n

z
n

\sum \nolimits _{n = - \infty }^\infty {{A_n}{z^n}}

, convergent in an annulus

{

ρ
>

|
z
|

>
1

/

ρ

}

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

for some

ρ
>
1

\rho > 1

, which satisfies an algebraic differential equation (ADE) there, but such that neither of the series

∑

n
=
0

∞

A
n

z
n

\sum \nolimits _{n = 0}^\infty {{A_n}{z^n}}

and

∑

n
=
−
∞

−
1

A
n

z
n

\sum \nolimits _{n = - \infty }^{ - 1} {{A_n}{z^n}}

satisfies any ADE.
Nonvanishing meromorphic univalent functions
Authors: Abu-Muhanna Yusuf , Schober Glenn
Published: June 2010

This note studies the best constants

s
s

such that the function

k
(
z
)
=
z
+
2
+
1

/

z

k(z) = z + 2 + 1/z

solves the linear coefficient problems

max
Re
⁡
{
s

b
0

+

b
n

}

\max \operatorname {Re} \{ s{b_0} + {b_n}\}

and

max
Re
⁡
{
s

b
0

−

b
n

}

\max \operatorname {Re} \{ s{b_0} - {b_n}\}

over nonvanishing functions in the class

Σ
\Sigma

.
A weak Krasnosel′skiĭ theorem in 𝑅^{𝑑}
Authors: Breen Marilyn
Published: June 2010

Let

S
S

be a compact, locally starshaped set in

R
d

{R^d}

, and let

k
k

be a fixed integer,

0
≤
k
≤
d

0 \leq k \leq d

. If every

d
−
k
+
1

d - k + 1

points of

S
S

are clearly visible via

S
S

from a common point, then for every

k
k

-flat

F
′

F’

there exists a translate

F
F

of

F
′

F’

such that the following holds: To each point

s
0

s_0

in

S
∼
F

S \sim F

there correspond a point

s
m

{s_m}

in

F
F

and a polygonal path

⋃

{

[

s

i
−
1

,

s
i

]
:
1
≤
i
≤
m

}

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

in

S
∩
aff
⁡
(

s
0

∪
F
)

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

with
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d i s t left-parenthesis s Subscript i Baseline comma upper F right-parenthesis greater-than d i s t left-parenthesis s Subscript i minus 1 Baseline comma upper F right-parenthesis comma 1 less-than-or-equal-to i less-than-or-equal-to m">

dist
⁡
(

s
i

,
F
)
>
dist
⁡
(

s

i
−
1

,
F
)
,
1
≤
i
≤
m

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

. If

k
=
0

k = 0

or

k
=
d
−
1

k = d - 1

, then each point of

S
S

sees via

S
S

some point of

F
F

. Moreover, if

k
=
1

k = 1

, then

F
F

can be chosen so that

F
∩
S

F \cap S

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

We give a complete characterization of all inequalities of the type
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript i Superscript alpha Baseline left-parenthesis upper K right-parenthesis upper W Subscript j Superscript beta Baseline left-parenthesis upper K right-parenthesis upper W Subscript k Superscript gamma Baseline left-parenthesis upper K right-parenthesis greater-than-or-equal-to c">

W
i
α

(
K
)

W
j
β

(
K
)

W
k
γ

(
K
)
≥
c

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

, where

K
K

is an arbitrary convex body of Euclidean

d
d

-space,

W
l

(
K
)
,
l
=
i
,
j
,
k

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

, denotes the

l
l

th quermassintegral of

K
K

and

α
,
β
,
γ

\alpha ,\beta ,\gamma

and

c
c

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

d
−
1

d - 1

basic inequalities, the Fenchel-Alexandrov inequalities.
Some remarks on real-valued measurable cardinals
Authors: Szymański Andrzej
Published: June 2010

We consider the set

[
ω

]
ω

{[\omega ]^\omega }

and the cofinality of the set

κ

⋋

^\kappa \leftthreetimes

assuming that some cardinals are endowed in total measures.
Vector-valued stochastic processes. V. Optional and predictable variation of stochastic measures and stochastic processes
Authors: Dinculeanu Nicolae
Published: June 2010

Let

μ
\mu

be a stochastic measure, with values in a Banach space

E
E

, with finite variation

|

μ

|

|\mu |

. If

μ
\mu

is optional (resp. predictable), then

|

μ

|

|\mu |

is also optional (resp. predictable) provided

E
E

is separable, or the dual of a separable space, or has the Radon-Nikodym property. Let

A
A

be a right continuous stochastic process with values in

E
E

, with finite variation

|

A

|

|A|

. If

A
A

is measurable (resp. optional, predictable), then

|

A

|

|A|

, the continuous part

|

A

|

c

|A{|^c}

and the discrete part

|

A

|

d

|A{|^d}

have the same property.
Concerning periodic points in mappings of continua
Authors: Ingram W. T.
Published: June 2010

In this paper we present some conditions which are sufficient for a mapping to have periodic points. Theorem. If

f
f

is a mapping of the space

X
X

into

X
X

and there exist subcontinua

H
H

and

K
K

of

X
X

such that (1) every subcontinuum of

K
K

has the fixed point property, (2)

f
[
K
]

f[K]

and every subcontinuum of

f
[
H
]

f[H]

are in class

W
W

, (3)

f
[
K
]

f[K]

contains

H
H

, (4)

f
[
H
]

f[H]

contains

H
∪
K

H \cup K

, and (5) if

n
n

is a positive integer such that

(
f

|

H

)

−
n

(
K
)

{(f|H)^{ - n}}(K)

intersects

K
K

, then

n
=
2

n = 2

, then

K
K

contains periodic points of

f
f

of every period greater than 1. Also included is a fixed point lemma: Lemma. Suppose

f
f

is a mapping of the space

X
X

into

X
X

and

K
K

is a subcontinuum of

X
X

such that

f
[
K
]

f[K]

contains

K
K

. If (1) every subcontinuum of

K
K

has the fixed point property, and (2) every subcontinuum of

f
[
K
]

f[K]

is in class

W
W

, then there is a point

x
x

of

K
K

such that

f
(
x
)
=
x

f(x) = x

. Further we show that: If

f
f

is a mapping of

[
0
,
1
]

[0,1]

into

[
0
,
1
]

[0,1]

and

f
f

has a periodic point which is not a power of 2, then

lim
{
[
0
,
1
]
,
f
}

\lim \{ [0,1],f\}

contains an indecomposable continuum. Moreover, for each positive integer

i
i

, there is a mapping of

[
0
,
1
]

[0,1]

into

[
0
,
1
]

[0,1]

with a periodic point of period

2
i

{2^i}

and having a hereditarily decomposable inverse limit.
Invariants of finite abelian groups acting on the algebra of two 2×2 generic matrices
Authors: Huh Chan
Published: June 2010

In this paper, we discuss the finite generation problem for the invariant subalgebras of finite abelian groups which act linearly on the

2
×
2

2 \times 2

generic matrix algebra, and we obtain some conditions on the groups to ensure that their invariant subalgebras are finitely generated.
A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations
Authors: Lasiecka I. , Triggiani R.
Published: June 2010

We consider the solution operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper L u right-parenthesis left-parenthesis t right-parenthesis equals upper A integral Subscript 0 Superscript t Baseline upper G left-parenthesis t minus tau right-parenthesis upper A Superscript negative 1 Baseline upper B u left-parenthesis tau right-parenthesis d tau">

(
L
u
)
(
t
)
=
A

∫
0
t

G
(
t
−
τ
)

A

−
1

B
u
(
τ
)
d
τ

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

\]
corresponding to the abstract equation

x
˙

=
A
x
+
B
u

\dot x = Ax + Bu

on a reflexive Banach space

X
X

, where the linear operator

A
:
X
⊃

D

(
A
)
→
X

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

is the infinitesimal generator of a (strongly continuous) group

G
(
t
)

G(t)

of bounded operators on

X
X

, and

B
:
U
⊃

D

(
B
)
→
X

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

is a generally unbounded linear operator with

A

−
1

B
∈

L

(
U
,
X
)
,
U

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

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

A
A

to be boundedly invertible). Let

0
>
T
>
∞

0 > T > \infty

be given. We prove the following theorem: if

L
L

is continuous
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p Baseline left-parenthesis 0 comma upper T semicolon upper U right-parenthesis right-arrow upper L Superscript p Baseline left-parenthesis 0 comma upper T semicolon upper X right-parenthesis comma 1 greater-than p greater-than normal infinity">

L
p

(
0
,
T
;
U
)
→

L
p

(
0
,
T
;
X
)
,
1
>
p
>
∞

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

, then in fact

L
:

L:

continuous

L
p

(
0
,
T
;
U
)
→
C
[
0
,
T
]
;
X

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

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

G
(
t
)

G(t)

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

Ω
⊂

R
n

,
∂
Ω
=
Γ

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

, with nonhomogeneous term of class

L
2

(
0
,
T
;

L
2

(
Γ
)
)

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

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

T
=
∞

T = \infty

is also studied.
Most quasidiagonal operators are not block-diagonal
Authors: Herrero Domingo A.
Published: June 2010

The set of all block-diagonal operators is a dense first category subset of the class

(
Q
D
)

(QD)

of all quasidiagonal operators. On the other hand, the subset of all irreducible quasidiagonal operators with thin spectra, that are similar to block-diagonal ones, includes a

G
δ

{G_\delta }

-dense subset of

(
Q
D
)

(QD)

.
Positively expansive maps and growth of fundamental groups
Authors: Hiraide K.
Published: June 2010

In this note we will show that a positively expansive map of an arbitrary closed topological manifold is topologically conjugate to an expanding infra-nil-endomorphism.
Linear spaces, absolute retracts, and the compact extension property
Authors: van der Bijl Jos , van Mill Jan
Published: June 2010

We formulate a "partial realization" property and prove that this property is equivalent to the compact extension property. In addition, we prove that a linear space

L
L

has the compact extension property if and only if

L
L

is admissible if and only if

L
L

has the

σ
\sigma

-compact extension property. This implies that for a

σ
\sigma

-compact linear space

L
L

, the following statements are equivalent: (1)

L
L

is an absolute retract, (2)

L
L

has the compact extension property, and (3)

L
L

is admissible. Finally, we prove that if there exists a linear space which is not an absolute retract then there is an admissible linear space which is not an absolute retract.
Critical values of Fredholm maps
Authors: Church P. T. , Timourian J. G.
Published: June 2010

For a

C
r

{C^r}

Fredholm map

A
:
X
→
Y

A:X \to Y

, let

T
k

(
A
)

{T_k}\left ( A \right )

be the set of

x
∈
X

x \in X

such that codim Range

D
A

(
x
)

≥
k

DA\left ( x \right ) \geq k

. The homotopy of

Y
−
A

(

T
k

(
A
)

)

Y - A\left ( {{T_k}\left ( A \right )} \right )

is related to that of

Y
Y

, examples are given, and a factorization result is proved.
An area theorem for a one-dimensional semidirect extension of homogeneous groups
Authors: Damek Ewa
Published: June 2010

Let

N
N

be a homogeneous group [3] and let

{

δ
a

:
a
∈
A
=

R
+

}

\{ {\delta _a}:a \in A = {R^ + }\}

be the group of dilations of

N
N

. We prove an area theorem for harmonic functions w.r.t. a class of second-order left-invariant hypoelliptic differential operators

L
L

on the semidirect product

S
=
N
A

S = NA

with

a
x

a

−
1

=

δ
a

(
x
)
,
a
∈
A
,
x
∈
N

ax{a^{ - 1}} = {\delta _a}(x),a \in A,x \in N

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

For

1
≤
s
>
∞

1 \leq s > \infty

and crescents

1

^{1}

G
G

, with harmonic measure

ω
\omega

, the author examines the collection of bounded point evaluations,

bpe
⁡

(

P
s

(
ω
)

)

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

, (resp. analytic bounded point evaluations,

abpe
⁡

(

P
s

(
ω
)

)

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

) for polynomials with respect to the

L
s

(
ω
)

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

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

H
s

(
G
)

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

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b p e left-parenthesis upper P Superscript s Baseline left-parenthesis omega right-parenthesis right-parenthesis equals a b p e left-parenthesis upper P Superscript s Baseline left-parenthesis omega right-parenthesis right-parenthesis equals upper G">

bpe
⁡

(

P
s

(
ω
)

)

=
abpe
⁡

(

P
s

(
ω
)

)

=
G

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

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

H
s

(
G
)

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

, then (with a mild restriction on

∂
G

\partial G

)
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b p e left-parenthesis upper P Superscript s Baseline left-parenthesis omega right-parenthesis right-parenthesis equals a b p e left-parenthesis upper P Superscript s Baseline left-parenthesis omega right-parenthesis right-parenthesis equals i n t left-parenthesis upper G overbar Superscript logical-and Baseline right-parenthesis">

bpe
⁡

(

P
s

(
ω
)

)

=
abpe
⁡

(

P
s

(
ω
)

)

=
int
⁡
(

G
¯

∧

)

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

(Theorem 7).
A Lie property in group rings
Authors: Giambruno Antonino , Sehgal Sudarshan K.
Published: June 2010

Let

A
A

be an additive subgroup of a group ring

R
R

over a field

K
K

. Denote by

[
A
,
R
]

[A,R]

the additive subgroup generated by the Lie products

[
a
,
r
]
=
a
r
−
r
a
,
a
∈
A
,
r
∈
R

[a,r] = ar - ra,a \in A,r \in R

. Inductively, let

[
A
,

R
n

]
=
[
[
A
,

R

n
−
1

]
,
R
]

[A,{R_n}] = [[A,{R_{n - 1}}],R]

. We prove that

[
A
,

R
n

]
=
0

[A,{R_n}] = 0

for some

n
⇒
[
A
,
R
]
R

n \Rightarrow [A,R]R

is a nilpotent ideal.
Isometries homotopic to the identity
Authors: Norris Douglas A.
Published: June 2010

The types of surfaces which admit nontrivial isometries homotopic to the identity are classified up to diffeomorphism. In dimension three this is done for complete manifolds of constant negative curvature. Three-dimensional visibility manifolds that admit nontrivial isometries homotopic to the identity are shown to be diffeomorphic to a product

L
×

R
1

L \times {R^1}

.
𝜅-metrizable spaces, stratifiable spaces and metrization
Authors: Suzuki J. , Tamano K. , Tanaka Y.
Published: June 2010

It is shown that every

κ
\kappa

-metrizable CW-complex is metrizable. Examples are given showing that a stratifiable

κ
\kappa

-metrizable space and an additively

κ
\kappa

-metrizable space need not be metrizable.
Invariants for a class of torsion-free abelian groups
Authors: Arnold D. , Vinsonhaler C.
Published: June 2010

In this note we present a complete set of quasi-isomorphism invariants for strongly indecomposable abelian groups of the form

G
=
G
(

A
1

,
…
,

A
n

)

G = G({A_1}, \ldots ,{A_n})

. Here

A
1

,
…
,

A
n

{A_1}, \ldots ,{A_n}

are subgroups of the rationals

Q
Q

and

G
G

is the kernel of

f
:

A
1

⊕
⋯
⊕

A
n

→
Q

f:{A_1} \oplus \cdots \oplus {A_n} \to Q

, where

f
(

a
1

,
…
,

a
n

)
=
Σ

a
i

f({a_1}, \ldots ,{a_n}) = \Sigma {a_i}

. The invariants are the collection of numbers

rank

∩
{
G
[
σ
]

|

σ
∈
M
}

{\text {rank}} \cap \{ G[\sigma ]|\sigma \in M\}

, where

M
M

ranges over all subsets of the type lattice generated by

{

type

(

A
i

)

}

\left \{ {{\text {type}}({A_i})} \right \}

. Our results generalize the classical result of Baer for finite rank completely decomposable groups, as well as a result of F. Richman on a subset of the groups of the form

G
(

A
1

,
…
,

A
n

)

G({A_1}, \ldots ,{A_n})

.
Tree-like continua and exactly 𝑘-to-1 functions
Authors: Heath Jo
Published: June 2010

To answer a question of Nadler and Ward,

k
k

-to-

1
1

maps from tree-like continua onto tree-like continua are constructed, for

k
>
2

k > 2

. It is shown that certain arc-like continua cannot be the domain of any

2
2

-to-

1
1

map and that certain tree-like continua cannot be the image of any

2
2

-to-

1
1

map (defined on continua) but it is unknown if any indecomposable arc-like continuum can be the domain or any tree-like continuum the image of a

2
2

-to-

1
1

map.
Tame theories with hyperarithmetic homogeneous models
Authors: Millar Terrence
Published: June 2010

A tame theory is a decidable first-order theory with only countably many countable models, and all complete types recursive. It is shown here that the recursive complexity of countable homogeneous models of tame theories is unbounded in the hyperarithmetic hierarchy.
An elementary proof of a theorem of Schaefer, Wolff and Arendt
Authors: Huijsmans C. B.
Published: June 2010

An elementary proof of the following result, due to Schaefer, Wolff, and Arendt is given: if

T
T

is a lattice homomorphism on a Banach lattice

E
E

with spectrum

σ
(
T
)
=
{
1
}

\sigma (T) = \{ 1\}

, then

T
=
I

T = I

, the identity mapping on

E
E

.
Short-time asymptotics for the trace of one- and multi-dimensional Schrödinger semigroups
Authors: Papanicolaou Vassilis G.
Published: June 2010

Using Brownian motion we derive the leading asymptotic behaviour, as

t
↓
0

t \downarrow 0

, of the (normalized) trace of

e

t
L

−

e

t

L
H

{e^{tL}} - {e^{t{L^H}}}

, where

L
L

is the operator

Δ

/

2
+
q
(
x
)

\Delta /2 + q(x)

on

R

d

{{\mathbf {R}}^d}

(with zero boundary condition at infinity),

H
H

is a hyperplane of

R

d

{{\mathbf {R}}^d}

and

L
H

{L^H}

is the direct sum of

Δ

/

2
+
q
(
x
)

\Delta /2 + q(x)

acting on

H
+

{H^ + }

, with Dirichlet boundary condition on

H
H

(and 0 at infinity), and the same operator acting on

H
−

{H^ - }

(

H
+

{H^ + }

and

H
−

{H^ - }

are the two half-spaces defined by

H
H

). The function

q
q

is taken bounded and continuous on

R

d

{{\mathbf {R}}^d}

and, if

d
≥
2

d \geq 2

, we also assume that

q
q

is integrable on

R

d

{{\mathbf {R}}^d}

(in fact we need a little less than that). We also show how to get higher order terms in our expansion, but in this case

q
q

is required to be smoother. In the one-dimensional case our result extends a result of Deift and Trubowitz (see the [D-T, Appendix]), since they proved a similar formula under the additional assumption that

q
(
x
)
→
0

q(x) \to 0

as

|

x

|

→
∞

|x| \to \infty

. The asympotic formula we give implies that

q
q

can be recovered from certain spectral properties of

L
L

and

L
H

{L^H}

.
A 4-color theorem for surfaces of genus 𝑔
Authors: Berman Kenneth A. , Paul Jerome L.
Published: June 2010

For

G

\mathcal {G}

a set of graphs, define the bounded chromatic number

χ
B

(

G

)

{\chi _B}(\mathcal {G})

(resp. the bounded path chromatic number

χ
p

(

G

)

{\chi _p}(\mathcal {G})

) to be the minimum number of colors

c
c

for which there exists a constant

M
M

such that every graph

G
∈

G

G \in \mathcal {G}

can be vertex

c
c

-colored so that all but

M
M

vertices of

G
G

are properly colored (resp. the length of the longest monochromatic path in

G
G

is at most

M
M

). For

G

\mathcal {G}

the set of toroidal graphs, Albertson and Stromquist [1] conjectured that the bounded chromatic number is 4. For any fixed

g
≥
0

g \geq 0

, let

S

g

{\mathcal {S}_g}

denote the set of graphs of genus

g
g

. The Albertson-Stromquist conjecture can be extended to the conjecture that

χ
B

(

S

g

)
=
4

{\chi _B}({\mathcal {S}_g}) = 4

for all

g
≥
0

g \geq 0

. In this paper we show that

4
≤

χ
B

(

S

g

)
≤
6

4 \leq {\chi _B}({\mathcal {S}_g}) \leq 6

. We also show that the bounded path chromatic number

χ
p

(

S

g

)

{\chi _p}({\mathcal {S}_g})

equals 4 for all

g
≥
0

g \geq 0

. Let

μ
c

(
g
,
n
)
(

π
c

(
g
,
n
)
)

{\mu _c}(g,n)({\pi _c}(g,n))

denote the minimum

l
l

such that every graph of genus

g
g

on

n
n

vertices can be

c
c

-colored without forcing

l
+
1

l + 1

monochromatic edges (a monochromatic path of length

l
+
1

l + 1

). We also obtain bounds for

μ
c

(
g
,
n
)

{\mu _c}(g,n)

and

π
c

(
g
,
n
)

{\pi _c}(g,n)

.
Sieved orthogonal polynomials and discrete measures with jumps dense in an interval
Authors: Van Assche Walter , Magnus Alphonse P.
Published: June 2010

We investigate particular classes of sieved Jacobi polynomials for which the weight function vanishes at the zeros of a Chebyshev polynomial of the first kind. These polynomials are then used to give a proof, using only orthogonal polynomials on

[
−
1
,
1
]

[-1,1]

, that the discrete orthogonal polynomials introduced by Lubinsky have converging recurrence coefficients. We construct similar discrete measures with jumps dense in

[
−
1
,
1
]

[-1,1]

and use sieved ultraspherical polynomials to show that their recurrence coefficients converge.
Elliptic curves of bounded degree and height
Authors: Silverman Joseph H.
Published: June 2010

We show that there are only finitely many elliptic curves of bounded degree and height, provided that one takes a naive height defined in terms of minimal Weierstrass equations. We show that the corresponding statement is false if instead one uses the Faltings-Parshin modular height.
Microlocal hypo-analyticity and hypo-analytic pseudodifferential operators
Authors: Berhanu S.
Published: June 2010

Let

A
A

be a hypo-analytic pseudodifferential operator on a hypoanalytic manifold

Ω
\Omega

of maximal dimension. It is proved here that the operator

A
A

decreases the hypo-analytic wave front set of a distribution.
Even quadratic forms with cube-free discriminant
Authors: James Donald G.
Published: June 2010

A formula is given for the number of genera of even lattices with rank

n
n

, signature

s
s

, and discriminant

(
−
1

)

(
n
−
s
)

/

2

d
2

D

{( - 1)^{(n - s)/2}}{d^2}D

, when

d
D

dD

is odd and square-free. In the indefinite case, an orthogonal splitting of these lattices into simple components is also determined.
On the primitivity of group rings of amalgamated free products
Authors: Balogun Bola O.
Published: June 2010

In this paper we prove the primitivity of the group ring

F
[
G
]

F[G]

where

F
F

is a field and

G
G

belongs to a certain class of free products of groups with amalgamation studied recently by the author. Thus our result generalizes a result of Formanek on free products of groups.
Stone-Čech remainders which make continuous images normal
Authors: Fleissner William , Levy Ronnie
Published: June 2010

If

f
f

is a continuous surjection from a normal space

X
X

onto a regular space

Y
Y

, then there are a space

Z
Z

and a perfect map

b
f
:
Z
→
Y

bf:Z \to Y

extending

f
f

such that

X
⊂
Z
⊂
β
X

X \subset Z \subset \beta X

. If

f
f

is a continuous surjection from normal

X
X

onto Tychonov

Y
Y

and

β
X
∖
X

\beta X\backslash X

is sequential, then

Y
Y

is normal. More generally, if

f
f

is a continuous surjection from normal

X
X

onto regular

Y
Y

and

β
X
∖
X

\beta X\backslash X

has the property that countably compact subsets are closed (this property is called

C
C

-closed), then

Y
Y

is normal. There is an example of a normal space

X
X

such that

β
X
∖
X

\beta X\backslash X

is

C
C

-closed but not sequential. If

X
X

is normal and

β
X
∖
X

\beta X\backslash X

is first countable, then

β
X
∖
X

\beta X\backslash X

is locally compact.
Monotone closures of commutative 𝐶*-algebras
Authors: Hamana Masamichi
Published: June 2010

We show by an example that an analogue of Pedersen’s up-down-up theorem does not hold for monotone complete

C
∗

{C^*}

-algebras.
Maximum rank of powers of a matrix of a given pattern
Authors: Poljak Svatopluk
Published: June 2010

The pattern of a matrix is the structure of its zero and nonzero entries. For any prescribed pattern we determine the maximum possible rank of each power of a matrix with a given pattern. The approach is based on a combinatorial result that may be of some interest independently.
Immersed projective planes in lens spaces
Authors: Carter J. Scott
Published: June 2010

The minimum number of triple points of a substantial generic immersion of a projection plane in a lens space of type

(
2
k
,
q
)

(2k,q)

depends quadratically on

k
k

.
The Diophantine equation 𝑥²=4𝑞ⁿ+4𝑞^{𝑚}+1
Authors: Le Mao Hua
Published: June 2010

Let

q
=

p
r

q = {p^r}

be a prime power. In this paper, we prove that the equation

x
2

=
4

q
n

+
4

q
m

+
1
,
m
>
n
,

g

.c

.d

.

(
m
,
n
)
=
1

{x^2} = 4{q^n} + 4{q^m} + 1,m > n,{\text {g}}{\text {.c}}{\text {.d}}{\text {.}}(m,n) = 1

has no positive integer solution

(
m
,
n
,
x
)

(m,n,x)

with

m
>
2

m > 2

.
A note on joint hyponormality
Authors: McCullough Scott , Paulsen Vern
Published: June 2010

We describe certain cones of polynomials in two variables naturally associated to the class(es) of operators

T
T

for which the tuple

(
T
,

T
2

,
…
,

T
n

)

(T,{T^2}, \ldots ,{T^n})

is jointly (weakly) hyponormal. As an application we give an example of an operator

T
T

such that the tuple

(
T
,

T
2

)

(T,{T^2})

is jointly but not weakly hyponormal. Further, we show that there exists a polynomially hyponormal operator which is not subnormal if and only if there exists a weighted shift with the same property.
Countably additive full conditional probabilities
Authors: Armstrong Thomas E.
Published: June 2010

Armstrong-Sudderth showed that a method of Carlson could be adapted to construct examples of countably additive full conditional probabilities. Here it is shown that all countably additive full conditional probabilities arise in this fashion. It is also shown that if the dimensionally ordered family of measures associated by Renyi to a full conditional probability contains an unbounded measure the full conditional probability fails to be countably additive. This leads to the notion of proper full conditional probabilities which if well founded in the sense of Krauss agree with the class of full conditional probabilities arising from Carlson’s construction. Section 3 explores the connection between coherence of countable betting systems and countable additivity of full conditional probabilities.
Some geometric aspects of hyperbolic boundary value problems
Authors: Conrey J. Brian , Smiley Michael W.
Published: June 2010

Let

L
L

denote the linear operator associated with the wave equation when it is subjected to boundary conditions in both space and time. The properties of invertibility or partial invertibility of

L
L

, and compactness of the (partial) inverse when it exists, are characterized in terms of the space time domain

Ω
×
(
0
,
T
)

\Omega \times (0,T)

, for all rectangular domains

Ω
⊂

R

n

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

.