bluuebunny/crossref_metadata_2025 huggingface dataset:
Trees, Gleason spaces, and coabsolutes of 𝛽𝑁∼𝑁
Authors: Williams Scott W.
Published: June 2010

For a regular Hausdorff space

X
X

, let

E

(
X
)

\mathcal {E}(X)

denote its absolute, and call two spaces

X
X

and

Y
Y

coabsolute (

G

\mathcal {G}

-absolute) when

E

(
X
)

\mathcal {E}(X)

and

E

(
Y
)

\mathcal {E}(Y)

(

β

E

(
X
)

\beta \mathcal {E}(X)

and

β

E

(
Y
)

\beta \mathcal {E}(Y)

) are homeomorphic. We prove

X
X

is

G

\mathcal {G}

-absolute with a linearly ordered space iff the POSET of proper regular-open sets of

X
X

has a cofinal tree; a Moore space is

G

\mathcal {G}

-absolute with a linearly ordered space iff it has a dense metrizable subspace; a dyadic space is

G

\mathcal {G}

-absolute with a linearly ordered space iff it is separable and metrizable; if

X
X

is a locally compact noncompact metric space, then

β
X
∼
X

\beta X \sim X

is coabsolute with a compact linearly ordered space having a dense set of

P
P

-points; CH implies but is not implied by "if

X
X

is a locally compact noncompact space of

π
\pi

-weight at most

2
ω

{2^\omega }

and with a compatible complete uniformity, then

β
X
∼
X

\beta X \sim X

and

β
N
∼
N

\beta N \sim N

are coabsolute."
Singular elliptic operators of second order with purely discrete spectra
Authors: Lewis Roger T.
Published: June 2010

The Friedrichs extension of a second order singular elliptic operator is considered on a weighted

L

w
2

(
Ω
)

\mathcal {L}_w^2(\Omega )

space. The region

Ω
\Omega

is not necessarily bounded. Necessary conditions and sufficient conditions on the coefficients that will insure a discrete spectrum are given with a certain degree of sharpness achieved. The boundary conditions include the Dirichlet, Neumann, and mixed Dirichlet-Neumann boundary value problems.
Cellular maps between polyhedra
Authors: Henderson James P.
Published: June 2010

A compact subset

X
X

of a polyhedron

P
P

is cellular in

P
P

if there is a pseudoisotopy of

P
P

shrinking precisely

X
X

to a point. A proper surjection

f
:
P
→
Q

f:P\rightarrow Q

is cellular if each point inverse of

f
f

is cellular in

P
P

. We give certain conditions under which cellular maps between polyhedra are approximable by homeomorphisms. An example of a cellular map which is not approximable by homeomorphisms is also given.
Invariant operators and univalent functions
Authors: Harmelin Reuven
Published: June 2010

Necessary and sufficient conditions for univalence of meromorphic functions in certain domains in the complex plane are established in terms of some differential operators of degrees

≥
3

\ge 3

, possessing the same invariance property as the Schwarzian derivative. Those operators include the derivatives of the Schwarzian derivative and Aharonov’s invariants. Conditions for the existence of quasiconformal extensions are also achieved.
Algebraic and geometric models for 𝐻₀-spaces
Authors: Aguadé J. , Zabrodsky A.
Published: June 2010

For every

H
0

{H_0}

-space (i.e. a space whose rationalization is an

H
H

-space) we construct a space

J
J

depending only on

H
∗

(
X
;

Z

)

{H^\ast }(X;{\mathbf {Z}})

and a rational homotopy equivalence

J
→
X

J \to X

(i.e.

J
J

is a universal space to the left of all

H
0

{H_0}

-spaces having the same integral cohomology ring as

X
X

is constructed generalizing the James reduced product. We study also the integral cohomology of

H
0

{H_0}

-spaces and we prove that under certain conditions it contains an algebra with divided powers.
No division implies chaos
Authors: Li Tien Yien , Misiurewicz Michał , Pianigiani Giulio , Yorke James A.
Published: June 2010

Let

I
I

be a closed interval in

R
1

{R^1}

and

f
:
I
→
I

f:I \to I

be continuous. Let

x
0

∈
I

{x_0} \in I

and
\[

x

i
+
1

=
f
(

x
i

)

for

i
>
0.

{x_{i + 1}} = f({x_i})\quad {\text {for}}\;i > 0.

\]
We say there is no division for

(

x
1

,

x
2

,
…
,

x
n

)

({x_1},{x_2}, \ldots ,{x_n})

if there is no

a
∈
I

a \in I

such that

x
j

>
a

{x_j} > a

for all

j
j

even and

x
j

>
a

{x_j} > a

for all

j
j

odd. The main result of this paper proves the simple statement: no division implies chaos. Also given here are some converse theorems, detailed estimates of the existing periods, and examples which show that, under our conditions, one cannot do any better.
If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal
Authors: Fleissner William G.
Published: June 2010

We formulate an axiom, HYP, and from it construct a normal, metacompact, nonmetrizable Moore space. HYP unifies two better known axioms. The Continuum Hypothesis implies HYP; the nonexistence of an inner model with a measurable cardinal implies HYP. As a consequence, it is impossible to replace Nyikos’ "provisional" solution to the normal Moore space problem with a solution not involving large cardinals. Finally, we discuss how this space continues a process of lowering the character for normal, not collectionwise normal spaces.
Equivariant minimal models
Authors: Triantafillou Georgia V.
Published: June 2010

Let

G
G

be a finite group. We give an algebraicization of rational

G
G

-homotopy theory analogous to Sullivan’s theory of minimal models in ordinary homotopy theory.
On extending free group actions on spheres and a conjecture of Iwasawa
Authors: Connolly Frank , Geist Robert
Published: June 2010

A transfer map for Reidemeister torsion is defined and used to determine whether free actions of

Z

/

k

\mathbf {Z}/k

on

S

2
n
+
1

S^{2n+1}

,

n
>
1

n > 1

, extend to free actions of

Z

/

h
k

\mathbf {Z}/hk

. It is shown that for

k
k

odd, every free

Z

/

k

\mathbf {Z}/k

action on

S

2
n
+
1

S^{2n+1}

,

n
>
1

n > 1

, extends to a free

Z

/

2
k

\mathbf {Z}/2k

action. For prime

p
p

, extension of an arbitrary free

Z

/

p

\mathbf {Z}/p

action to a free

Z

/

p

2

\mathbf {Z}/p^{2}

action is reduced to a long-standing conjecture of Iwasawa.
On the group of real analytic diffeomorphisms of a compact real analytic manifold
Authors: Leslie J.
Published: June 2010

In this paper we exhibit a Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold. Further, we show that a variant of the Kupka-Smale theorem holds for the underlying topology of the above mentioned Lie group.
Unstable bordism groups and isolated singularities
Authors: Ellis David
Published: June 2010

An isolated singularity of an embedded submanifold can be topologically smoothed if and only if a certain obstruction element in

π
∗

(
M
G
)

{\pi _ \ast }(MG)

vanishes, where

G
G

is the group of the normal bundle. In fact this obstruction lies in a certain subgroup which is referred to here as the unstable

G
G

-bordism group. In this paper some of the unstable

S
O

SO

-bordism groups are computed; the obstruction to smoothing the complex cone on an oriented submanifold

X
⊂

C

P
n

X \subset \mathbf {C}{P^n}

at

∞
\infty

is computed in terms of the characteristic numbers of

X
X

. Examples of nonsmoothable complex cone singularities are given using these computations.
Markov processes with identical hitting probabilities
Authors: Glover Joseph
Published: June 2010

Let

(
X
(
t
)
,

P
x

)

(X(t),{P^x})

and

(
Y
(
t
)
,

Q
x

)

(Y(t),{Q^x})

be transient Hunt processes on a state space

E
E

satisfying the hypothesis of absolute continuity (Meyer’s hypothesis (L)). Let

T
(
K
)

T(K)

be the first entrance time into a set

K
K

, and assume
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P Superscript x Baseline left-parenthesis upper T left-parenthesis upper K right-parenthesis greater-than normal infinity right-parenthesis equals upper Q Superscript x Baseline left-parenthesis upper T left-parenthesis upper K right-parenthesis greater-than normal infinity right-parenthesis">

P
x

(
T
(
K
)
>
∞
)
=

Q
x

(
T
(
K
)
>
∞
)

{P^x}(T(K) > \infty ) = {Q^x}(T(K) > \infty )

for all compact sets

K
⊆
E

K \subseteq E

. There exists a strictly increasing continuous additive functional of

X
(
t
)
,
A
(
t
)

X(t),A(t)

, so that if

T
(
t
)
=

inf

{
s
:
A
(
s
)
>
t
}

T(t) = {\text {inf}}\{s:A(s) > t\}

, then

(
X
(
T
(
t
)
)
,

P
x

)

(X(T(t)),{P^x})

and

(
Y
(
t
)
,

Q
x

)

(Y(t),{Q^x})

have the same joint distributions. An analogous result is stated if

X
X

and

Y
Y

are right processes (with an additional hypothesis). These theorems generalize the Blumenthal-Getoor-McKean Theorem and have interpretations in terms of potential theory.
Normal subgroups of 𝐷𝑖𝑓𝑓^{Ω}(𝑅ⁿ)
Authors: Mascaró Francisca
Published: June 2010

Let

Ω
\Omega

be a volume element on

R

n

{{\mathbf {R}}^n}

.

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

is the group of

Ω
\Omega

-preserving diffeomorphisms of

R

n

{{\mathbf {R}}^n}

.

Diff

W
Ω

(

R

n

)

{\text {Diff}}_W^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements whose set of nonfixed points has finite

Ω
\Omega

-volume.

Diff

f
Ω

(

R

n

)

{\text {Diff}}_f^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements whose support has finite

Ω
\Omega

-volume.

Diff

c
Ω

(

R

n

)

{\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements with compact support.

Diff

co

Ω

(

R

n

)

{\text {Diff}}_{{\text {co}}}^\Omega ({{\mathbf {R}}^n})

is the subgroup of all elements compactly

Ω
\Omega

-isotopic to the identity. We prove, in the case

vo

l

Ω

R

n

>
∞

{\text {vo}}{{\text {l}}_{\Omega }}{{\mathbf {R}}^n} > \infty

and for

n

⩾

3

{\text {n}} \geqslant {\text {3}}

that any subgroup of

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

,

N
N

, is normal if and only if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Diff Subscript co Superscript normal upper Omega Baseline left-parenthesis bold upper R Superscript n Baseline right-parenthesis subset-of upper N subset-of Diff Subscript c Superscript normal upper Omega Baseline left-parenthesis bold upper R Superscript n Baseline right-parenthesis">

Diff

co

Ω

(

R

n

)
⊂
N
⊂

Diff

c
Ω

(

R

n

)

{\text {Diff}}_{{\text {co}}}^\Omega ({{\mathbf {R}}^n}) \subset N \subset {\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

. If

vo

l

Ω

R

n

=
∞

{\text {vo}}{{\text {l}}_{\Omega }}{{\mathbf {R}}^n} = \infty

, any subgroup of

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

,

N
N

, satisfying
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Diff Subscript co Superscript normal upper Omega Baseline left-parenthesis bold upper R Superscript n Baseline right-parenthesis subset-of upper N subset-of Diff Subscript c Superscript normal upper Omega Baseline left-parenthesis bold upper R Superscript n Baseline right-parenthesis">

Diff

co

Ω

(

R

n

)
⊂
N
⊂

Diff

c
Ω

(

R

n

)

{\text {Diff}}_{{\text {co}}}^\Omega ({{\mathbf {R}}^n}) \subset N \subset {\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

is normal, for

n
⩾

3

n \geqslant {\text {3}}

, there are no normal subgroups between

Diff

W
Ω

(

R

n

)

{\text {Diff}}_W^\Omega ({{\mathbf {R}}^n})

and

Dif

f

Ω

(

R

n

)

{\text {Dif}}{{\text {f}}^\Omega }({{\mathbf {R}}^n})

and for

n
⩾
4

n \geqslant 4

there are no normal subgroups between

Diff

c
Ω

(

R

n

)

{\text {Diff}}_c^\Omega ({{\mathbf {R}}^n})

and

Diff

f
Ω

(

R

n

)

{\text {Diff}}_f^\Omega ({{\mathbf {R}}^n})

.
The number of factorizations of numbers less than 𝑥 into factors less than 𝑦
Authors: Hensley Douglas
Published: June 2010

Let

K
(
x
,
y
)

K(x,y)

be the number in the title. There is a function

f
(
r
)

f(r)

, concave and decreasing with

f
(
0
)
=
2

f(0) = 2

and

f
′

(
0
)
=
0

f’(0) = 0

such that if

r
=

log
⁡
x

/

log
⁡
y

r = \sqrt {\log x} /\log y

then as

x
→
∞

x \to \infty

with

r
r

fixed,
\[

K
(
x
,
y
)
=
x
exp

(

f
(
r
)

log
⁡
x

+
O

(
log
⁡
log
⁡
x
)

2

)

K(x,y) = x \exp \,\left ({f(r)\,\sqrt {\log x} + O\,{{(\log \log x)}^2}} \right )

\]
. The proof uses a uniform version of Chernoff’s theorem on large deviations from the sample mean of a sum of

N
N

independent random variables.
Generic algebras
Authors: Isbell John
Published: June 2010

The familiar (merely) generic algebras in a variety

V

\mathcal {V}

are those which separate all the different operations of

V

\mathcal {V}

, or equivalently lie in no proper Birkhoff subcategory. Stronger notions are considered, the strongest being canonicalness of a (small) subcategory

A

\mathcal {A}

of

V

\mathcal {V}

, defined: the structure functor takes inclusion

A

⊂

V

\mathcal {A} \subset \mathcal {V}

to an isomorphism of varietal theories. Intermediate are dominance and exemplariness: lying in no proper varietal subcategory, respectively full subcategory. It is shown that, modulo measurable cardinals, every finitary variety has a canonical set (subcategory) of one or two algebras, the possible second one being the empty algebra. Without reservation, every variety with rank has a dominant set of one or two algebras (the second as before). Finally, in modules over a ring

R
R

, the generic module

R
R

is shown to be (a) dominant if exemplary, and (b) dominant if

R
R

is countable or right artinian. However, power series rings

R
R

and some others are not dominant

R
R

-modules.
Characterizations of simply connected rotationally symmetric manifolds
Authors: Choi Hyeong In
Published: June 2010

We prove that a simply connected, complete Riemannian manifold

M
M

is rotationally symmetric at

p
p

if and only if the exponential image of every linear subspace of

M
p

{M_p}

is a smooth, closed, totally geodesic submanifold of

M
M

. This result is in essence Schur’s theorem at one point

p
p

, as it becomes apparent in the proof.
A general maximal operator and the 𝐴_{𝑝}-condition
Authors: Leckband M. A. , Neugebauer C. J.
Published: June 2010

A rearrangement inequality for a general maximal operator

M
f
(
x
)
=

sup

x
∈
Q

∫

f

ϕ

Q

d
ν

Mf(x) = {\sup _{x \in Q}}\int {f\phi _{Q}\,d\nu }

is established. This is then applied to the Hardy-Littlewood maximal operator with weights.
An algebraic classification of certain simple even-dimensional knots
Authors: Kearton C.
Published: June 2010

The simple

2
q

2q

-knots,

q
⩾
4

q \geqslant 4

, for which

H
q

(

K
~

)

{H_q}(\tilde {K})

contains no

Z

{\mathbf {Z}}

-torsion, are classified by means of Hermitian duality pairings on their homology and homotopy modules.
A necessary and sufficient condition for the asymptotic version of Ahlfors’ distortion property
Authors: Rodin Burton , Warschawski S. E.
Published: June 2010

Let

f
f

be a conformal map of

R
=
{
w
=
u
+
i
v
∈

C

|

φ
0

(
u
)
>
v
>

φ
1

(
u
)
}

R = \{w = u + iv \in {\mathbf {C}}|{\varphi _0}(u) > v > {\varphi _1}(u)\}

onto

S
=
{
z
=
x
+
i
y
∈

C

|

0
>
y
>
1
}

S = \{z = x + iy \in {\mathbf {C}}|0 > y > 1\}

where the

φ
j

∈

C
0

(
−
∞
,
∞
)

{\varphi _j} \in {C^0}( - \infty ,\infty )

and

Re
⁡
f
(
w
)
→
±
∞

\operatorname {Re} f(w) \to \pm \infty

as

Re
⁡
w
→
±
∞

\operatorname {Re} w \to \pm \infty

. There are well-known results giving conditions on

R
R

sufficient for the distortion property
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R e f left-parenthesis u plus i v right-parenthesis equals integral Subscript 0 Superscript u Baseline left-parenthesis phi 1 minus phi 0 right-parenthesis Superscript negative 1 Baseline d u plus const period plus o left-parenthesis 1 right-parenthesis">

Re
⁡
f
(
u
+
i
v
)
=

∫
0
u

(

φ
1

−

φ
0

)

−
1

d
u
+

const

.
+
o
(
1
)

\operatorname {Re} f(u + iv) = \int _0^u ({\varphi _1} - {\varphi _0})^{- 1}du + {\text {const}}. + o(1)

, where

o
(
1
)
→
0

o(1) \to 0

as

u
→
+
∞

u \to + \infty

. In this paper the authors give a condition on

R
R

which is both necessary and sufficient for

f
f

to have this property.
The dispersion of the coefficients of univalent functions
Authors: Hamilton D. H.
Published: June 2010

The Hayman

T
a

{T_a}

function for the asymptotic distribution of the coefficients of univalent functions has a continuous derivative which is closely related to the asymptotic behavior of coefficient differences.
Products of powers of nonnegative derivatives
Authors: Mařík Jan , Weil Clifford E.
Published: June 2010

This paper contains some results concerning functions that can be written as

f
1

β
1

⋯

f
n

β
n

f_1^{{\beta _1}} \cdots f_n^{{\beta _n}}

, where

n
n

is an integer greater than

1
1

,

f
j

{f_j}

are nonnegative derivatives and

β
j

{\beta _j}

are positive numbers. If we choose

β
1

=
⋯
=

β
n

=
1

{\beta _1} = \cdots = {\beta _n} = 1

, we obtain theorems about products of nonnegative derivatives.
Branched coverings. I
Authors: Stong R. E.
Published: June 2010

This paper analyzes the possible cobordism classes

[
M
]
−
(
deg

ϕ
)
[
N
]

[M] - (\deg \;\phi )[N]

for

ϕ
:
M
→
N

\phi : M \to N

a smooth branched covering of closed smooth manifolds. It is assumed that the branch set is a codimension

2
2

submanifold. The results are a fairly complete description in the unoriented case, a partial description in the oriented case, and a detailed analysis of the case in which

N
N

is a sphere.
Domain Bloch constants
Authors: Minda C. David
Published: June 2010

The classical Bloch constant

B

\mathcal {B}

is defined for holomorphic functions

f
f

defined on

B

=
{
z
:

|

z

|

>
1
}

{\mathbf {B}} = \{z:|z| > 1\}

and normalized by

|

f
′

(
0
)

|

=
1

|f’(0)| = 1

. Let

R
f

{R_f}

denote the Riemann surface of

f
f

and

B
f

{B_f}

the set of branch points. Then

B

\mathcal {B}

can be regarded as a lower bound for the radius of the largest disk contained in

R
f

∖

B
f

{R_f}\backslash {B_f}

. The metric on

R
f

{R_f}

used to measure the size of disks on

R
f

{R_f}

is obtained by lifting the euclidean metric from

C

{\mathbf {C}}

to

R
f

{R_f}

. The surface

R
f

{R_f}

can also be regarded as spread over

B

{\mathbf {B}}

and the hyperbolic metric lifted to

R
f

{R_f}

. One may then ask for the radius of the largest hyperbolic disk on

R
f

∖

B
f

{R_f}\backslash {B_f}

. A lower bound for this radius is called a domain Bloch constant. The determination of domain Bloch constants is nontrivial for nonconstant analytic functions

f
:

B

→
X

f:{\mathbf {B}} \to X

, where

X
X

is a hyperbolic Riemann surface. Upper and lower bounds for domain Bloch constants are given. Also, domain Bloch constants are given an interpretation as a radius of local schlichtness.
Efficient computation in groups and simplicial complexes
Authors: Stillwell John C.
Published: June 2010

Using HNN extensions of the Boone-Britton group, a group

E
E

is obtained which simulates Turing machine computation in linear space and cubic time. Space in

E
E

is measured by the length of words, and time by the number of substitutions of defining relators and conjugations by generators required to convert one word to another. The space bound is used to derive a PSPACE-complete problem for a topological model of computation previously used to characterize NP-completeness and RE-completeness.
Conjugate Fourier series on certain solenoids
Authors: Hewitt Edwin , Ritter Gunter
Published: June 2010

We consider an arbitrary noncyclic subgroup of the additive groupQ{\mathbf {Q}}of rational numbers, denoted byQa{{\mathbf {Q}}_{\mathbf {a}}}, and its compact character groupΣa{\Sigma _{\mathbf {a}}}. For1>p>∞1 > p > \infty, an abstract form of Marcel Riesz’s theorem on conjugate series is known. ForffinLp(Σa){\mathfrak {L}_p}({\Sigma _{\mathbf {a}}}), there is a functionf~\tilde {f}inLp(Σa){\mathfrak {L}_p}({\Sigma _{\mathbf {a}}})whose Fourier transform(f~)^(α)(\tilde {f})^{\hat {}}(\alpha )atα\alphainQa{{\mathbf {Q}}_{\mathbf {a}}}is−isgnαf^(α)- i\,\operatorname {sgn}\,\alpha \hat {f}(\alpha ). We show in this paper how to constructf~\tilde {f}explicitly as a pointwise limit almost everywhere onΣa{\Sigma _{\mathbf {a}}}of certain harmonic functions, as was done by Riesz for the circle group. Some extensions of this result are also presented.
Highest weight modules over graded Lie algebras: resolutions, filtrations and character formulas
Authors: Rocha-Caridi Alvany , Wallach Nolan R.
Published: June 2010

In this paper the study of multiplicities in Verma modules for Kac-Moody algebras is initiated. Our analysis comprises the case when the integral root system is Euclidean of rank two. Complete results are given in the case of rank two, Kac-Moody algebras, affirming the Kazhdan-Lusztig conjectures for the case of infinite dihedral Coxeter groups. The main tools in this paper are the resolutions of standard modules given in [21] and a generalization to the case of Kac-Moody Lie algebras of Jantzen’s character sum formula for a quotient of two Verma modules (one of the main results of this article). Finally, a precise analogy is drawn between the rank two, Kac-Moody algebras and the Witt algebra (the Lie algebra of vector fields on the circle).
Conservation laws of free boundary problems and the classification of conservation laws for water waves
Authors: Olver Peter J.
Published: June 2010

The two-dimensional free boundary problem for incompressible irrotational water waves without surface tension is proved to have exactly eight nontrivial conservation laws. Included is a discussion of what constitutes a conservation law for a general free boundary problem, and a characterization of conservation laws for two-dimensional free boundary problems involving a harmonic potential proved using elementary methods from complex analysis.
The Atiyah-Singer invariant, torsion invariants, and group actions on spheres
Authors: Smith Donald E.
Published: June 2010

This paper deals with the classification of cyclic group actions on spheres using the Atiyah-Singer invariant and Reidemeister-type torsion. Our main tool is the computation of the group of relative homotopy triangulations of the product of a disk and a lens space. These results are applied to obtain lower bounds on the image of an equivariant

J
J

-homomorphism.
Rotation hypersurfaces in spaces of constant curvature
Authors: do Carmo M. , Dajczer M.
Published: June 2010

Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the belicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space

H
3

H^3

; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in

H
3

H^3

.
Isomorphism types in wreath products and effective embeddings of periodic groups
Authors: Hickin Kenneth K. , Phillips Richard E.
Published: June 2010

For any finitely generated group

Y
,
ω
(
Y
)

Y,\omega (Y)

denotes the Turing degree of the word problem of

Y
Y

. Let

G
G

be any non-Abelian

2
2

-generator group and

B
B

an infinite group generated by

k
⩾
1

k \geqslant 1

elements. We prove that if

τ
\tau

is any Turing degree with

τ
⩾
1.

u.b.

{

ω
(
G
)
,
ω
(
B
)

}

\tau \geqslant 1.{\text {u.b.}}\{ {\omega (G),\omega (B)} \}

then the unrestricted wreath product

W
=
G

Wr

B

W = G{\text {Wr}}\,B

has a

(

k
+
1

)

( {k + 1} )

-generator subgroup

H
H

with

ω
(
H
)
=
τ

\omega (H) = \tau

. If

B
B

is also periodic, then

W
W

has a

k
k

-generator subgroup

H
H

such that

τ
=
1.

u.b.

{

ω
(
B
)
,
ω
(
H
)

}

\tau = 1.{\text {u.b.}}\{ {\omega (B),\omega (H)} \}

. Easy consequences include:

G

Wr

Z

G{\text {Wr}}\,{\mathbf {Z}}

has

2

ℵ
0

{2^{{\aleph _0}}}

pairwise nonembeddable

2
2

-generator subgroups and if

B
B

is periodic then

G

Wr

B

G{\text {Wr}}\,B

has

2

ℵ
0

{2^{{\aleph _0}}}

pairwise nonembeddable

k
k

-generator subgroups. Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in

2
2

-generator periodic groups.
Stop rule inequalities for uniformly bounded sequences of random variables
Authors: Hill Theodore P. , Kertz Robert P.
Published: June 2010

If

X

0

,

X
1

,
…

{X_{0}},{X_1},\ldots

is an arbitrarily-dependent sequence of random variables taking values in

[
0
,
1
]

[0,1]

and if

V
(

X
0

,

X
1

,
…
)

V({X_0},{X_1},\ldots )

is the supremum, over stop rules

t
t

, of

E

X
t

E{X_t}

, then the set of ordered pairs

{
(
x
,
y
)
:
x
=
V
(

X
0

,

X
1

,
…
,

X
n

)

\{ (x,y):x = V({X_0},{X_1},\ldots ,{X_n})

and

y
=
E
(

max

j

⩽

n

X
j

)

y = E({\max _{j\, \leqslant \,n}}{X_j})

for some

X
0

,
…
,

X
n

}

{X_0},\ldots ,{X_n}\}

is precisely the set
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript n Baseline equals left-brace left-parenthesis x comma y right-parenthesis colon x less-than-or-slanted-equals y less-than-or-slanted-equals x left-parenthesis 1 plus n left-parenthesis 1 minus x Superscript 1 slash n Baseline right-parenthesis right-parenthesis semicolon 0 less-than-or-slanted-equals x less-than-or-slanted-equals 1 right-brace semicolon">

C
n

=
{
(
x
,
y
)
:
x
⩽
y
⩽
x

(

1
+
n

(
1
−

x

1

/

n

)

)
;
0
⩽
x
⩽
1
}
;

{C_n} = \{ (x,y):x \leqslant y \leqslant x\,( {1 + n\,(1 - {x^{1/n}})} );0 \leqslant x \leqslant 1\} ;

\]
and the set of ordered pairs

{
(
x
,
y
)
:
x
=
V
(

X

0

,

X
1

,
…
)

\{ (x,y):x = V({X_{0}},{X_1},\ldots )

and

y
=
E
(

sup
n

X
n

)

y = E({\sup _n}\;{X_n})

for some

X
0

,

X
1

,
…
}

{X_0},{X_1},\ldots \}

is precisely the set
\[

C
=

⋃

n
=
1

∞

C
n

.

C = \bigcup \limits _{n = 1}^\infty {{C_n}} .

\]
As a special case, if

X
0

,

X
1

,
…

{X_0},{X_1},\ldots

is a martingale with

E

X
0

=
x

E{X_0} = x

, then

E
(

max

j
⩽
n

X
)
⩽
x
+
n
x
(
1
−

x

1

/

n

)

E({\max _{j \leqslant n}} X) \leqslant x + nx(1 - {x^{1\,/\,n}})

and

E
(

sup
n

X
n

)
⩽
x
−
x
ln

x

E({\sup _n}\;{X_n}) \leqslant x - x\ln \;x

, and both inequalities are sharp.
Convergence of linear and nonlinear Padé approximants from series of orthogonal polynomials
Authors: Lubinsky D. S. , Sidi A.
Published: June 2010

Analogues of the Nuttall-Pommerenke theorem and Wallin-type theorems for classical Padé approximants, are proved for linear and nonlinear Padé approximants formed from series of orthogonal polynomials, corresponding to a distribution

d
α
(
x
)

d\alpha (x)

with at most finitely many sign changes.
A note on Michael’s problem concerning the Lindelöf property in the Cartesian products
Authors: Alster K.
Published: June 2010

In this note we present a sketch of a negative solution of the Michael’s conjecture which says that if the product

Y
×
X

Y \times X

is Lindelöf for every hereditarily Lindelöf space

Y
Y

, then

Y
×

X
ω

Y \times {X^\omega }

is Lindelöf for every hereditarily Lindelöf space

Y
Y

.
Measure and category approximations for 𝐶-sets
Authors: Srivatsa V. V.
Published: June 2010

The class of

C
C

-sets in a Polish space is the smallest

σ
\sigma

-field containing the Borel sets and closed under operation

(

A

)

(\mathcal {A})

. In this article we show that any

C
C

-set in the product of two Polish spaces can be approximated (in measure and category), uniformly over all sections, by sets generated by rectangles with one side a

C
C

-set and the other a Borel set. Such a formulation unifies many results in the literature. In particular, our methods yield a simpler proof of a selection theorem for

C
C

-sets with

G
δ

{G_\delta }

-sections due to Burgess [4].
Conformally flat manifolds with nilpotent holonomy and the uniformization problem for 3-manifolds
Authors: Goldman William M.
Published: June 2010

A conformally flat manifold is a manifold with a conformal class of Riemannian metrics containing, for each point

x
x

, a metric which is flat in a neighborhood of

x
x

. In this paper we classify closed conformally flat manifolds whose fundamental group (more generally, holonomy group) is nilpotent or polycyclic of rank

3
3

. Specifically, we show that such conformally flat manifolds are covered by either the sphere, a flat torus, or a Hopf manifold—in particular, their fundamental groups contain abelian subgroups of finite index. These results are applied to show that certain

T
2

{T^2}

-bundles over

S
1

{S^1}

(namely, those whose attaching map has infinite order) do not have conformally flat structures. Apparently these are the first examples of

3
3

-manifolds known not to admit conformally flat structures.
The structure of 𝜔₁-separable groups
Authors: Eklof Paul C.
Published: June 2010

A classification theorem is proved for

ω
1

{\omega _1}

-separable

ω
1

{\omega _1}

-free abelian groups of cardinality

ω
1

{\omega _1}

assuming Martin’s Axiom

(
MA
)

(\text {MA})

and

2

ℵ
0

>

ℵ
1

{2^{\aleph _0}} > {\aleph _1}

. As a consequence, several structural results about direct sum decompositions of

ω
1

{\omega _1}

-separable groups are proved. These results are proved independent of

ZFC
\text {ZFC}

, and, in addition, another structural property is proved undecidable in

ZFC

+

MA

+

2

ℵ
0

>

ℵ
1

{\text {ZFC}} + {\text {MA}} + {2^{\aleph _0}} > {\aleph _1}

. The problem of classifying these groups in a model of

2

ℵ
0

=

ℵ
1

{2^{\aleph _0}} = {\aleph _1}

is also investigated.
A generalization of 𝐹-spaces and some topological characterizations of GCH
Authors: Swardson Mary Anne
Published: June 2010

Several topological characterizations involving

F
F

-spaces of the continuum hypothesis are due to R. G Woods and E. K. van Douwen. We extend this work by defining a space

X
X

to be an

F
α

{F_\alpha }

-space if the union of

>
α

> \alpha

cozero-sets is

C

∗

{C^{\ast }}

-embedded in

X
X

and by giving, for every infinite cardinal

α
\alpha

, topological characterizations involving

F
α

{F_\alpha }

-spaces of the cardinal equality

2
α

=

α
+

{2^\alpha } = {\alpha ^ + }

.
Nonimmersions and nonembeddings of quaternionic spherical space forms
Authors: Kobayashi Teiichi
Published: June 2010

We determine the orders of the canonical elements in

K
O

KO

-rings of quaternionic spherical space forms

S

4
n
+
3

/

Q
k

{S^{4n + 3}}/{Q_k}

and apply them to prove the nonexistence theorems of immersions and embeddings of

S

4
n
+
3

/

Q
k

{S^{4n + 3}}/{Q_k}

in Euclidean spaces.
All three-manifolds are pullbacks of a branched covering 𝑆³ to 𝑆³
Authors: Hilden Hugh M. , Lozano María Teresa , Montesinos José María
Published: June 2010

There are two main results in this paper. First, we show that every closed orientable

3
3

-manifold can be constructed by taking a pair of disjoint bounded orientable surfaces in

S
3

{S^3}

, call them

F
1

{F_1}

and

F
2

{F_2}

; taking three copies of

S
3

{S^3}

; splitting the first along

F
1

{F_1}

, the second along

F
1

{F_1}

and

F
2

{F_2}

, and the third along

F
2

{F_2}

; and then pasting in the natural way. Second, we show that given any closed orientable

3
3

-manifold

M
3

{M^3}

there is a

3
3

-fold irregular branched covering space,

p
:

M
3

→

S
3

p:{M^3} \to {S^3}

, such that

p
:

M
3

→

S
3

p:{M^3} \to {S^3}

is the pullback of the

3
3

-fold irregular branched covering space

q
:

S
3

→

S
3

q:{S^3} \to {S^3}

branched over a pair of unknotted unlinked circles.
Qualitative differentiation
Authors: Evans Michael J. , Larson Lee
Published: June 2010

Qualitative derivates and derivatives, as well as qualitative symmetric derivates and derivatives, are studied in the paper. Analogues of several results known for ordinary derivates and derivatives are obtained in the qualitative setting.
On certain sums of Fourier-Stieltjes coefficients
Authors: Twomey J. B.
Published: June 2010

We obtain estimates for certain sums of Fourier-Stieltjes (and hence also Fourier) coefficients of continuous functions

f
f

of bounded variation in terms of the modulus of continuity of

f
f

. As a consequence of one of our results we obtain an improvement on a theorem of Zygmund on the absolute convergence of Fourier series of functions of bounded variation. We also consider absolutely continuous functions and show by examples that a number of the results we obtain are "best possible".
Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem
Authors: Hilliker David Lee , Straus E. G.
Published: June 2010

In 1887 Runge [13] proved that a binary Diophantine equation

F
(
x
,
y
)
=
0

F(x,y) = 0

, with

F
F

irreducible, in a class including those in which the leading form of

F
F

is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge’s method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions

x
x

and

y
y

. Runge did not give such a computation. Here we first deduce Runge’s Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge’s Theorem in which the solutions

x
x

and

y
y

of the polynomial equation

F
(
x
,
y
)
=
0

F(x,y) = 0

are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions

(
x
,
y
)
∈

Z

2

(x,y) \in {{\mathbf {Z}}^2}

in terms of the height of

F
F

and the degrees in

x
x

and

y
y

of

F
F

.
The rank of a Hardy field
Authors: Rosenlicht Maxwell
Published: June 2010

A Hardy field is a field of germs of real-valued functions on positive half-lines that is closed under differentiation. Its rank is the rank of the associated ordered abelian group, the value group of the canonical valuation of the field. The properties of this rank are worked out, its relevance to asymptotic expansions indicated, examples provided, and applications given to the order of growth of solutions of ordinary differential equations.
Additivity of measure implies additivity of category
Authors: Bartoszyński Tomek
Published: June 2010

In this paper it is proved that

2
ω

{2^\omega }

-additivity of category follows from

2
ω

{2^\omega }

-additivity of measure, and a combinatorial characterization of additivity of measure is found.
Weak restricted and very restricted operators on 𝐿²
Authors: Ash J. Marshall
Published: June 2010

A battlement is a real function with values in

{
0
,
1
}

\{ 0,1\}

that looks like a castle battlement. A commuting with translation linear operator

T
T

mapping step functions on

R

{\mathbf {R}}

into the set of all measurable functions on

R

{\mathbf {R}}

and satisfying

∥
T
b

∥
2

⩽
C
∥
b

∥
2

\parallel Tb{\parallel _2} \leqslant C\parallel b{\parallel _2}

for all battlements

b
b

is bounded on

L
2

(

R

)

{L^2}({\mathbf {R}})

. This remains true if the underlying space is the circle but is demonstrably false if the underlying space is the integers. Michael Cowling’s theorem that linear commuting with translation operators are bounded on

L
2

{L^2}

if they are weak restricted

(
2
,
2
)

(2,2)

is reproved and an application of this result to sums of exponentials is given.
Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems
Authors: Cosner Chris
Published: June 2010

We consider solutions to the nonlinear eigenvalue problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis asterisk right-parenthesis upper A left-parenthesis x comma ModifyingAbove u With right-arrow right-parenthesis ModifyingAbove u With right-arrow plus lamda f left-parenthesis x comma ModifyingAbove u With right-arrow right-parenthesis equals 0 in normal upper Omega comma ModifyingAbove u With right-arrow equals 0 on partial-differential normal upper Omega comma ModifyingAbove u With right-arrow equals 0 comma on partial-differential normal upper Omega comma ModifyingAbove u With right-arrow equals 0 comma">

(
∗
)

A
(
x
,

u
→

)

u
→

+
λ
f
(
x
,

u
→

)
=
0

in

Ω
,

u
→

=
0

on

∂
Ω
,

u
→

 = 

0
,

on

∂
Ω
,

u
→

=
0
,

(*)\quad A(x,\vec u)\vec u + \lambda f(x,\vec u) = 0\:\quad {\text {in}}\,\Omega ,\quad \vec u = 0\:\quad {\text {on}}\,\partial \Omega ,\quad \vec u{\text { = }}0,\quad {\text {on}}\partial \Omega ,\quad \vec {u} = 0,

\]
where (*) is a quasilinear strongly coupled second order elliptic system of partial differential equations and

Ω
⊆

R

n

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

is a smooth bounded domain. We obtain lower bounds for

λ
\lambda

in the case where

f
(
x
,

u
→

)

f(x,\vec u)

has linear growth, and relations between

λ
,
Ω

\lambda ,\Omega

, and ess sup

|

u
→

|

|\vec u|

when

f
(
x
,

u
→

)

f(x,\vec u)

has sub- or superlinear growth. The estimates are based on integration by parts and application of certain Sobolev inequalities. We briefly discuss extensions to higher order systems.
Classifying torsion-free subgroups of the Picard group
Authors: Brunner Andrew M. , Frame Michael L. , Lee Youn W. , Wielenberg Norbert J.
Published: June 2010

Torsion-free subgroups of finite index in the Picard group are the fundamental groups of hyperbolic

3
3

-manifolds. The Picard group is a polygonal product of finite groups. Recent work by Karrass, Pietrowski and Solitar on the subgroups of a polygonal product make it feasible to calculate all the torsion-free subgroups of any finite index. This computation is carried out here for index 12 and 24, where there are, respectively, 2 and 17 nonisomorphic subgroups. The manifolds are identified by using surgery.
Some properties of viscosity solutions of Hamilton-Jacobi equations
Authors: Crandall M. G. , Evans L. C. , Lions P.-L.
Published: June 2010

Recently M. G. Crandall and P. L. Lions introduced the notion of "viscosity solutions" of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a " viscosity solution" admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper.
Jordan domains and the universal Teichmüller space
Authors: Flinn Barbara Brown
Published: June 2010

Let

L
L

denote the lower half plane and let

B
(
L
)

B(L)

denote the Banach space of analytic functions

f
f

in

L
L

with

‖
f
‖

L

>
∞

{\left \| f \right \|_L} > \infty

, where

‖
f
‖

L

{\left \| f \right \|_L}

is the suprenum over

z
∈
L

z \in L

of the values

|

f
(
z
)

|

(
t
e
x
t

I
m

z

)
2

\left | {f(z)} \right |{(text{Im} z)^2}

. The universal Teichmüller space,

T
T

, is the subset of

B
(
L
)

B(L)

consisting of the Schwarzian derivatives of conformal mappings of

L
L

which have quasiconformal extensions to the extended plane. We denote by

J
J

the set
\[

{

S
f

:
f

is conformal in 

L

and 

f
(
L
)

is a Jordan domain

}

,

\left \{ {{S_f}:f{\text {is conformal in }}L{\text {and }}f(L){\text {is a Jordan domain}}} \right \},

\]
which is a subset of

B
(
L
)

B(L)

contained in the Schwarzian space

S
S

. In showing

S
−

T
¯

≠
∅

S - \bar T \ne \emptyset

, Gehring actually proves

S
−

J
¯

≠
∅

S - \bar J \ne \emptyset

. We give an example which demonstrates that

J
−

T
¯

≠
∅

J - \bar T \ne \emptyset

.
Monotone decompositions of IUC continua
Authors: Collins W. Dwayne
Published: June 2010

For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which "improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum

M
M

has property IUC provided each proper subcontinuum of

M
M

with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.
A sharp form of the Ahlfors’ distortion theorem, with applications
Authors: Hamilton D. H.
Published: June 2010

The constant appearing in the asymptotic version of the Ahlfors’ distortion theorem is

1
1

. Also it is shown that for mean

1
1

-valent functions

f
=
z
+

a
2

z
2

⋯

‖

a

n
+
1

|

−

|

a
n

‖

⩽
1

f = z + {a_2}{z^2} \cdots \left \| {{a_{n + 1}}| - |{a_n}} \right \| \leqslant 1

for

n
⩾
N
(
f
)

n \geqslant N(f)

.
Contraction operators quasisimilar to a unilateral shift
Authors: Alexander V. T.
Published: June 2010

Let

U
n

{U_n}

denote the unilaterial shift of finite multiplicity

n
n

. It is shown that a contraction operator

T
T

is quasisimilar to

U
n

{U_n}

if and only if

T
T

is of Class

C
1

{C_1}

., the canonical isometry

V
V

associated with

T
T

is pure and

T
T

is

n
n

-cyclic with analytically independent vectors. For this, the notions of operators of analytic type and analytic independence of vectors are introduced. A characterization of the cyclic vectors of the Backward Shift is also presented.
Stable rank 2 reflexive sheaves on 𝑃³ with small 𝑐₂ and applications
Authors: Chang Mei-Chu
Published: June 2010

We investigate the moduli spaces of stable rank two reflexive sheaves on

P

3

{{\mathbf {P}}^3}

with small Chern classes. As an application to curves of low degree in

P

3

{{\mathbf {P}}^3}

, we prove the curve has maximal rank and that the corresponding Hilbert scheme is irreducible and unirational.
Operators with 𝐶*-algebra generated by a unilateral shift
Authors: Conway John B. , McGuire Paul
Published: June 2010

If

T
T

is an operator on a Hilbert space

H

\mathcal {H}

, this paper gives necessary and sufficient conditions on

T
T

such that

C

∗

(
T
)

{C^{\ast } }(T)

, the

C

∗

{C^{\ast } }

-algebra generated by

T
T

, is generated by a unilateral shift of some multiplicity. This result is then specialized to the cases in which

T
T

is a hyponormal or subnormal operator. In particular, it is shown how to prove a recent conjecture of C. R. Putnam as a consequence of our result.
Liouville theorems, partial regularity and Hölder continuity of weak solutions to quasilinear elliptic systems
Authors: Meier Michael
Published: June 2010

This paper describes the connections between Liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth. It is shown that equivalence does in general hold only in some restricted sense. A complete correspondence can be established in certain cases, e.g. for small solutions and for minima of quadratic variational integrals.
Fine and parabolic limits for solutions of second-order linear parabolic equations on an infinite slab
Authors: Mair B. A.
Published: June 2010

This paper investigates the boundary behaviour of positive solutions of the equation

L
u
=
0

Lu = 0

, where

L
L

is a uniformly parabolic second-order differential operator in divergence form having Hölder-continuous coefficients on

X
=

R

n

×
(
0
,
T
)

X = {{\mathbf {R}}^n} \times (0,T)

, where

0
>
T
>
∞

0 > T > \infty

. In particular, the notion of semithinness for the potential theory on

X
X

associated with

L
L

is introduced, and the relationships between fine, semifine and parabolic convergence at points of

R

n

×
{
0
}

{{\mathbf {R}}^n} \times \{ 0 \}

are studied. The abstract Fatou-Naim-Doob theorem is used to deduce that every positive solution of

L
u
=
0

Lu = 0

on

X
X

has parabolic limits Lebesgue-almost-everywhere on

R

n

×
{
0
}

{{\mathbf {R}}^n} \times \{ 0 \}

. Furthermore, a Carleson-type result is obtained for solutions defined on a union of parabolic regions.
On the spectrum of 𝐶₀-semigroups
Authors: Prüss Jan
Published: June 2010

In this paper we give characterizations of the spectrum of a

C
0

{C_0}

-semigroup

e

A
t

{e^{At}}

in terms of certain solution properties of the differential equation

(
∗
)

u
′

=
A
u
+
f

(\ast )\;u’ = Au + f

and, in case

X
X

is a Hilbert space, also in terms of properties of

(
λ
−
A

)

−
1

{(\lambda - A)^{ - 1}}

. We give several applications of these results including a study of the existence of dichotomic projections for

(
∗
)

(\ast )

.
On some nonextendable derivations of the gauge-invariant CAR algebra
Authors: Price Geoffrey L.
Published: June 2010

We provide examples of some approximately inner, commutative

∗
\ast

-derivations which are generators on the gauge-invariant CAR algebra but which have no closed densely-defined extensions to the CAR. Necessary conditions are given for a class of generators on the GICAR algebra to extend to closed

∗
\ast

-derivations on the CAR.
Cyclic vectors in the Dirichlet space
Authors: Brown Leon , Shields Allen L.
Published: June 2010

We study the Hilbert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if

|

f
(
z
)

|

⩾

|

g

(
z
)

|

|f(z)| \geqslant |g\,(z)|

at all points and if

g
g

is cyclic, then

f
f

is cyclic. Theorems 3-5 give a sufficient condition (

f
f

is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function

f
f

to be cyclic.
Closed timelike geodesics
Authors: Galloway Gregory J.
Published: June 2010

It is shown that every stable free

t
t

-homotopy class of closed timelike curves in a compact Lorentzian manifold contains a longest curve which must be a closed timelike geodesic. This result enables one to obtain a Lorentzian analogue of a classical theorem of Synge. A criterion for stability is presented, and a theorem of Tipler is derived as a special case of the result stated above.
Slice links in 𝑆⁴
Authors: Cochran Tim
Published: June 2010

We produce necessary and sufficient conditions of a homotopy-theoretic nature for a link of

2
2

-spheres in

S
4

{S^4}

to be slice (i.e., cobordant to the unlink). We give algebraic conditions on the link group sufficient to guarantee sliceness, generalizing the known results for boundary links. The notion of a "stable link" is introduced and shown to be useful in constructing cobordisms in dimension

4
4

.
Contributions from conjugacy classes of regular elliptic elements in 𝑆𝑝(𝑛,𝑍) to the dimension formula
Authors: Eie Min King
Published: June 2010

The dimension of the space of cusp forms on the degree

n
n

Siegel upper half-space can be obtained from the Selberg trace formula; in this paper we compute the contribution from the conjugacy classes of regular elliptic elements in

Sp
⁡
(
n
,

Z

)

\operatorname {Sp}(n,{\mathbf {Z}})

using Weyl’s character formula for representations of

GL

(
n
,

C

)

{\text {GL}}(n,{\mathbf {C}})

.
Central limit theorem for products of random matrices
Authors: Berger Marc A.
Published: June 2010

Using the semigroup product formula of

P
\text {P}

. Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.