We show that given a stable weighted configuration on the asymptotic
boundary of a
locally compact Hadamard space, there is a polygon with Gauss
map prescribed by the given weighted configuration.
Moreover, the same result holds for
semistable configurations on arbitrary Euclidean buildings.

It is shown that the group of compactly
supported, measure-preserving homeomorphisms of a
connected, second countable manifold is locally contractible in the direct limit topology.
Furthermore, this group is weakly homotopically equivalent to the more general group of
compactly supported homeomorphisms.

In this paper we study affine completeness of generalised dihedral
groups. We give a formula for the number of unary compatible
functions on these groups, and we characterise for every $k \in~\N$
the $k$-affine complete generalised dihedral groups. We find that
the direct product of a $1$-affine complete group with itself need not
be $1$-affine complete. Finally, we give an example of a nonabelian
solvable affine complete group. For nilpotent groups we find a
strong necessary condition for $2$-affine completeness.

We show that each point of the principal eigencurve of the
nonlinear problem
$$
-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.

This paper is concerned with the structure of
inner $E_0$-semigroups. We show that any inner
$E_0$-semigroup acting on an infinite factor
$M$ is completely determined by a continuous
tensor product system of Hilbert spaces in
$M$ and that the product system associated
with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.

It is known that the derivative of
a Blaschke product whose zero sequence lies in a Stolz angle
belongs to all the Bergman spaces $A^p$ with $0<p<3/2$.
The question
of whether this result is best possible remained open.
In this paper,
for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain
a number of conditions which are equivalent to the membership of $B'$ in the
space $A^p$ ($p>1$). As a consequence,
we prove that there exists a Blaschke product $B$
with zeros on a radius such that $B'\notin A^{3/2}$.

Free analogues of the logarithmic Sobolev inequality compare the relative
free Fisher information with the relative free entropy. In the present paper
such an inequality is obtained for measures on the circle. The method is
based on a random matrix approximation procedure, and a large deviation
result concerning the eigenvalue distribution of special unitary matrices is
applied and discussed.

This note shows that any set of cofibrations containing the standard
set of generating projective cofibrations determines a cofibrantly
generated proper closed model structure on the category of simplicial
presheaves on a small Grothendieck site, for which the weak
equivalences are the local weak equivalences in the usual sense.

We construct vector-valued modular forms of weight 2 associated to
Jacobi-like forms with respect to a symmetric tensor representation of
$\G$ by using the method of Kuga and Shimura as well as the
correspondence between Jacobi-like forms and sequences of modular forms.
As an application, we obtain vector-valued modular forms determined by
theta functions and by pseudodifferential operators.

We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal
if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed
that Littlewood polynomials with particularly high minimum modulus on
the unit
circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a
skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle,
as well as providing a new proof of the known result that a self-reciprocal
Littlewood polynomial must have a zero on the unit circle.

In this paper, we find a lower bound on the number of cyclic function
fields of prime degree~$l$ whose class numbers are divisible by a
given
integer $n$. This generalizes a previous result of D. Cardon and R.
Murty
which gives a lower bound on the number of quadratic function fields
with
class numbers divisible by $n$.

The purpose of this note is to show that the homologically trivial
cycles contructed by Clemens and their generalisations
due to Paranjape can be detected by the technique of
spreading out. More precisely, we associate to these cycles (and the
ambient varieties in which they live) certain families which arise
naturally and which are defined over $\bbC$ and show that these
cycles, along with their relations, can be detected in the singular
cohomology of the total space of these families.