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673  The Generalized Cuspidal Cohomology Problem Bart, Anneke; Scannell, Kevin P.
Let $\Gamma \subset \SO(3,1)$ be a lattice.
The well known bending deformations, introduced by
\linebreak Thurston
and Apanasov, can be used
to construct nontrivial curves of representations of $\Gamma$
into $\SO(4,1)$ when $\Gamma \backslash \hype{3}$ contains
an embedded totally geodesic surface. A tangent vector to such a
curve is given by a nonzero group cohomology class
in $\H^1(\Gamma, \mink{4})$. Our main result generalizes this
construction of cohomology to the context of ``branched''
totally geodesic surfaces.
We also consider a natural generalization of the famous
cuspidal cohomology problem for the Bianchi groups
(to coefficients in nontrivial representations), and
perform calculations in a finite range.
These calculations lead directly to an interesting example of a
link complement in $S^3$
which is not infinitesimally rigid in $\SO(4,1)$.
The first order deformations of this link complement are supported
on a piecewise totally geodesic $2$complex.


691  Hypoelliptic BiInvariant Laplacians on Infinite Dimensional Compact Groups Bendikov, A.; SaloffCoste, L.
On a compact connected group $G$, consider the infinitesimal
generator $L$ of a central symmetric Gaussian convolution
semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution
and smooth function spaces, we prove that $L$ is hypoelliptic if and only if
$(\mu_t)_{t>0} $ is absolutely continuous with respect to Haar measure
and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that
$\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds
if and only if any Borel measure $u$ which is solution of $Lu=0$
in an open set $\Omega$ can be represented by a continuous
function in $\Omega$. Examples are discussed.


726  On Value Distribution Theory of Second Order Periodic ODEs, Special Functions and Orthogonal Polynomials Chiang, YikMan; Ismail, Mourad E. H.
We show that the value distribution (complex oscillation) of
solutions of certain periodic second order ordinary differential
equations studied by Bank, Laine and Langley is closely
related to confluent hypergeometric functions, Bessel functions
and Bessel polynomials. As a result, we give a complete
characterization of the zerodistribution in the sense of
Nevanlinna theory of the solutions for two classes of the ODEs.
Our approach uses special functions and their asymptotics. New
results concerning finiteness of the number of zeros
(finitezeros) problem of Bessel and Coulomb wave functions with
respect to the parameters are also obtained as a consequence. We
demonstrate that the problem for the remaining class of ODEs not
covered by the above ``special function approach" can be
described by a classical Heine problem for differential
equations that admit polynomial solutions.


768  Decomposability of von Neumann Algebras and the Mazur Property of Higher Level Hu, Zhiguo; Neufang, Matthias
The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal nonzero
projections in $\m$. In this paper, we explore the close
connection between $\dec(\m)$ and the cardinal level of the Mazur
property for the predual $\m_*$ of $\m$, the study of which was
initiated by the second author. Here, our main focus is on
those von Neumann algebras whose preduals constitute such
important Banach algebras on a locally compact group $G$ as the
group algebra $\lone$, the Fourier algebra $A(G)$, the measure
algebra $M(G)$, the algebra $\luc^*$, etc. We show that for
any of these von Neumann algebras, say $\m$, the cardinal number
$\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$
are completely encoded in the underlying group structure. In fact,
they can be expressed precisely by two dual cardinal
invariants of $G$: the compact covering number $\kg$ of $G$ and
the least cardinality $\bg$ of an open basis at the identity of
$G$. We also present an application of the Mazur property of higher
level to the topological centre problem for the Banach algebra
$\ag^{**}$.


796  MordellWeil Groups and the Rank of Elliptic Curves over Large Fields Im, BoHae
Let $K$ be a number field, $\overline{K}$ an algebraic closure of
$K$ and $E/K$ an elliptic curve
defined over $K$. In this paper, we prove that if $E/K$ has a
$K$rational point $P$ such that $2P\neq O$ and $3P\neq O$, then
for each $\sigma\in \Gal(\overline{K}/K)$, the MordellWeil group
$E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of
$\overline{K}$ under $\sigma$ has infinite rank.


820  Diametrically Maximal and Constant Width Sets in Banach Spaces Moreno, J. P.; Papini, P. L.; Phelps, R. R.
We characterize diametrically maximal and constant width
sets in $C(K)$, where $K$ is any compact Hausdorff space. These
results are applied to prove that the sum of two diametrically
maximal sets needs not be diametrically maximal, thus solving a
question raised in a paper by Groemer. A~characterization of
diametrically maximal sets in $\ell_1^3$ is also given, providing
a negative answer to Groemer's problem in finite dimensional
spaces. We characterize constant width sets in $c_0(I)$, for
every $I$, and then we establish the connections between the Jung
constant of a Banach space and the existence of constant width
sets with empty interior. Porosity properties of families of sets
of constant width and rotundity properties of diametrically
maximal sets are also investigated. Finally, we present some
results concerning nonreflexive and Hilbert spaces.


843  On the OneLevel Density Conjecture for Quadratic Dirichlet LFunctions Õzlük, A. E.; Snyder, C.
In a previous article, we studied the distribution of ``lowlying"
zeros of the family of quad\ratic Dirichlet $L$functions assuming
the Generalized Riemann Hypothesis for all Dirichlet
$L$functions. Even with this very strong assumption, we were
limited to using weight functions whose Fourier transforms are
supported in the interval $(2,2)$. However, it is widely believed
that this restriction may be removed, and this leads to what has
become known as the OneLevel Density Conjecture for the zeros of
this family of quadratic $L$functions. In this note, we make use
of Weil's explicit formula as modified by Besenfelder to prove an
analogue of this conjecture.


859  Nonstandard Ideals from Nonstandard Dual Pairs for $L^1(\omega)$ and $l^1(\omega)$ Read, C. J.
The Banach convolution algebras $l^1(\omega)$
and their continuous counterparts $L^1(\bR^+,\omega)$
are much
studied, because (when the submultiplicative weight function
$\omega$ is radical) they are pretty much the prototypic examples
of commutative radical Banach algebras. In cases of ``nice''
weights $\omega$, the only closed ideals they have are the obvious,
or ``standard'', ideals. But in the
general case, a brilliant but very difficult paper of Marc Thomas
shows that nonstandard ideals exist in $l^1(\omega)$. His
proof was successfully exported to the continuous case
$L^1(\bR^+,\omega)$ by Dales and McClure, but remained
difficult. In this paper we first present a small improvement: a
new and easier proof of the existence of nonstandard ideals in
$l^1(\omega)$ and $L^1(\bR^+,\omega)$. The new proof is based on
the idea of a ``nonstandard dual pair'' which we introduce.
We are then able to make a much larger improvement: we
find nonstandard ideals in $L^1(\bR^+,\omega)$ containing functions
whose supports extend all the way down to zero in $\bR^+$, thereby solving
what has become a notorious problem in the area.


877  Functorial Decompositions of Looped Coassociative Co$H$ Spaces Selick, P.; Theriault, S.; Wu, J.
Selick and Wu gave a functorial decomposition of
$\Omega\Sigma X$ for pathconnected, $p$local \linebreak$\CW$\nbdcom\plexes $X$
which obtained the smallest nontrivial functorial retract $A^{\min}(X)$
of $\Omega\Sigma X$. This paper uses methods developed by
the second author in order to extend such functorial
decompositions to the loops on coassociative co$H$ spaces.
