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1201  Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups Arvanitoyeorgos, Andreas; Dzhepko, V. V.; Nikonorov, Yu. G.
A Riemannian manifold $(M,\rho)$ is called Einstein if the metric
$\rho$ satisfies the condition \linebreak$\Ric (\rho)=c\cdot \rho$ for some
constant $c$. This paper is devoted to the investigation of
$G$invariant Einstein metrics, with additional symmetries,
on some homogeneous spaces $G/H$ of classical groups.
As a consequence, we obtain new invariant Einstein metrics on some
Stiefel manifolds $\SO(n)/\SO(l)$.
Furthermore, we show that for any positive integer $p$ there exists a
Stiefel manifold $\SO(n)/\SO(l)$
that admits at least $p$
$\SO(n)$invariant Einstein metrics.


1214  Close Lattice Points on Circles Cilleruelo, Javier; Granville, Andrew
We classify the sets of four lattice points that all lie on a
short arc of a circle that has its center at the origin;
specifically on arcs of length $tR^{1/3}$ on a circle of radius
$R$, for any given $t>0$. In particular we prove that any arc of
length $ (40 + \frac{40}3\sqrt{10} )^{1/3}R^{1/3}$ on a circle of
radius $R$, with $R>\sqrt{65}$, contains at most three lattice
points, whereas we give an explicit infinite family of $4$tuples
of lattice points, $(\nu_{1,n},\nu_{2,n},\nu_{3,n},\nu_{4,n})$,
each of which lies on an arc of length $ (40 +
\frac{40}3\sqrt{10})^{\smash{1/3}}R_n^{\smash{1/3}}+o(1)$ on a circle of
radius $R_n$.


1239  Periodicity in Rank 2 Graph Algebras Davidson, Kenneth R.; Yang, Dilian
Kumjian and Pask introduced an aperiodicity condition
for higher rank graphs.
We present a detailed analysis of when this occurs
in certain rank 2 graphs.
When the algebra is aperiodic, we give another proof
of the simplicity of $\mathrm{C}^*(\mathbb{F}^+_{\theta})$.
The periodic $\mathrm{C}^*$algebras are characterized, and it is shown
that $\mathrm{C}^*(\mathbb{F}^+_{\theta}) \simeq
\mathrm{C}(\mathbb{T})\otimes\mathfrak{A}$
where $\mathfrak{A}$ is a simple $\mathrm{C}^*$algebra.


1262  On the Local Lifting Properties of Operator Spaces Dong, Z.
In this paper, we mainly study operator spaces which have the
locally lifting property (LLP). The dual of any ternary ring of operators is shown to
satisfy the strongly local reflexivity, and this is used to prove
that strongly local reflexivity holds also for operator spaces
which have the LLP. Several homological characterizations of the
LLP and weak expectation property are given. We also prove that for any operator space
$V$, $V^{**}$ has the LLP if and only if $V$ has the LLP and
$V^{*}$ is exact.


1279  Tail Bounds for the Stable Marriage of Poisson and Lebesgue Hoffman, Christopher; Holroyd, Alexander E.; Peres, Yuval
Let $\Xi$ be a discrete set in $\rd$. Call the elements of $\Xi$
{\em centers}. The wellknown Voronoi tessellation partitions
$\rd$ into polyhedral regions (of varying volumes) by allocating
each site of $\rd$ to the closest center. Here we study
allocations of $\rd$ to $\Xi$ in which each center attempts to
claim a region of equal volume $\alpha$.


1300  Monodromy Groups and SelfInvariance Hubard, Isabel; Orbani\'c, Alen; Weiss, Asia Ivi\'c
For every polytope $\mathcal{P}$ there is the universal regular
polytope of the same rank as $\mathcal{P}$ corresponding to the
Coxeter group $\mathcal{C} =[\infty, \dots, \infty]$. For a given
automorphism $d$ of $\mathcal{C}$, using monodromy groups, we
construct a combinatorial structure $\mathcal{P}^d$. When
$\mathcal{P}^d$ is a polytope isomorphic to $\mathcal{P}$ we say that
$\mathcal{P}$ is selfinvariant with respect to $d$, or
$d$invariant. We develop algebraic tools for investigating these
operations on polytopes, and in particular give a criterion on the
existence of a $d$\nobreakdashauto\morphism of a given order. As an application,
we analyze properties of selfdual edgetransitive polyhedra and
polyhedra with two flagorbits. We investigate properties of medials
of such polyhedra. Furthermore, we give an example of a selfdual
equivelar polyhedron which contains no polarity (duality of order
2). We also extend the concept of Petrie dual to higher dimensions,
and we show how it can be dealt with using selfinvariance.


1325  Uniqueness of Shalika Models Nien, Chufeng
Let $\BF_q$ be a finite field of $q$ elements, $\CF$ a $p$adic field,
and $D$ a quaternion division algebra over $\CF$. This paper proves
uniqueness of Shalika models for $\GL_{2n}(\BF_q) $ and $\GL_{2n}(D)$,
and reobtains uniqueness of Shalika models for $\GL_{2n}(\CF)$ for
any $n\in \BN$.


1341  Simultaneous Polynomial Approximations of the Lerch Function Rivoal, Tanguy
We construct bivariate polynomial approximations of the Lerch
function that for certain specialisations of the variables and
parameters turn out to be HermitePad\'e approximants either of
the polylogarithms or of Hurwitz zeta functions. In the former
case, we recover known results, while in the latter the results
are new and generalise some recent works of Beukers and Pr\'evost.
Finally, we make a detailed comparison of our work with Beukers'.
Such constructions are useful in the arithmetical study of the
values of the Riemann zeta function at integer points and of the
KubotaLeopold $p$adic zeta function.


1357  On a Class of Landsberg Metrics in Finsler Geometry Shen, Zhongmin
In this paper, we study a long existing open problem on Landsberg
metrics in Finsler geometry. We consider Finsler metrics defined by a
Riemannian metric and a $1$form on a manifold. We show that a
regular Finsler metric in this form is Landsbergian if and only if it
is Berwaldian. We further show that there is a twoparameter family of
functions, $\phi=\phi(s)$, for which there are a Riemannian metric
$\alpha$ and a $1$form $\beta$ on a manifold $M$ such that the scalar
function $F=\alpha \phi (\beta/\alpha)$ on $TM$ is an almost regular
Landsberg metric, but not a Berwald metric.


1375  Stable Discrete Series Characters at Singular Elements Spallone, Steven
Write $\Theta^E$ for the stable discrete series character associated
with an irreducible finitedimensional representation $E$ of a connected
real reductive group $G$. Let $M$ be the centralizer of the split
component of a maximal torus $T$, and denote by $\Phi_M(\gm,\Theta^E)$
Arthur's extension of $ D_M^G(\gm)^{\lfrac 12}
\Theta^E(\gm)$ to $T(\R)$. In this paper we give a simple
explicit expression for
$\Phi_M(\gm,\Theta^E)$ when $\gm$ is elliptic in $G$. We do not assume $\gm$ is regular.


1383  Integral Representation for $U_{3} \times \GL_{2}$ Wambach, Eric
Gelbart and PiatetskiiShapiro constructed
various integral
representations of RankinSel\berg type for groups $G \times
\GL_{n}$,
where $G$
is of split rank $n$. Here we show that their method
can equally well be applied
to the product $U_{3} \times \GL_{2}$, where $U_{3}$
denotes the quasisplit
unitary group in three variables. As an application, we describe which
cuspidal automorphic representations of $U_{3}$ occur
in the Siegel induced
residual spectrum of the quasisplit $U_{4}$.


1407  Traces, CrossRatios and 2Generator Subgroups of $\SU(2,1)$ Will, Pierre
In this work, we investigate how to decompose a pair $(A,B)$ of
loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under
the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are
involutions. The main result is a decomposability criterion, which
is expressed in terms of traces of elements of the group $\langle
A,B\rangle$.


1437  Author Index  Index des auteurs 2009, for 2009  pour
No abstract.

