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3  Connected Components of Moduli Stacks of Torsors via Tamagawa Numbers Behrend, Kai; Dhillon, Ajneet
Let $X$ be a smooth projective geometrically connected curve over
a finite field with function field $K$. Let $\G$ be a connected semisimple group
scheme over $X$. Under certain hypotheses we prove the equality of
two numbers associated with $\G$.
The first is an arithmetic invariant, its Tamagawa number. The second
is a geometric invariant, the number of connected components of the moduli
stack of $\G$torsors on $X$. Our results are most useful for studying
connected components as much is known about Tamagawa numbers.


29  The Minimal Resolution Conjecture for Points on the Cubic Surface Casanellas, M.
In this paper we prove that a generalized version of the Minimal
Resolution Conjecture given by Musta\c{t}\v{a} holds for certain
general sets of points on a smooth cubic surface $X \subset
\PP^3$. The main tool used is Gorenstein liaison theory and, more
precisely, the relationship between the free resolutions of two linked schemes.


50  Composition operators on $\mu$Bloch spaces Chen, Huaihui; Gauthier, Paul
Given a positive continuous function $\mu$ on the
interval $0<t\le1$, we consider the space of socalled $\mu$Bloch
functions on the unit ball. If $\mu(t)=t$, these are the classical
Bloch functions. For $\mu$, we define a metric $F_z^\mu(u)$ in
terms of which we give a characterization of $\mu$Bloch functions.
Then, necessary and sufficient conditions are obtained in order that
a composition operator be a bounded or compact operator between
these generalized Bloch spaces. Our results extend those of Zhang
and Xiao.


76  Ascent Properties of Auslander Categories Christensen, Lars Winther; Holm, Henrik
Let $R$ be a homomorphic image of a Gorenstein local ring. Recent
work has shown that there is a bridge between Auslander categories
and modules of finite Gorenstein homological dimensions over $R$.


109  The Ample Cone of the Kontsevich Moduli Space Coskun, Izzet; Harris, Joe; Starr, Jason
We produce ample (resp.\ NEF, eventually free) divisors in the
Kontsevich space $\Kgnb{0,n} (\mathbb P^r, d)$ of $n$pointed,
genus $0$, stable maps to $\mathbb P^r$, given such divisors in
$\Kgnb{0,n+d}$. We prove that this produces all ample (resp.\ NEF,
eventually free) divisors in $\Kgnb{0,n}(\mathbb P^r,d)$.
As a consequence, we construct a contraction of the boundary
$\bigcup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,dk}$ in
$\Kgnb{0,0}(\mathbb P^r,d)$, analogous to a contraction of
the boundary $\bigcup_{k=3}^{\lfloor n/2 \rfloor}
\tilde{\Delta}_{k,nk}$ in $\kgnb{0,n}$ first constructed by Keel
and McKernan.


124  Characterizing Complete Erd\H os Space Dijkstra, Jan J.; Mill, Jan van
The space now known as {\em complete Erd\H os
space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the
closed subspace of the Hilbert space $\ell^2$ consisting of all
vectors such that every coordinate is in the convergent sequence
$\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G.
Oversteegen we present simple and useful topological
characterizations of $\cerdos$.
As an application we determine the class
of factors of $\cerdos$. In another application we determine
precisely which of the spaces that can be constructed in the Banach
spaces $\ell^p$ according to the `Erd\H os method' are homeomorphic
to $\cerdos$. A novel application states that if $I$ is a
Polishable $F_\sigma$ideal on $\omega$, then $I$ with the Polish
topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$,
$\Z\times2^\omega$, or $\cerdos$. This last result answers a
question that was asked
by Stevo Todor{\v{c}}evi{\'c}.


141  On the Littlewood Problem Modulo a Prime Green, Ben; Konyagin, Sergei
Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow
\mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f}
\Vert_1 \leq 1$. Then
$\min_{x \in \Zp} f(x) = O(\log p)^{1/3 + \epsilon}$.
One should think of $f$ as being ``approximately continuous''; our
result is then an ``approximate intermediate value theorem''.
As an immediate consequence we show that if $A \subseteq \Zp$ is a
set of cardinality $\lfloor p/2\rfloor$, then
$\sum_r \widehat{1_A}(r) \gg (\log p)^{1/3  \epsilon}$. This
gives a result on a ``mod $p$'' analogue of Littlewood's wellknown
problem concerning the smallest possible $L^1$norm of the Fourier
transform of a set of $n$ integers.


165  Exponents of Diophantine Approximation in Dimension Two Laurent, Michel
Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,
\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a
quadruple $\Omega(\Theta)$ of exponents that measure the quality
of approximation to $\Theta$ both by rational points and by
rational lines. The two ``uniform'' components of $\Omega(\Theta)$
are related by an equation due to Jarn\'\i k, and the four
exponents satisfy two inequalities that refine Khintchine's
transference principle. Conversely, we show that for any quadruple
$\Omega$ fulfilling these necessary conditions, there exists
a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.


190  Bounded Hankel Products on the Bergman Space of the Polydisk Lu, Yufeng; Shang, Shuxia
We consider the problem of determining for which square integrable
functions $f$ and $g$ on the polydisk the densely defined Hankel
product $H_{f}H_g^\ast$ is bounded on the Bergman space of the
polydisk. Furthermore, we obtain similar results for the mixed
Haplitz products $H_{g}T_{\bar{f}}$ and $T_{f}H_{g}^{*}$, where $f$
and $g$ are square integrable on the polydisk and $f$ is analytic.


205  Representations of NonNegative Polynomials, Degree Bounds and Applications to Optimization Marshall, M.
Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability $1$. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.


222  Klyachko Models for General Linear Groups of Rank 5 over a $p$Adic Field Nien, Chufeng
This paper shows the existence and uniqueness of Klyachko models for
irreducible unitary representations of $\GL_5(\CF)$, where $\CF$ is
a $p$adic field. It is an extension of the work of Heumos and Rallis on $\GL_4(\CF)$.


241  Operator Integrals, Spectral Shift, and Spectral Flow Azamov, N. A.; Carey, A. L.; Dodds, P. G.; Sukochev, F. A.
We present a new and simple approach to the theory of multiple
operator integrals that applies to unbounded operators affiliated with general \vNa s.
For semifinite \vNa s we give applications
to the Fr\'echet differentiation of operator functions that sharpen existing results,
and establish the BirmanSolomyak representation of the spectral
shift function of M.\,G.\,Krein
in terms of an average of spectral measures in the type II setting.
We also exhibit a surprising connection between the spectral shift
function and spectral flow.


264  On $\BbZ$Modules of Algebraic Integers Bell, J. P.; Hare, K. G.
Let $q$ be an algebraic integer of degree $d \geq 2$.
Consider the rank of the multiplicative subgroup of $\BbC^*$ generated
by the conjugates of $q$.
We say $q$ is of {\em full rank} if either the rank is $d1$ and $q$
has norm $\pm 1$, or the rank is $d$.
In this paper we study some properties of $\BbZ[q]$ where $q$ is an
algebraic integer of full rank.
The special cases of when $q$ is a Pisot number and when $q$ is a Pisotcyclotomic number
are also studied.
There are four main results.
\begin{compactenum}[\rm(1)]
\item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive
integer,
then there are only finitely many $m$ such that
$\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$.
\item If $q$ and $r$ are algebraic integers of degree $d$ of full rank
and $\BbZ[q^n] = \BbZ[r^n]$ for
infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$,
where
$r'$ is some conjugate of $r$ and $\omega$ is some root of unity.
\item Let $r$ be an algebraic integer of degree at most $3$.
Then there are at most $40$ Pisot numbers $q$ such that
$\BbZ[q] = \BbZ[r]$.
\item There are only finitely many Pisotcyclotomic numbers of any fixed
order.
\end{compactenum}


282  Closed Ideals in Some Algebras of Analytic Functions Bouya, Brahim
We obtain a complete description of closed ideals of the algebra
$\cD\cap \cL$, $0<\alpha\leq\frac{1}{2}$, where $\cD$ is the
Dirichlet space and $\cL$ is the algebra of analytic functions
satisfying the Lipschitz condition of order $\alpha$.


299  \v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$ Dawson, Robert J. MacG.; Moszy\'{n}ska, Maria
A set in a metric space is called a \v{Ceby\v{s}ev set} if
it has a unique ``nearest neighbour'' to each point of the space. In
this paper we generalize this notion, defining a set to be
\v{Ceby\v{s}ev relative to} another set if every point in the
second set has a unique ``nearest neighbour'' in the first. We are
interested in \v{C}eby\v{s}ev sets in some hyperspaces over $\R$,
endowed with the Hausdorff metric, mainly the hyperspaces of compact
sets, compact convex sets, and strictly convex compact sets.


315  Injective Representations of Infinite Quivers. Applications Enochs, E.; Estrada, S.; Rozas, J. R. Garc\'{\i}a
In this article we study injective representations of infinite
quivers. We classify the indecomposable injective representations of
trees and describe Gorenstein injective and projective
representations of barren trees.


336  The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes Garaev, M. Z.
Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ any
strictly increasing sequence of positive integers satisfying $s_n\le
n^{15/14+o(1)}.$ In this paper we give a version of the large sieve
inequality for the sequence $\lambda^{s_n}.$ In particular, we
obtain nontrivial estimates of the associated trigonometric sums
``on average" and establish equidistribution properties of the
numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$,
modulo $p$ for most primes $p.$


351  Multiplication of Polynomials on Hermitian Symmetric spaces and LittlewoodRichardson Coefficients Graham, William; Hunziker, Markus
Let $K$ be a complex reductive algebraic group and $V$ a
representation of $K$. Let $S$ denote the ring of polynomials on
$V$. Assume that the action of $K$ on $S$ is multiplicityfree. If
$\lambda$ denotes the isomorphism class of an irreducible
representation of $K$, let $\rho_\lambda\from K \rightarrow
GL(V_{\lambda})$ denote the corresponding irreducible representation
and $S_\lambda$ the $\lambda$isotypic component of $S$. Write
$S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of
$S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible
constituent of $V_\lambda\otimes V_\mu$, is it true that
$S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors
investigate this question for representations arising in the context
of Hermitian symmetric pairs. It is shown that the answer is yes in
some cases and, using an earlier result of Ruitenburg, that in the
remaining classical cases, the answer is yes provided that a
conjecture of Stanley on the multiplication of Jack polynomials is
true. It is also shown how the conjecture connects multiplication in
the ring $S$ to the usual LittlewoodRichardson rule.


373  An Infinite Order Whittaker Function McKee, Mark
In this paper we construct a flat smooth section of an induced space
$I(s,\eta)$ of $SL_2(\mathbb{R})$ so that the attached Whittaker function
is not of finite order.
An asymptotic method of classical analysis is used.


382  Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra $A(G)$ Miao, Tianxuan
Let $\mathcal{A}$ be a Banach algebra with a bounded right
approximate identity and let $\mathcal B$ be a closed ideal of
$\mathcal A$. We study the relationship between the right identities
of the double duals ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$ under
the Arens product. We show that every right identity of ${\mathcal
B}^{**}$ can be extended to a right identity of ${\mathcal A}^{**}$ in
some sense. As a consequence, we answer a question of Lau and
\"Ulger, showing that for the Fourier algebra $A(G)$ of a locally
compact group $G$, an element $\phi \in A(G)^{**}$ is in $A(G)$ if and
only if $A(G) \phi \subseteq A(G)$ and $E \phi = \phi $ for all right
identities $E $ of $A(G)^{**}$. We also prove some results about the
topological centers of ${\mathcal B}^{**}$ and ${\mathcal A}^{**}$.


395  $L$Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications Moriyama, Tomonori
Let $\Pi$ be a generic cuspidal automorphic representation of
$\GSp(2)$ defined over a totally real algebraic number field $\gfk$
whose archimedean type is either a (limit of) large discrete series
representation or a certain principal series representation. Through
explicit computation of archimedean local zeta integrals, we prove the
functional equation of tensor product $L$functions $L(s,\Pi \times
\sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$
of $\GL(2)$. We also give an application to the spinor $L$function
of $\Pi$.


427  On Reducibility and Unitarizability for Classical $p$Adic Groups, Some General Results Tadi\'c, Marko
The aim of this paper is to prove two general results on parabolic
induction of classical $p$adic groups (actually, one of them holds also
in the archimedean case), and to obtain from them some consequences about
irreducible unitarizable representations. One of these consequences is a
reduction of the unitarizability problem for these groups. This
reduction is similar to the reduction of the unitarizability problem
to the case of real infinitesimal
character for real reductive groups.


451  A Subalgebra Intersection Property for Congruence Distributive Varieties Valeriote, Matthew A.
We prove that if a finite algebra $\m a$ generates a congruence
distributive variety, then the subalgebras of the powers of $\m a$
satisfy a certain kind of intersection property that fails for
finite idempotent algebras that locally exhibit affine or unary
behaviour. We demonstrate a connection between this property and the
constraint satisfaction problem.


465  On Partitions into Powers of Primes and Their Difference Functions Woodford, Roger
In this paper, we extend the approach first outlined by Hardy and
Ramanujan for calculating the asymptotic formulae for the number of
partitions into $r$th powers of primes, $p_{\mathbb{P}^{(r)}}(n)$,
to include their difference functions. In doing so, we rectify an
oversight of said authors, namely that the first difference function
is perforce positive for all values of $n$, and include the
magnitude of the error term.


481  Uniform Distribution of Fractional Parts Related to Pseudoprimes Banks, William D.; Garaev, Moubariz Z.; Luca, Florian; Shparlinski, Igor E.
We estimate exponential sums with the Fermatlike quotients
$$
f_g(n) = \frac{g^{n1}  1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n1}1}{P(n)},
$$
where $g$ and $n$ are positive integers, $n$ is composite, and
$P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$
and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base
$g$, and if $n$ is a Carmichael number, this is true for all $g$
coprime to $n$. Nevertheless, our bounds imply that the fractional
parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on
average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We
also obtain similar results with the functions ${\widetilde f}_g(n)
= gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.


503  Subspaces of de~Branges Spaces Generated by Majorants Baranov, Anton; Woracek, Harald
For a given de~Branges space $\mc H(E)$ we investigate
de~Branges subspaces defined in terms of majorants
on the real axis. If $\omega$ is a nonnegative function on $\mathbb R$,
we consider the subspace
\[
\mc R_\omega(E)=\clos_{\mc H(E)} \big\{F\in\mc H(E):
\text{ there exists } C>0:
E^{1} F\leq C\omega \mbox{ on }{\mathbb R}\big\}
.
\]
We show that $\mc R_\omega(E)$ is a de~Branges subspace and
describe all subspaces of this form. Moreover,
we give a criterion for the existence of positive minimal majorants.


518  Global Units Modulo Circular Units: Descent Without Iwasawa's Main Conjecture Belliard, JeanRobert
Iwasawa's classical asymptotical formula relates the orders of the $p$parts $X_n$ of the ideal
class groups along a $\mathbb{Z}_p$extension $F_\infty/F$ of a number
field $F$ to Iwasawa structural invariants $\la$ and $\mu$
attached to the inverse limit $X_\infty=\varprojlim X_n$.
It relies on ``good" descent properties satisfied by
$X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic, it is known
that the $p$parts of the orders of the global units modulo
circular units $U_n/C_n$ are asymptotically equivalent to the
$p$parts of the ideal class numbers. This suggests that these
quotients $U_n/C_n$, so to speak unit class groups, also satisfy
good descent properties. We show this directly, i.e., without using Iwasawa's Main Conjecture.


534  Girsanov Transformations for NonSymmetric Diffusions Chen, ChuanZhong; Sun, Wei
Let $X$ be a diffusion process, which is assumed to be
associated with a (nonsymmetric) strongly local Dirichlet form
$(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For
$u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet
space, we investigate some properties of the Girsanov transformed
process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of
$X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$.
We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to
be in duality with respect to the measure $e^{2u}m$. We also
construct a counterexample, which shows that this condition may
not be satisfied and hence $(Y,\widehat{Y})$ may not be dual
processes. Then we present a sufficient condition under which $Y$
is associated with a semiDirichlet form. Moreover, we give an
explicit representation of the semiDirichlet form.


548  Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces Girouard, Alexandre
We study the effect of two types of degeneration of a Riemannian
metric on the first eigenvalue of the Laplace operator on
surfaces. In both cases we prove that the first eigenvalue of the
round sphere is an optimal asymptotic upper bound. The first type of
degeneration is concentration of the density to a point within a
conformal class. The second is degeneration of the
conformal class to the boundary of the moduli space on the torus and
on the Klein bottle. In the latter, we follow the outline proposed
by N. Nadirashvili in 1996.


566  Convex Subordination Chains in Several Complex Variables Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.
In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\z\,\z\)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.


583  Algebraic Properties of a Family of Generalized Laguerre Polynomials Hajir, Farshid
We study the algebraic properties of Generalized Laguerre Polynomials
for negative integral values of the parameter. For integers $r,n\geq
0$, we conjecture that $L_n^{(1nr)}(x) = \sum_{j=0}^n
\binom{nj+r}{nj}x^j/j!$ is a $\Q$irreducible polynomial whose
Galois group contains the alternating group on $n$ letters. That this
is so for $r=n$ was conjectured in the 1950's by Grosswald and proven
recently by Filaseta and Trifonov. It follows from recent work of
Hajir and Wong that the conjecture is true when $r$ is large with
respect to $n\geq 5$. Here we verify it in three situations: i) when
$n$ is large with respect to $r$, ii) when $r \leq 8$, and iii) when
$n\leq 4$. The main tool is the theory of $p$adic Newton Polygons.


604  First Countable Continua and Proper Forcing Hart, Joan E.; Kunen, Kenneth
Assuming the Continuum Hypothesis,
there is a compact, first countable, connected space of weight $\aleph_1$
with no totally disconnected perfect subsets.
Each such space, however, may be destroyed by
some proper forcing order which does not add reals.


617  Square Integrable Representations and the Standard Module Conjecture for General Spin Groups Kim, Wook
In this paper we study square integrable representations and
$L$functions for quasisplit general spin groups over a $p$adic
field. In the first part, the holomorphy of $L$functions in a half
plane is proved by using a variant form of Casselman's square
integrability criterion and the LanglandsShahidi method. The
remaining part focuses on the proof of the standard module
conjecture. We generalize Mui\'c's idea via the LanglandsShahidi method
towards a proof of the conjecture. It is used in the work of M. Asgari
and F. Shahidi on generic transfer for general spin groups.


641  Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions Maeda, Sadahiro; Udagawa, Seiichi
For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.


656  Generalized Polynomials and Mild Mixing McCutcheon, Randall; Quas, Anthony
An unsettled conjecture of V. Bergelson and I. H\aa land proposes that
if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving
system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ are
generalized polynomials (functions built out of regular polynomials
via iterated use of the greatest integer or floor function) having the
property that no $p_i$, nor any $p_ip_j$, $i\neq j$, is constant on a
set of positive density, then for any measurable sets
$A_0,A_1,\dots
,A_k$, there exists a zerodensity set $E\subset \z$ such that
\[\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots
\cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).\] We formulate and prove a
faithful version of this conjecture for mildly mixing systems and
partially characterize, in the degree two case, the set of families
$\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem.


674  A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$ Pollack, David; Pollack, Robert
We give a constructive proof, in the special case of ${\rm GL}_3$, of
a theorem of Ash and Stevens which compares overconvergent cohomology
to classical cohomology. Namely, we show that every ordinary
classical Heckeeigenclass can be lifted uniquely to a rigid analytic
eigenclass. Our basic method builds on the ideas of M. Greenberg; we
first form an arbitrary lift of the classical eigenclass to a
distributionvalued cochain. Then, by appropriately iterating the
$U_p$operator, we produce a cocycle whose image in cohomology is the
desired eigenclass. The constructive nature of this proof makes it
possible to perform computer computations to approximate these
interesting overconvergent eigenclasses.


691  Prehomogeneity on QuasiSplit Classical Groups and Poles of Intertwining Operators Yu, Xiaoxiang
Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit,
connected, reductive classical group $G$ defined over a nonArchimedean
field and $A$ is the standard intertwining operator attached to a
tempered representation of $G$ induced from $M$. In this paper we
determine all the cases in which $\Lie(N)$ is
prehomogeneous under $\Ad(m)$ when $N$ is nonabelian, and give necessary
and sufficient conditions for $A$ to have a pole at $0$.


708  Regular Homeomorphisms of Finite Order on Countable Spaces Zelenyuk, Yevhen
We present a structure theorem for a broad class of homeomorphisms of
finite order on countable zero dimensional spaces. As applications we
show the following.
\begin{compactenum}[\rm(a)]
\item Every countable nondiscrete topological group not containing an
open Boolean subgroup can be partitioned into infinitely many dense
subsets.


721  SubRiemannian Geometry on the Sphere $\mathbb{S}^3$ Calin, Ovidiu; Chang, DerChen; Markina, Irina
We discuss the subRiemannian
geometry induced by two noncommutative
vector fields which are left invariant
on the Lie group $\mathbb{S}^3$.


740  On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings Caprace, PierreEmmanuel; Haglund, Frédéric
Given a complete CAT(0) space $X$ endowed with a geometric action of a group $\Gamma$, it is known that if
$\Gamma$ contains a free abelian group of rank $n$, then $X$ contains a geometric flat of dimension $n$. We
prove the converse of this statement in the special case where $X$ is a convex subcomplex of the CAT(0)
realization of a Coxeter group $W$, and $\Gamma$ is a subgroup of $W$. In particular a convex cocompact subgroup
of a Coxeter group is Gromovhyperbolic if and only if it does not contain a free abelian group of rank 2. Our
result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits
buildings.


762  The Hilbert Coefficients of the Fiber Cone and the $a$Invariant of the Associated Graded Ring D'Cruz, Clare; Puthenpurakal, Tony J.
Let $(A,\m)$ be a Noetherian local ring with infinite residue
field and let $I$ be an ideal in $A$ and let $F(I) =
\bigoplus_{n \geq 0}I^n/\m I^n$ be the fiber cone of $I$.
We prove certain relations among the Hilbert coefficients $f_0(I),f_1(I), f_2(I)$ of $F(I)$
when the $a$invariant of the associated graded ring $G(I)$ is negative.


779  Residual Spectra of Split Classical Groups and their Inner Forms Grbac, Neven
This paper is concerned with the residual spectrum of the
hermitian quaternionic classical groups $G_n'$ and $H_n'$ defined
as algebraic groups for a quaternion algebra over an algebraic
number field. Groups $G_n'$ and
$H_n'$ are not quasisplit. They are inner forms of the split
groups $\SO_{4n}$ and $\Sp_{4n}$. Hence, the parts of the residual
spectrum of $G_n'$ and $H_n'$ obtained in this paper are compared
to the corresponding parts for the split groups $\SO_{4n}$ and
$\Sp_{4n}$.


807  Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients Hong, Sunggeum; Kim, Joonil; Yang, Chan Woo
We prove the $L^p(\mathbb{R}^d)$ ($1<p\le \infty$) boundedness of
the maximal operators associated with a family of vector polynomials
given by the form
$\{(2^{k_1}\mathfrak{p}_1(t),\dots,2^{k_d}\mathfrak{p}_d(t)):
t\in\mathbb{R} \}.$ Furthermore, by using the lifting argument, we
extend this result to a general class of vector polynomials whose
coefficients are of the form constant times $2^k$.


828  Twisted GrossZagier Theorems Howard, Benjamin
The theorems of GrossZagier and Zhang relate the N\'eronTate
heights of complex multiplication points on the modular curve $X_0(N)$
(and on Shimura curve analogues) with the central derivatives of
automorphic $L$function. We extend these results to include certain
CM points on modular curves of the form
$X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues).
These results are motivated by applications to Hida theory
that can be found in the companion article
"Central derivatives of $L$functions in Hida families", Math.\ Ann.\
\textbf{399}(2007), 803818.


888  Face Ring Multiplicity via CMConnectivity Sequences Novik, Isabella; Swartz, Ed
The multiplicity conjecture of Herzog, Huneke, and Srinivasan
is verified for the face rings of the following classes of
simplicial complexes: matroid complexes, complexes of dimension
one and two,
and Gorenstein complexes of dimension at most four.
The lower bound part of this conjecture is also established for the
face rings of all doubly CohenMacaulay complexes whose 1skeleton's
connectivity does not exceed the codimension plus one as well as for
all $(d1)$dimensional $d$CohenMacaulay complexes.
The main ingredient of the proofs is a new interpretation
of the minimal shifts in the resolution of the face ring
$\field[\Delta]$ via the CohenMacaulay connectivity of the
skeletons of $\Delta$.


904  The Face Semigroup Algebra of a Hyperplane Arrangement Saliola, Franco V.
This article presents a study of an algebra spanned by the faces of a
hyperplane arrangement. The quiver with relations of the algebra is
computed and the algebra is shown to be a Koszul algebra.
It is shown that the algebra depends only on the intersection lattice of
the hyperplane arrangement. A complete system of primitive orthogonal
idempotents for the algebra is constructed and other algebraic structure
is determined including: a description of the projective indecomposable
modules, the Cartan invariants, projective resolutions of the simple
modules, the Hochschild homology and cohomology, and the Koszul dual
algebra. A new cohomology construction on posets is introduced, and it is
shown that the face semigroup algebra is isomorphic to the cohomology
algebra when this construction is applied to the intersection lattice of
the hyperplane arrangement.


930  Prolongations and Computational Algebra Sidman, Jessica; Sullivant, Seth
We explore the geometric notion of prolongations in the setting of
computational algebra, extending results of Landsberg and Manivel
which relate prolongations to equations for secant varieties. We also
develop methods for computing prolongations that are combinatorial in
nature. As an application, we use prolongations to derive a new
family of secant equations for the binary symmetric model in
phylogenetics.


950  Infinitesimal Invariants in a Function Algebra Tange, Rudolf
Let $G$ be a reductive connected linear algebraic group
over an algebraically closed field of positive
characteristic and let $\g$ be its Lie algebra.
First we extend a wellknown result about the Picard group of a
semisimple group to reductive groups.
Then we prove that if the derived group is simply connected
and $\g$ satisfies a
mild condition, the algebra $K[G]^\g$ of regular functions
on $G$ that are invariant under the action of $\g$ derived
from the conjugation action is a unique factorisation domain.


961  Transfert des intégrales orbitales pour les algèbres de Lie classiques Bernon, Florent
Dans cet article, on consid\`ere un groupe semisimple $\rmG$ classique
r\'eel et connexe. On suppose de plus que $\rmG$ poss\`ede un
sousgroupe de Cartan compact. On d\'efinit une famille de
sousalg\`ebres de Lie associ\'ee \`a $\g = \Lie(\rmG)$, de m\^eme rang
que $\g$ dont tous les facteurs simples sont de rang $1$ ou~$2$.
Soit $\g'$ une telle sousalg\`ebre de Lie. On construit alors une
application de transfert des int\'egrales orbitales de $\g'$ dans
l'espace des int\'egrales orbitales de $\g$. On montre que cette
application est d\'efinie d\`es que $\g$ ne poss\`ede pas de facteur
simple r\'eel de type $\CI$ de rang sup\'erieur ou \'egal \`a $3$.
Si de plus, $\g$ ne poss\`ede pas de facteur simple de type $\BI$ de
rang sup\'erieur \`a $3$, on montre la surjectivit\'e de cette
application de transfert.


1050  Examples of CalabiYau 3Folds of $\mathbb{P}^{7}$ with $\rho=1$ Bertin, MarieAmélie
We give some examples of CalabiYau $3$folds with $\rho=1$ and
$\rho=2$, defined over $\mathbb{Q}$ and constructed as
$4$codimensional subvarieties of $\mathbb{P}^7$ via commutative
algebra methods. We explain how to deduce their Hodge diamond and
top Chern classes from computer based computations over some
finite field $\mathbb{F}_{p}$. Three of our examples (of degree
$17$ and $20$) are new. The two others (degree $15$ and $18$) are
known, and we recover their wellknown invariants with our
method. These examples are built out of GulliksenNeg{\aa}rd and
KustinMiller complexes of locally free sheaves.


1073  On the $2$Rank of the Hilbert Kernel of Number Fields Griffiths, Ross; Lescop, Mikaël
Let $E/F$ be a quadratic extension of
number fields. In this paper, we show that the genus formula for
Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the
$2$rank of the Hilbert kernel of $E$ provided that the $2$primary
Hilbert kernel of $F$ is trivial. However, since the original genus
formula is not explicit enough in a very particular case, we first
develop a refinement of this formula in order to employ it in the
calculation of the $2$rank of $E$ whenever $F$ is totally real with
trivial $2$primary Hilbert kernel. Finally, we apply our results to
quadratic, biquadratic, and triquadratic fields which include
a complete $2$rank formula for the family of fields
$\Q(\sqrt{2},\sqrt{\delta})$ where $\delta$ is a squarefree integer.


1092  Minimal Transitive Factorizations of Permutations into Cycles Irving, John
We introduce a new approach to an enumerative problem
closely linked with the geometry of branched coverings,
that is, we study the number $H_{\alpha}(i_2,i_3,\dots)$ of ways a
given permutation (with cycles described by the partition $\a$) can be
decomposed into a product of exactly $i_2$ 2cycles, $i_3$ 3cycles,
etc., with certain minimality and transitivity conditions imposed on the factors. The method is to
encode such factorizations as planar maps with certain descent structure and apply a new combinatorial
decomposition to make their enumeration more manageable. We apply our technique to determine
$H_{\alpha}(i_2,i_3,\dots)$ when $\a$ has one or two parts, extending earlier work of Goulden and Jackson.
We also show how these methods are readily modified to count inequivalent factorizations, where
equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits us to
generalize recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of
their analysis.


1118  Petits points d'une surface Pontreau, Corentin
Pour toute sousvari\'et\'e g\'eom\'etriquement irr\'eductible $V$
du grou\pe multiplicatif
$\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de
translat\'es de tores exceptionnels
inclus dans $V$, tous les points sont de hauteur minor\'ee par une
certaine quantit\'e $q(V)^{1}>0$. On conna\^it de plus une borne
sup\'erieure pour la somme des degr\'es de ces translat\'es de
tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$.
Ceci n'est pas le cas si l'on exige une minoration quasioptimale
pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e.


1151  Covering Maps and Periodic Functions on Higher Dimensional Sierpinski Gaskets Ruan, HuoJun; Strichartz, Robert S.
We construct covering maps from infinite blowups of the
$n$dimensional Sierpinski gasket $SG_n$ to certain compact
fractafolds based on $SG_n$. These maps are fractal analogs of the
usual covering maps from the line to the circle. The construction
extends work of the second author in the case $n=2$, but a
different method of proof is needed, which amounts to solving a
Sudokutype puzzle. We can use the covering maps to define the
notion of periodic function on the blowups. We give a
characterization of these periodic functions and describe the
analog of Fourier series expansions. We study covering maps onto
quotient fractalfolds. Finally, we show that such covering maps
fail to exist for many other highly symmetric fractals.


1182  Periodic and Almost Periodic Functions on Infinite Sierpinski Gaskets Strichartz, Robert S.
We define periodic functions on infinite blowups of the Sierpinski
gasket as lifts of functions defined on certain compact fractafolds
via covering maps. This is analogous to defining periodic functions
on the line as lifts of functions on the circle via covering maps. In
our setting there is only a countable set of covering maps. We
give two different characterizations of periodic functions in terms of
repeating patterns. However, there is no discrete group action that
can be used to characterize periodic functions. We also give a
Fourier series type description in terms of periodic eigenfunctions of
the Laplacian. We define almost periodic functions as uniform limits
of periodic functions.


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.

