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3  Locally Compact Pro$C^*$Algebras Amini, Massoud
Let $X$ be a locally compact noncompact Hausdorff topological space. Consider
the algebras $C(X)$, $C_b(X)$, $C_0(X)$, and $C_{00}(X)$ of respectively arbitrary,
bounded, vanishing at infinity, and compactly supported continuous functions on $X$.
Of these, the second and third are $C^*$algebras, the fourth is a normed algebra,
whereas the first is only a topological algebra (it is indeed a pro$C^\ast$algebra).
The interesting fact about these algebras is that if one of them is given, the
others can be obtained using functional analysis tools. For instance, given the
$C^\ast$algebra $C_0(X)$, one can get the other three algebras by
$C_{00}(X)=K\bigl(C_0(X)\bigr)$, $C_b(X)=M\bigl(C_0(X)\bigr)$, $C(X)=\Gamma\bigl(
K(C_0(X))\bigr)$, where the right hand sides are the Pedersen ideal, the
multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of
$C_0(X)$, respectively. In this article we consider the possibility of these
transitions for general $C^\ast$algebras. The difficult part is to start with a
pro$C^\ast$algebra $A$ and to construct a $C^\ast$algebra $A_0$ such that
$A=\Gamma\bigl(K(A_0)\bigr)$. The pro$C^\ast$algebras for which this is
possible are called {\it locally compact\/} and we have characterized them using
a concept similar to that of an approximate identity.


23  Ternary Diophantine Equations via Galois Representations and Modular Forms Bennett, Michael A.; Skinner, Chris M.
In this paper, we develop techniques for solving ternary Diophantine
equations of the shape $Ax^n + By^n = Cz^2$, based upon the theory of
Galois representations and modular forms. We subsequently utilize
these methods to completely solve such equations for various choices
of the parameters $A$, $B$ and $C$. We conclude with an application
of our results to certain classical polynomialexponential equations,
such as those of RamanujanNagell type.


55  $\mathbb{Z}[\sqrt{14}]$ is Euclidean Harper, Malcolm
We provide the first unconditional proof that the ring $\mathbb{Z}
[\sqrt{14}]$ is a Euclidean domain. The proof is generalized to
other real quadratic fields and to cyclotomic extensions of
$\mathbb{Q}$. It is proved that if $K$ is a real quadratic field
(modulo the existence of two special primes of $K$) or if $K$ is a
cyclotomic extension of $\mathbb{Q}$ then:


71  Euclidean Rings of Algebraic Integers Harper, Malcolm; Murty, M. Ram
Let $K$ be a finite Galois extension of the field of rational numbers
with unit rank greater than~3. We prove that the ring of integers of
$K$ is a Euclidean domain if and only if it is a principal ideal
domain. This was previously known under the assumption of the
generalized Riemann hypothesis for Dedekind zeta functions. We now
prove this unconditionally.


77  HighDimensional Graphical Networks of SelfAvoiding Walks Holmes, Mark; Járai, Antal A.; Sakai, Akira; Slade, Gordon
We use the lace expansion to analyse networks of mutuallyavoiding
selfavoiding walks, having the topology of a graph. The networks are
defined in terms of spreadout selfavoiding walks that are permitted
to take large steps. We study the asymptotic behaviour of networks in
the limit of widely separated network branch points, and prove
Gaussian behaviour for sufficiently spreadout networks on
$\mathbb{Z}^d$ in dimensions $d>4$.


115  Estimates of Hausdorff Dimension for the NonWandering Set of an Open Planar Billiard Kenny, Robert
The billiard flow in the plane has a simple geometric definition; the
movement along straight lines of points except where elastic
reflections are made with the boundary of the billiard domain. We
consider a class of open billiards, where the billiard domain is
unbounded, and the boundary is that of a finite number of strictly
convex obstacles. We estimate the Hausdorff dimension of the
nonwandering set $M_0$ of the discrete time billiard ball map, which
is known to be a Cantor set and the largest invariant set. Under
certain conditions on the obstacles, we use a wellknown coding of
$M_0$ \cite{Morita} and estimates using convex fronts related to the
derivative of the billiard ball map \cite{StAsy} to estimate the
Hausdorff dimension of local unstable sets. Consideration of the
local product structure then yields the desired estimates, which
provide asymptotic bounds on the Hausdorff dimension's convergence to
zero as the obstacles are separated.


134  Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States Li, ChiKwong; Sourour, Ahmed Ramzi
Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical
ranges on the algebra $M_n$ of all $n\times n$ complex matrices,
the spatial numerical range
$$
W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\},
$$
where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range
$$
V(A) = \{ f(A) : f \in \mathcal{S} \},
$$
where $\mathcal{S}$ is the set of states on the normed algebra
$M_n$ under the operator norm induced by $\nu$. For a symmetric
norm $\nu$, we identify all linear maps on $M_n$ that preserve
either one of the two norm numerical ranges or the set of states or
vector states. We also identify the numerical radius isometries,
{\it i.e.}, linear maps that preserve the (one) numerical radius
induced by either numerical range. In particular, it is shown that
if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms,
then the linear maps that preserve either numerical range or either
set of states are ``inner'', {\it i.e.}, of the form $A\mapsto
Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a
permutation matrix and the numerical radius isometries are
unimodular scalar multiples of such inner maps. For the $\ell_1$
and the $\ell_\infty$ norms, the results are quite different.


168  On a Certain Residual Spectrum of $\Sp_8$ Pogge, James Todd
Let $G=\Sp_{2n}$ be the symplectic group defined over a number
field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental
problem in the theory of automorphic forms is to decompose the
right regular representation of $G(\mathbb{A})$ acting on the
Hilbert space $L^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)$. Main
contributions have been made by Langlands. He described, using his
theory of Eisenstein series, an orthogonal decomposition of this
space of the form: $L_{\dis}^2 \bigl( G(F)\setminus G(\mathbb{A})
\bigr)=\bigoplus_{(M,\pi)} L_{\dis}^2(G(F) \setminus G(\mathbb{A})
\bigr)_{(M,\pi)}$, where $(M,\pi)$ is a Levi subgroup with a
cuspidal automorphic representation $\pi$ taken modulo conjugacy
(Here we normalize $\pi$ so that the action of the maximal split
torus in the center of $G$ at the archimedean places is trivial.)
and $L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$
is a space of residues of Eisenstein series associated to
$(M,\pi)$. In this paper, we will completely determine the space
$L_{\dis}^2\bigl(G(F)\setminus G(\mathbb{A})\bigr)_{(M,\pi)}$, when
$M\simeq\GL_2\times\GL_2$. This is the first result on the
residual spectrum for nonmaximal, nonBorel parabolic subgroups,
other than $\GL_n$.


194  Selmer Groups of Elliptic Curves with Complex Multiplication Saikia, A.
Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a
number field $F$ with complex multiplication by the ring of integers in $K$.
Let $p$ be a rational prime that splits as $\mathfrak{p}_{1}\mathfrak{p}_{2}$
in $K$. Let $E_{p^{n}}$ denote the $p^{n}$division points on $E$. Assume
that $F(E_{p^{n}})$ is abelian over $K$ for all $n\geq 0$. This paper proves
that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty}$Selmer group of
$E$ over $F(E_{p^{\infty}})$ is a finitely generated free $\Lambda$module,
where $\Lambda$ is the Iwasawa algebra of $\Gal\bigl(F(E_{p^{\infty}})/
F(E_{\mathfrak{p}_{1}^{\infty}\mathfrak{p}_{2}})\bigr)$. It also gives a simple
formula for the rank of the Pontrjagin dual as a $\Lambda$module.


209  A Central Limit Theorem and Law of the Iterated Logarithm for a Random Field with Exponential Decay of Correlations Schmuland, Byron; Sun, Wei
In \cite{P69}, Walter Philipp wrote that ``\dots the law of the
iterated logarithm holds for any process for which the BorelCantelli
Lemma, the central limit theorem with a reasonably good remainder and
a certain maximal inequality are valid.'' Many authors \cite{DW80},
\cite{I68}, \cite{N91}, \cite{OY71}, \cite{Y79} have followed this
plan in proving the law of the iterated logarithm for sequences (or
fields) of dependent random variables.


225  Complex Uniform Convexity and Riesz Measure Blower, Gordon; Ransford, Thomas
The norm on a Banach space gives rise to a subharmonic function on the
complex plane for which the distributional Laplacian gives a Riesz measure.
This measure is calculated explicitly here for Lebesgue $L^p$ spaces and the
von~NeumannSchatten trace ideals. Banach spaces that are $q$uniformly
$\PL$convex in the sense of Davis, Garling and TomczakJaegermann are
characterized in terms of the mass distribution of this measure. This gives
a new proof that the trace ideals $c^p$ are $2$uniformly $\PL$convex for
$1\leq p\leq 2$.


246  Éléments unipotents réguliers des sousgroupes de Levi Bonnafé, Cédric
We investigate the structure of the centralizer of a regular unipotent element
of a Levi subgroup of a reductive group. We also investigate the structure of
the group of components of this centralizer in relation with the notion of
cuspidal local system defined by Lusztig. We determine its unipotent radical,
we prove that it admits a Levi complement, and we get some properties on its Weyl
group. As an application, we prove some results which were announced in previous
paper on regular unipotent elements.
Nous \'etudions la structure du centralisateur d'un \'el\'ement unipotent
r\'egulier d'un sousgroupe de Levi d'un groupe r\'eductif, ainsi que la structure
du groupe des composantes de ce centralisateur en relation avec la notion de
syst\`eme local cuspidal d\'efinie par Lusztig. Nous d\'eterminons son radical
unipotent, montrons l'existence d'un compl\'ement de Levi et \'etudions la
structure de son groupe de Weyl. Comme application, nous d\'emontrons des
r\'esultats qui \'etaient annonc\'es dans un pr\'ec\'edent article de l'auteur
sur les \'el\'ements unipotents r\'eguliers.


277  Spectral Properties of the Commutator of Bergman's Projection and the Operator of Multiplication by an Analytic Function Dostanić, Milutin R.
It is shown that the singular values of the operator $aPPa$, where $P$ is
Bergman's projection over a bounded domain $\Omega$ and $a$ is a function
analytic on $\bar{\Omega}$, detect the length of the boundary of $a(\Omega)$.
Also we point out the relation of that operator and the spectral asymptotics
of a Hankel operator with an antianalytic symbol.


293  Structure of modules induced from simple modules with minimal annihilator Khomenko, Oleksandr; Mazorchuk, Volodymyr
We study the structure of generalized Verma modules over a
semisimple complex finitedimensional Lie algebra, which are
induced from simple modules over a parabolic subalgebra. We consider
the case when the annihilator of the starting simple module is a
minimal primitive ideal if we restrict this module to the Levi factor of
the parabolic subalgebra. We show that these modules correspond to
proper standard modules in some parabolic generalization of the
BernsteinGelfandGelfand category $\Oo$ and prove that the blocks of
this parabolic category are equivalent to certain blocks of the
category of HarishChandra bimodules. From this we derive, in
particular, an irreducibility criterion for generalized Verma modules.
We also compute the composition multiplicities of those simple
subquotients, which correspond to the induction from simple modules
whose annihilators are minimal primitive ideals.


310  The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order Llibre, Jaume; Schlomiuk, Dana
In this article we determine the global geometry of the planar
quadratic differential systems with a weak focus of third order. This
class plays a significant role in the context of Hilbert's 16th
problem. Indeed, all examples of quadratic differential systems with
at least four limit cycles, were obtained by perturbing a system in
this family. We use the algebrogeometric concepts of divisor and
zerocycle to encode global properties of the systems and to give
structure to this class. We give a theorem of topological
classification of such systems in terms of integervalued affine
invariants. According to the possible values taken by them in this
family we obtain a total of $18$ topologically distinct phase
portraits. We show that inside the class of all quadratic systems
with the topology of the coefficients, there exists a neighborhood of
the family of quadratic systems with a weak focus of third order and
which may have graphics but no polycycle in the sense of \cite{DRR}
and no limit cycle, such that any quadratic system in this
neighborhood has at most four limit cycles.


344  Predual of the Multiplier Algebra of $A_p(G)$ and Amenability Miao, Tianxuan
For a locally compact group $G$ and $1<p<\infty$, let $A_p(G)$ be the
HerzFig\`aTalamanca algebra and let $PM_p(G)$ be its dual Banach space.
For a Banach $A_p(G)$module $X$ of $PM_p(G)$, we prove that the multiplier
space $\mathcal{M}\xxbigl(A_p(G),X^*\xxbigr)$ is the dual Banach space of $Q_X$,
where $Q_X$ is the norm closure of the linear span $A_p(G) X$ of $u f$ for
$u\in A_p(G)$ and $f\in X$ in the dual of $\mathcal{M}\xxbigl(A_p(G),X^*\xxbigr)$.
If $p=2$ and $PF_p(G)\subseteq X$, then $A_p(G)X$ is closed in $X$ if and only
if $G$ is amenable. In particular, we prove that the multiplier algebra $MA_p(G)$
of $A_p(G)$ is the dual of $Q$, where $Q$ is the completion of $L^1(G)$ in the
$\VcdotV _M$norm. $Q$ is characterized by the following: $f\in Q$ if
an only if there are $u_i\in A_p(G)$ and $f_i\in PF_p(G)$ $(i=1,2,\dots)$ with
$\sum_{i=1}^{\infty}\Vert u_i\Vert_{A_p(G)}\Vert f_i\Vert_{PF_p(G)}<\infty$ such
that $f=\sum_{i=1}^{\infty}u_i f_i$ on $MA_p(G)$. It is also proved that if
$A_p(G)$ is dense in $MA_p(G)$ in the associated $w^*$topology, then the
multiplier norm and $\VcdotV _{A_p(G)}$norm are equivalent on
$A_p(G)$ if and only if $G$ is amenable.


356  NonAbelian Generalizations of the Erd\H osKac Theorem Murty, M. Ram; Saidak, Filip
Let $a$ be a natural number greater than $1$.
Let $f_a(n)$ be the order of $a$ mod $n$.
Denote by $\omega(n)$ the number of distinct
prime factors of $n$. Assuming a weak form
of the generalised Riemann hypothesis, we prove
the following conjecture of Erd\"os and Pomerance:
The number of $n\leq x$ coprime to $a$ satisfying
$$\alpha \leq \frac{\omega(f_a(n))  (\log \log n)^2/2
}{ (\log \log n)^{3/2}/\sqrt{3}} \leq \beta $$
is asymptotic to
$$\left(\frac{ 1 }{ \sqrt{2\pi}} \int_{\alpha}^{\beta}
e^{t^2/2}dt\right)
\frac{x\phi(a) }{ a}, $$
as $x$ tends to infinity.


373  An Elementary Proof of a Weak Exceptional Zero Conjecture Orton, Louisa
In this paper we extend Darmon's theory of ``integration on $\uh_p\times \uh$''
to cusp forms $f$ of higher even weight. This enables us to prove a ``weak
exceptional zero conjecture'': that when the $p$adic $L$function of $f$ has
an exceptional zero at the central point, the $\mathcal{L}$invariant arising is
independent of a twist by certain Dirichlet characters.


406  Theta Series, Eisenstein Series and Poincaré Series over Function Fields Pál, Ambrus
We construct analogues of theta series, Eisenstein series and
Poincar\'e series for function fields of one variable over finite
fields, and prove their basic properties.


431  Group Actions and Singular Martingales II, The Recognition Problem Rosenblatt, Joseph; Taylor, Michael
We continue our investigation in [RST] of a martingale formed by picking a
measurable set $A$ in a compact group $G$, taking random rotates of $A$, and
considering measures of the resulting intersections, suitably normalized. Here
we concentrate on the inverse problem of recognizing $A$ from a small amount of
data from this martingale. This leads to problems in harmonic analysis on $G$,
including an analysis of integrals of products of Gegenbauer polynomials.


449  The Best Constants Associated with Some Weak Maximal Inequalities in Ergodic Theory Demeter, Ciprian
We introduce a new device of measuring the degree of the failure of convergence
in the ergodic theorem along subsequences of integers. Relations with other types
of bad behavior in ergodic theory and applications to weighted averages are also
discussed.


472  InfiniteDimensional Polyhedrality Fonf, Vladimir P.; Veselý, Libor
This paper deals with generalizations of the notion of a polytope to infinite
dimensions. The most general definition is the following: a bounded closed
convex subset of a Banach space is called a polytope if each of its
finitedimensional affine sections is a (standard) polytope.


495  Coinvariant Algebras of Finite Subgroups of $\SL(3,C)$ Gomi, Yasushi; Nakamura, Iku; Shinoda, Kenichi
For most of the finite subgroups of $\SL(3,\mathbf{C})$, we give explicit formulae for
the Molien series of the coinvariant algebras, generalizing McKay's formulae
\cite{M99} for subgroups of $\SU(2)$. We also study the $G$orbit Hilbert scheme
$\Hilb^G(\mathbf{C}^3)$ for any finite subgroup $G$ of $\SO(3)$, which is known to be a
minimal (crepant) resolution of the orbit space $\mathbf{C}^3/G$. In this case the fiber
over the origin of the HilbertChow morphism from $\Hilb^G(\mathbf{C}^3)$ to $\mathbf{C}^3/G$
consists of finitely many smooth rational curves, whose planar dual graph is
identified with a certain subgraph of the representation graph of $G$. This is
an $\SO(3)$ version of the McKay correspondence in the $\SU(2)$ case.


529  Asymptotics for Minimal Discrete Riesz Energy on Curves in $\R^d$ MartínezFinkelshtein, A.; Maymeskul, V.; Rakhmanov, E. A.; Saff, E. B.
We consider the $s$energy
$$
E(\ZZ_n;s)=\sum_{i \neq j} K(\z_{i,n}z_{j,n}\;s)
$$
for point sets $\ZZ_n=\{ z_{k,n}:k=0,\dots,n\}$ on certain compact sets
$\Ga$ in $\R^d$ having finite onedimensional Hausdorff measure, where
$$
K(t;s)=
\begin{cases}
t^{s} ,& \mbox{if } s>0, \\
\ln t, & \mbox{if } s=0,
\end{cases}
$$
is the Riesz kernel. Asymptotics for the minimum $s$energy and the
distribution of minimizing sequences of points is studied. In
particular, we prove that, for $s\geq 1$, the minimizing nodes for a
rectifiable Jordan curve $\Ga$ distribute asymptotically uniformly with
respect to arclength as $n\to\infty$.


553  Cohomology Ring of Symplectic Quotients by Circle Actions Mohammadalikhani, Ramin
In this article we are concerned with how to compute the cohomology ring
of a symplectic quotient by a circle action using the information we have
about the cohomology of the original manifold and some data at the fixed
point set of the action. Our method is based on the TolmanWeitsman theorem
which gives a characterization of the kernel of the Kirwan map. First we
compute a generating set for the kernel of the Kirwan map for the case of
product of compact connected manifolds such that the cohomology ring of each
of them is generated by a degree two class. We assume the fixed point set is
isolated; however the circle action only needs to be ``formally Hamiltonian''.
By identifying the kernel, we obtain the cohomology ring of the symplectic
quotient. Next we apply this result to some special cases and in particular
to the case of products of two dimensional spheres. We show that the results
of Kalkman and HausmannKnutson are special cases of our result.


566  Geodesics in a Manifold with Heisenberg Group as Boundary Ni, Yilong
The Heisenberg group is considered as the boundary of a manifold. A class
of hypersurfaces in this manifold can be regarded as copies of the Heisenberg
group. The properties of geodesics in the interior and on the hypersurfaces
are worked out in detail. These properties are strongly related to those of
the Heisenberg group.


590  The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary Ni, Yilong
We study the Riemannian LaplaceBeltrami operator $L$ on a Riemannian
manifold with Heisenberg group $H_1$ as boundary. We calculate the heat
kernel and Green's function for $L$, and give global and small time
estimates of the heat kernel. A class of hypersurfaces in this
manifold can be regarded as approximations of $H_1$. We also restrict
$L$ to each hypersurface and calculate the corresponding heat kernel
and Green's function. We will see that the heat kernel and Green's
function converge to the heat kernel and Green's function on the
boundary.


612  Solvable Points on Projective Algebraic Curves Pál, Ambrus
We examine the problem of finding rational points defined over
solvable extensions on algebraic curves defined over general fields.
We construct nonsingular, geometrically irreducible projective curves
without solvable points of genus $g$, when $g$ is at least $40$, over
fields of arbitrary characteristic. We prove that every smooth,
geometrically irreducible projective curve of genus $0$, $2$, $3$ or
$4$ defined over any field has a solvable point. Finally we prove
that every genus $1$ curve defined over a local field of
characteristic zero with residue field of characteristic $p$ has a
divisor of degree prime to $6p$ defined over a solvable extension.


638  Multisymplectic Reduction for Proper Actions Śniatycki, Jędrzej
We consider symmetries of the Dedonder equation arising from
variational problems with partial derivatives. Assuming a proper
action of the symmetry group, we identify a set of reduced equations
on an open dense subset of the domain of definition of the fields
under consideration. By continuity, the Dedonder equation is
satisfied whenever the reduced equations are satisfied.


655  On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains Tao, Xiangxing; Wang, Henggeng
We consider the Neumann problem for the Schr\"odinger equations $\Delta u+Vu=0$,
with singular nonnegative potentials $V$ belonging to the reverse H\"older class
$\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given
boundary data $g$ in $H^p$ or $L^p$ for $1\epsilon<p\leq 2$, where $0<\epsilon
<\frac 1n$, it is shown that there is a unique solution, $u$, that solves the
Neumann problem for the given data and such that the nontangential maximal
function of $\nabla u$ is in $L^p(\partial\Omega)$. Moreover, the uniform
estimates are found.


673  Défaut de semistabilité des courbes elliptiques dans le cas non ramifié Cali, Élie
Let $\overline {\Q_2}$ be an algebraic closure of $\Q_2$ and $K$ be an unramified
finite extension of $\Q_2$ included in $\overline {\Q_2}$. Let $E$ be an elliptic
curve defined over $K$ with additive reduction over $K$, and having an integral
modular invariant. Let us denote by $K_{nr}$ the maximal unramified extension of
$K$ contained in $\overline {\Q_2}$. There exists a smallest finite extension $L$
of $K_{nr}$ over which $E$ has good reduction. We determine in this paper the
degree of the extension $L/K_{nr}$.


699  Bump Functions with Hölder Derivatives Gaspari, Thierry
We study the range of the gradients
of a $C^{1,\al}$smooth bump function defined on a Banach space.
We find that this set must satisfy two geometrical conditions:
It can not be too flat and it satisfies a strong compactness condition
with respect to an appropriate distance.
These notions are defined precisely below.
With these results we illustrate the differences with
the case of $C^1$smooth bump functions.
Finally, we give a sufficient condition on a subset of $X^{\ast}$ so that it is
the set of the gradients of a $C^{1,1}$smooth bump function.
In particular, if $X$ is an infinite dimensional Banach space
with a $C^{1,1}$smooth bump function,
then any convex open bounded subset of $X^{\ast}$ containing $0$ is the set
of the gradients of a $C^{1,1}$smooth bump function.


716  Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions Guardo, Elena; Van Tuyl, Adam
We study the Hilbert functions of fat points in $\popo$.
If $Z \subseteq \popo$ is an arbitrary fat point scheme, then
it can be shown that for every $i$ and $j$ the values of the Hilbert
function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for
$l \gg 0$. We show how to determine these eventual values
by using only the multiplicities of the points, and the
relative positions of the points in $\popo$. This enables
us to compute all but a finite number values of $H_{Z}$
without using the coordinates of points.
We also characterize the ACM fat point schemes
sing our description of the eventual behaviour. In fact,
n the case that $Z \subseteq \popo$ is ACM, then
the entire Hilbert function and its minimal free resolution
depend solely on knowing the eventual values of the Hilbert function.


742  Similarity Classification of CowenDouglas Operators Jiang, Chunlan
Let $\cal H$ be a complex separable Hilbert space
and ${\cal L}({\cal H})$ denote the collection of
bounded linear operators on ${\cal H}$.
An operator $A$ in ${\cal L}({\cal H})$
is said to be strongly irreducible, if
${\cal A}^{\prime}(T)$, the commutant of $A$, has no nontrivial idempotent.
An operator $A$ in ${\cal L}({\cal H})$ is said to a CowenDouglas
operator, if there exists $\Omega$, a connected open subset of
$C$, and $n$, a positive integer, such that
(a) ${\Omega}{\subset}{\sigma}(A)=\{z{\in}C; Az {\text {not invertible}}\};$
(b) $\ran(Az)={\cal H}$, for $z$ in $\Omega$;
(c) $\bigvee_{z{\in}{\Omega}}$\ker$(Az)={\cal H}$ and
(d) $\dim \ker(Az)=n$ for $z$ in $\Omega$.
In the paper, we give a similarity classification of strongly
irreducible CowenDouglas operators by using the $K_0$group of
the commutant algebra as an invariant.


776  Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices Lim, Yongdo
We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
CartanHadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$


794  SemiClassical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy Michel, Laurent
We study the semiclassical behavior as $h\rightarrow 0$ of the scattering
amplitude $f(\theta,\omega,\lambda,h)$ associated to a Schr\"odinger operator
$P(h)=\frac 1 2 h^2\Delta +V(x)$ with shortrange trapping
perturbations. First we realize a spatial localization in the general case
and we deduce a bound of the scattering amplitude on the real
line. Under an additional assumption on the resonances, we show that
if we modify the potential $V(x)$ in a domain lying behind the
barrier $\{x:V(x)>\lambda\}$, the scattering amplitude
$f(\theta,\omega,\lambda,h)$ changes by a term of order
$\O(h^{\infty})$. Under an escape assumption on the classical
trajectories incoming with fixed direction $\omega$, we obtain
an asymptotic development of $f(\theta,\omega,\lambda,h)$
similar to the one established in thenontrapping case.


825  Differentiability Properties of Optimal Value Functions Penot, JeanPaul
Differentiability properties of optimal value functions associated with
perturbed optimization problems require strong assumptions. We consider such
a set of assumptions which does not use compactness hypothesis but which
involves a kind of coherence property. Moreover, a strict differentiability
property is obtained by using techniques of Ekeland and Lebourg and a result
of Preiss. Such a strengthening is required in order to obtain genericity
results.


843  Type Decomposition and the Rectangular AFD Property for $W^*$TRO's Ruan, ZhongJin
We study the type decomposition and the rectangular AFD property for
$W^*$TRO's. Like von Neumann algebras, every $W^*$TRO can be
uniquely decomposed into the direct sum of $W^*$TRO's of
type $I$, type $II$, and type $III$.
We may further consider $W^*$TRO's of type $I_{m, n}$
with cardinal numbers $m$ and $n$, and consider $W^*$TRO's of
type $II_{\lambda, \mu}$ with $\lambda, \mu = 1$ or $\infty$.
It is shown that every separable stable $W^*$TRO
(which includes type $I_{\infty,\infty}$, type $II_{\infty,
\infty}$ and type $III$) is TROisomorphic to a von Neumann algebra.
We also introduce the rectangular version of the approximately finite
dimensional property for $W^*$TRO's.
One of our major results is to show that a separable $W^*$TRO
is injective if and only
if it is rectangularly approximately finite dimensional.
As a consequence of this result, we show that a dual operator space
is injective if and only if its operator predual is a rigid
rectangular ${\OL}_{1, 1^+}$ space (equivalently, a rectangular


871  Lie Elements and Knuth Relations Schocker, Manfred
A coplactic class in the symmetric group $\Sym_n$ consists of all
permutations in $\Sym_n$ with a given Schensted $Q$symbol, and may
be described in terms of local relations introduced by Knuth. Any
Lie element in the group algebra of $\Sym_n$ which is constant on
coplactic classes is already constant on descent classes. As a
consequence, the intersection of the Lie convolution algebra
introduced by Patras and Reutenauer and the coplactic algebra
introduced by Poirier and Reutenauer is the direct sum of all
Solomon descent algebras.


883  Kirillov Theory for a Class of Discrete Nilpotent Groups Tandra, Haryono; Moran, William
This paper is concerned with the Kirillov map for a class of
torsionfree nilpotent groups $G$. $G$ is assumed to be discrete,
countable and $\pi$radicable, with $\pi$ containing the primes
less than or equal to the nilpotence class of $G$. In addition,
it is assumed that all of the characters of $G$ have idempotent
absolute value. Such groups are shown to be plentiful.


897  Finding and Excluding $b$ary MachinType Individual Digit Formulae Borwein, Jonathan M.; Borwein, David; Galway, William F.
Constants with formulae of the form treated by D.~Bailey,
P.~Borwein, and S.~Plouffe (BBP formulae to a given base $b$) have
interesting computational properties, such as allowing single
digits in their base $b$ expansion to be independently computed,
and there are hints that they
should be normal numbers, {\em i.e.,} that their base $b$ digits
are randomly distributed. We study a formally limited subset of BBP
formulae, which we call Machintype BBP formulae, for which it
is relatively easy to determine whether or not a given constant
$\kappa$ has a Machintype BBP formula. In particular, given $b \in
\mathbb{N}$, $b>2$, $b$ not a proper power, a $b$ary Machintype
BBP arctangent formula for $\kappa$ is a formula of the form $\kappa
= \sum_{m} a_m \arctan(b^{m})$, $a_m \in \mathbb{Q}$, while when
$b=2$, we also allow terms of the form $a_m \arctan(1/(12^m))$. Of
particular interest, we show that $\pi$ has no Machintype BBP
arctangent formula when $b \neq 2$. To the best of our knowledge,
when there is no Machintype BBP formula for a constant then no BBP
formula of any form is known for that constant.


926  KHomology of the Rotation Algebras $A_{\theta}$ Hadfield, Tom
We study the Khomology of the rotation algebras
$A_{\theta}$ using the sixterm cyclic sequence
for the Khomology of a crossed product by
${\bf Z}$. In the case that $\theta$ is irrational,
we use Pimsner and Voiculescu's work on AFembeddings
of the $A_{\theta}$ to search for the missing
generator of the even Khomology.


945  Smoothness of Quotients Associated \\With a Pair of Commuting Involutions Helminck, Aloysius G.; Schwarz, Gerald W.
Let $\sigma$, $\theta$ be commuting involutions of the connected semisimple
algebraic group $G$ where $\sigma$, $\theta$ and $G$ are defined over
an algebraically closed field $\k$, $\Char \k=0$. Let $H:=G^\sigma$
and $K:=G^\theta$ be the fixed point groups. We have an action
$(H\times K)\times G\to G$, where $((h,k),g)\mapsto hgk\inv$, $h\in
H$, $k\in K$, $g\in G$. Let $\quot G{(H\times K)}$ denote the
categorical quotient $\Spec \O(G)^{H\times K}$. We determine when this
quotient is smooth. Our results are a generalization of those of
Steinberg \cite{Steinberg75}, Pittie \cite{Pittie72} and Richardson
\cite{Rich82b} in the symmetric case where $\sigma=\theta$ and $H=K$.


963  A BerryEsseen Type Theorem on Nilpotent Covering Graphs Ishiwata, Satoshi
We prove an estimate for the speed of convergence of the
transition probability for a symmetric random walk
on a nilpotent covering graph.
To obtain this estimate, we give a complete proof of
the Gaussian bound for the gradient of the Markov kernel.


983  Fubini's Theorem for Ultraproducts \\of Noncommutative $L_p$Spaces Junge, Marius
Let $(\M_i)_{i\in I}$, $(\N_j)_{j\in J}$ be families of von
Neumann algebras and $\U$, $\U'$ be ultrafilters in $I$, $J$,
respectively. Let $1\le p<\infty$ and $\nen$. Let $x_1$,\dots,$x_n$ in
$\prod L_p(\M_i)$ and $y_1$,\dots,$y_n$ in $\prod L_p(\N_j)$ be
bounded families. We show the following equality
$$
\lim_{i,\U} \lim_{j,\U'} \Big\ \summ_{k=1}^n x_k(i)\otimes
y_k(j)\Big\_{L_p(\M_i\otimes \N_j)} = \lim_{j,\U'} \lim_{i,\U}
\Big\ \summ_{k=1}^n x_k(i)\otimes y_k(j)\Big\_{L_p(\M_i\otimes \N_j)} .
$$
For $p=1$ this Fubini type result is related to the local
reflexivity of duals of $C^*$algebras. This fails for $p=\infty$.


1022  NonOrientable Surfaces and Dehn Surgeries Matignon, D.; Sayari, N.
Let $K$ be a knot in $S^3$. This paper is devoted to Dehn surgeries which create
$3$manifolds containing a closed nonorientable surface $\ch S$. We look at the
slope ${p}/{q}$ of the surgery, the Euler characteristic $\chi(\ch S)$ of the
surface and the intersection number $s$ between $\ch S$ and the core of the Dehn
surgery. We prove that if $\chi(\hat S) \geq 15  3q$, then $s=1$. Furthermore,
if $s=1$ then $q\leq 43\chi(\ch S)$ or $K$ is cabled and $q\leq 85\chi(\ch S)$.
As consequence, if $K$ is hyperbolic and $\chi(\ch S)=1$, then $q\leq 7$.


1034  Semiclassical Integrability,Hyperbolic Flows and the Birkhoff Normal Form Rouleux, Michel
We prove that a Hamiltonian
$p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a
nondegenerate critical
point $\rho_0$ of the energy, provided that the fundamental matrix
at $\rho_0$ has rationally independent eigenvalues, none purely imaginary.
This is done by using Birkhoff normal forms,
which turn out to be convergent in the $C^\infty$ sense.
We also give versions of the LewisSternberg normal form
near a hyperbolic fixed point of a canonical transformation.
Then we investigate the complex case, showing that when
$p$ is holomorphic near $\rho_0\in T^*{\bf C}^n$, then
$\re p$ becomes integrable in the complex domain for
real times, while the Birkhoff series and the Birkhoff transforms
may not converge, {\em i.e.,} $p$ may not be integrable. These normal forms
also hold in the semiclassical frame.


1068  Regular Embeddings of Generalized Hexagons Steinbach, Anja; Van Maldeghem, Hendrik
We classify the generalized hexagons which are laxly
embedded in projective space such that the embedding is flat and
polarized. Besides the standard examples related to the hexagons
defined over the algebraic groups of type $\ssG_2$, $^3\ssD_4$ and
$^6\ssD_4$ (and occurring in projective dimensions $5,6,7$), we
find new examples in unbounded dimension related to the mixed
groups of type $\ssG_2$.


1094  CycleLevel Intersection Theory for Toric Varieties Thomas, Hugh
This paper addresses the problem of constructing a
cyclelevel intersection theory for toric varieties.
We show that by making one global choice,
we can determine a cycle representative
for the intersection of an equivariant Cartier divisor with an invariant
cycle on a toric variety. For a toric variety
defined by a fan in $N$, the choice consists of giving an
inner product or a complete flag for $M_\Q=
\Qt \Hom(N,\mathbb{Z})$, or more
generally giving for each cone $\s$ in the fan a linear subspace of
$M_\Q$ complementary to $\s^\perp$, satisfying certain compatibility
conditions.
We show that these intersection cycles have properties analogous to the
usual intersections modulo rational equivalence.
If $X$ is simplicial (for instance, if $X$ is nonsingular),
we obtain a commutative ring structure
to the invariant cycles of $X$ with rational
coefficients. This ring structure determines cycles representing
certain characteristic classes of the toric variety.
We also discuss
how to define intersection cycles that require no choices,
at the expense of increasing
the size of the coefficient field.


1121  Division par un polynôme hyperbolique Chaumat, Jacques; Chollet, AnneMarie
On se donne un intervalle ouvert non vide $\omega$ de
$\mathbb R$, un ouvert connexe non vide $\Omega$ de $\mathbb R_s$ et
un polyn\^ome unitaire
\[
P_m(z, \lambda) = z^m + a_1(\lambda)z^{m1} = +\dots + a_{m1}(\lambda)
z + a_m(\lambda),
\]
de degr\'e $m>0$, d\'ependant du param\`etre $\lambda \in \Omega$. Un
tel polyn\^ome est dit $\omega$hyperbolique si, pour tout $\lambda
\in \Omega$, ses racines sont r\'eelles et appartiennent \`a $\omega$.


1145  On Log $\mathbb Q$Homology Planes and Weighted Projective Planes Daigle, Daniel; Russell, Peter
We classify normal affine surfaces with trivial MakarLimanov
invariant and finite Picard group of the smooth locus, realizing them
as open subsets of weighted projective planes.
We also show that such a surface admits, up to conjugacy,
one or two $G_a$actions.


1190  Meromorphic Functions Sharing the Same Zeros and Poles Frank, Günter; Hua, Xinhou; Vaillancourt, Rémi
In this paper, Hinkkanen's problem (1984) is completely solved,
{\em i.e.,} it is shown that any meromorphic function $f$ is determined
by its zeros and poles
and the zeros of $f^{(j)}$ for $j=1,2,3,4$


1228  On the Connectedness of Moduli Spaces of Flat Connections over Compact Surfaces Ho, NanKuo; Liu, ChiuChu Melissa
We study the connectedness of the moduli space
of gauge equivalence classes of flat $G$connections on a compact
orientable surface or a compact nonorientable surface for a class
of compact connected Lie groups. This class includes all the
compact, connected, simply connected Lie groups, and some
nonsemisimple classical groups.


1237  Central Sequence Algebras of a Purely Infinite Simple $C^{*}$algebra Kishimoto, Akitaka
We are concerned with a unital separable nuclear purely infinite
simple $C^{*}$algebra\ $A$ satisfying UCT with a Rohlin flow, as a
continuation of~\cite{Kismh}. Our first result (which is
independent of the Rohlin flow) is to characterize when two {\em
central} projections in $A$ are equivalent by a {\em central}
partial isometry. Our second result shows that the Ktheory of
the central sequence algebra $A'\cap A^\omega$ (for an $\omega\in
\beta\N\setminus\N$) and its {\em fixed point} algebra under the
flow are the same (incorporating the previous result). We will
also complete and supplement the characterization result of the
Rohlin property for flows stated in~ \cite{Kismh}.


1259  The Fourier Algebra for Locally Compact Groupoids Paterson, Alan L. T.
We introduce and investigate using Hilbert modules the properties
of the {\em Fourier algebra} $A(G)$ for
a locally compact groupoid $G$. We establish a duality theorem for
such groupoids in terms of multiplicative module maps. This includes
as a special case the classical duality theorem for locally compact
groups proved by P. Eymard.


1290  Equivariant Formality for Actions of Torus Groups Scull, Laura
This paper contains a comparison of several
definitions of equivariant formality for actions of torus groups. We
develop and prove some relations between the definitions. Focusing on
the case of the circle group, we use $S^1$equivariant minimal models
to give a number of examples of $S^1$spaces illustrating the
properties of the various definitions.


1308  Variations of Mixed Hodge Structures of Multiple Polylogarithms Zhao, Jianqiang
It is well known that multiple polylogarithms give rise to
good unipotent variations of mixed HodgeTate structures.
In this paper we shall {\em explicitly} determine these structures
related to multiple logarithms and some other multiple polylogarithms
of lower weights. The purpose of this explicit construction
is to give some important applications: First we study the limit of
mixed HodgeTate structures and make a conjecture relating the variations
of mixed HodgeTate structures of multiple logarithms to those of
general multiple {\em poly}\/logarithms. Then following
Deligne and Beilinson we describe an
approach to defining the singlevalued
real analytic version of the multiple polylogarithms which
generalizes the wellknown result of Zagier on
classical polylogarithms. In the process we find some interesting
identities relating singlevalued multiple polylogarithms of the
same weight $k$ when $k=2$ and 3. At the end of this paper,
motivated by Zagier's conjecture we pose
a problem which relates the special values of multiple
Dedekind zeta functions of a number field to the singlevalued
version of multiple polylogarithms.


1339  Author Index  Index des auteurs 2004, for 2004  pour
No abstract.

