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3  On Small Complete Sets of Functions Aizenberg, Lev; Vidras, Alekos
Using Local Residues and the Duality Principle a multidimensional
variation of the completeness theorems by T.~Carleman and A.~F.~Leontiev
is proven for the space of holomorphic functions defined on a suitable
open strip $T_{\alpha}\subset {\bf C}^2$. The completeness theorem is a
direct consequence of the Cauchy Residue Theorem in a torus. With
suitable modifications the same result holds in ${\bf C}^n$.


31  On RussellType Modular Equations Chan, Heng Huat; Liaw, WenChin
In this paper, we revisit Russelltype modular equations, a
collection of modular equations first studied systematically by
R.~Russell in 1887. We give a proof of Russell's main theorem and
indicate the relations between such equations and the constructions
of Hilbert class fields of imaginary quadratic fields. Motivated by
Russell's theorem, we state and prove its cubic analogue which
allows us to construct Russelltype modular equations in the theory
of signature~$3$.


47  Comparison of $K$Theory Galois Module Structure Invariants Chinburg, T.; Kolster, M.; Snaith, V. P.
We prove that two, apparently different, classgroup valued Galois
module structure invariants associated to the algebraic $K$groups
of rings of algebraic integers coincide. This comparison result is
particularly important in making explicit calculations.


92  A Stochastic Calculus Approach for the Brownian Snake Dhersin, JeanStéphane; Serlet, Laurent
We study the ``Brownian snake'' introduced by Le Gall, and also
studied by Dynkin, Kuznetsov, Watanabe. We prove that It\^o's
formula holds for a wide class of functionals. As a consequence,
we give a new proof of the connections between the Brownian snake
and superBrownian motion. We also give a new definition of the
Brownian snake as the solution of a wellposed martingale problem.
Finally, we construct a modified Brownian snake whose lifetime is
driven by a pathdependent stochastic equation. This process gives
a representation of some superprocesses.


119  Corrigendum to ``Spectral Theory for the Neumann Laplacian on Planar Domains with HornLike Ends'' Edward, Julian
Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997),
232262) are corrected. A nonstandard regularisation of the
auxiliary operator $A$ appearing in Mourre theory is used.


123  An Algorithm for Fat Points on $\mathbf{P}^2 Harbourne, Brian
Let $F$ be a divisor on the blowup $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.


141  Numerical Ranges Arising from Simple Lie Algebras Li, ChiKwong; Tam, TinYau
A unified formulation is given to various generalizations of the
classical numerical range including the $c$numerical range,
congruence numerical range, $q$numerical range and von Neumann
range. Attention is given to those cases having connections with
classical simple real Lie algebras. Convexity and inclusion
relation involving those generalized numerical ranges are
investigated. The underlying geometry is emphasized.


172  Cubic Base Change for $\GL(2)$ Mao, Zhengyu; Rallis, Stephen
We prove a relative trace formula that establishes the cubic base
change for $\GL(2)$. One also gets a classification of the image
of base change. The case when the field extension is nonnormal
gives an example where a trace formula is used to prove lifting
which is not endoscopic.


197  Sublinearity and Other Spectral Conditions on a Semigroup Radjavi, Heydar
Subadditivity, sublinearity, submultiplicativity, and other
conditions are considered for spectra of pairs of operators on a
Hilbert space. Sublinearity, for example, is a weakening of the
wellknown property~$L$ and means $\sigma(A+\lambda B) \subseteq
\sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The
effect of these conditions is examined on commutativity,
reducibility, and triangularizability of multiplicative semigroups
of operators. A sample result is that sublinearity of spectra
implies simultaneous triangularizability for a semigroup of compact
operators.


225  Localization in Categories of Complexes and Unbounded Resolutions Alonso Tarrío, Leovigildo; Jeremías López, Ana; Souto Salorio, María José
In this paper we show that for a Grothendieck category $\A$ and a
complex $E$ in $\CC(\A)$ there is an associated localization
endofunctor $\ell$ in $\D(\A)$. This means that $\ell$ is
idempotent (in a natural way) and that the objects that go to 0 by
$\ell$ are those of the smallest localizing (= triangulated and
stable for coproducts) subcategory of $\D(\A)$ that contains $E$.
As applications, we construct Kinjective resolutions for complexes
of objects of $\A$ and derive Brown representability for $\D(\A)$
from the known result for $\D(R\text{}\mathbf{mod})$, where $R$ is
a ring with unit.


248  Spectral Problems for NonLinear SturmLiouville Equations with Eigenparameter Dependent Boundary Conditions Binding, Paul A.; Browne, Patrick J.; Watson, Bruce A.
The nonlinear SturmLiouville equation
$$
(py')' + qy = \lambda(1  f)ry \text{ on } [0,1]
$$
is considered subject to the boundary conditions
$$
(a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j =
0,1.
$$
Here $a_0 = 0 = c_0$ and $p,r > 0$ and $q$ are functions depending
on the independent variable $x$ alone, while $f$ depends on $x$,
$y$ and $y'$. Results are given on existence and location of sets
of $(\lambda,y)$ bifurcating from the linearized eigenvalues, and
for which $y$ has prescribed oscillation count, and on completeness
of the $y$ in an appropriate sense.


265  On Orbit Closures of Symmetric Subgroups in Flag Varieties Brion, Michel; Helminck, Aloysius G.
We study $K$orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ and we
determine the isotropy groups. As a consequence, we describe the
closed (resp. affine) orbits in terms of $\theta$stable
(resp. $\theta$split) parabolic subgroups. We also describe the
decomposition of any $(K,P)$double coset in $G$ into
$(K,B)$double cosets, where $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$types.


293  Floer Homology for Knots and $\SU(2)$Representations for Knot Complements and Cyclic Branched Covers Collin, Olivier
In this article, using 3orbifolds singular along a knot with
underlying space a homology sphere $Y^3$, the question of existence
of nontrivial and nonabelian $\SU(2)$representations of the
fundamental group of cyclic branched covers of $Y^3$ along a knot
is studied. We first use Floer Homology for knots to derive an
existence result of nonabelian $\SU(2)$representations of the
fundamental group of knot complements, for knots with a
nonvanishing equivariant signature. This provides information on
the existence of nontrivial and nonabelian
$\SU(2)$representations of the fundamental group of cyclic
branched covers. We illustrate the method with some examples of
knots in $S^3$.


306  Characters of DepthZero, Supercuspidal Representations of the Rank2 Symplectic Group Cunningham, Clifton
This paper expresses the character of certain depthzero
supercuspidal representations of the rank2 symplectic group as the
Fourier transform of a finite linear combination of regular
elliptic orbital integralsan expression which is ideally suited
for the study of the stability of those characters. Building on
work of F.~Murnaghan, our proof involves Lusztig's Generalised
Springer Correspondence in a fundamental way, and also makes use of
some results on elliptic orbital integrals proved elsewhere by the
author using MoyPrasad filtrations of $p$adic Lie algebras. Two
applications of the main result are considered toward the end of
the paper.


332  Multiple Mixing and Rank One Group Actions del Junco, Andrés; Yassawi, Reem
Suppose $G$ is a countable, Abelian group with an element of
infinite order and let ${\cal X}$ be a mixing rank one action of
$G$ on a probability space. Suppose further that the F\o lner
sequence $\{F_n\}$ indexing the towers of ${\cal X}$ satisfies a
``bounded intersection property'': there is a constant $p$ such
that each $\{F_n\}$ can intersect no more than $p$ disjoint
translates of $\{F_n\}$. Then ${\cal X}$ is mixing of all orders.
When $G={\bf Z}$, this extends the results of Kalikow and Ryzhikov
to a large class of ``funny'' rank one transformations. We follow
Ryzhikov's joining technique in our proof: the main theorem follows
from showing that any pairwise independent joining of $k$ copies of
${\cal X}$ is necessarily product measure. This method generalizes
Ryzhikov's technique.


348  Singularités quasiordinaires toriques et polyèdre de Newton du discriminant González Pérez, P. D.
Nous \'etudions les polyn\^omes $F \in \C \{S_\tau\} [Y] $ \`a
coefficients dans l'anneau de germes de fonctions holomorphes au
point sp\'ecial d'une vari\'et\'e torique affine. Nous
g\'en\'eralisons \`a ce cas la param\'etrisation classique des
singularit\'es quasiordinaires. Cela fait intervenir d'une part
une g\'en\'eralization de l'algorithme de NewtonPuiseux, et
d'autre part une relation entre le poly\`edre de Newton du
discriminant de $F$ par rapport \`a $Y$ et celui de $F$ au moyen du
polytopefibre de Billera et Sturmfels~\cite{Sturmfels}. Cela nous
permet enfin de calculer, sous des hypoth\`eses de non
d\'eg\'en\'erescence, les sommets du poly\`edre de Newton du
discriminant a partir de celui de $F$, et les coefficients
correspondants \`a partir des coefficients des exposants de $F$ qui
sont dans les ar\^etes de son poly\`edre de Newton.


369  An Upper Bound on the Least Inert Prime in a Real Quadratic Field Granville, Andrew; Mollin, R. A.; Williams, H. C.
It is shown by a combination of analytic and computational
techniques that for any positive fundamental discriminant $D >
3705$, there is always at least one prime $p < \sqrt{D}/2$ such
that the Kronecker symbol $\left(D/p\right) = 1$.


381  Hardy Space Estimate for the Product of Singular Integrals Miyachi, Akihiko
$H^p$ estimate for the multilinear operators which are finite sums
of pointwise products of singular integrals and fractional
integrals is given. An application to Sobolev space and some
examples are also given.


412  Geometric and Potential Theoretic Results on Lie Groups Varopoulos, N. Th.
The main new results in this paper are contained in the geometric
Theorems 1 and~2 of Section~0.1 below and they are related to
previous results of M.~Gromov and of myself (\cf\
\cite{1},~\cite{2}). These results are used to prove some general
potential theoretic estimates on Lie groups (\cf\ Section~0.3) that
are related to my previous work in the area (\cf\
\cite{3},~\cite{4}) and to some deep recent work of G.~Alexopoulos
(\cf\ \cite{5},~\cite{21}).


438  On Some $q$Analogs of a Theorem of KostantRallis Wallach, N. R.; Willenbring, J.
In the first part of this paper generalizations of Hesselink's
$q$analog of Kostant's multiplicity formula for the action of a
semisimple Lie group on the polynomials on its Lie algebra are given
in the context of the KostantRallis theorem. They correspond to the
cases of real semisimple Lie groups with one conjugacy class of Cartan
subgroup. In the second part of the paper a $q$analog of the
KostantRallis theorem is given for the real group $\SL(4,\mathbb{R})$
(that is $\SO(4)$ acting on symmetric $4 \times 4$ matrices). This
example plays two roles. First it contrasts with the examples of the
first part. Second it has implications to the study of entanglement
of mixed 2 qubit states in quantum computation.


449  An Intertwining Result for $p$adic Groups Adler, Jeffrey D.; Roche, Alan
For a reductive $p$adic group $G$, we compute the supports of the Hecke
algebras for the $K$types for $G$ lying in a certain frequentlyoccurring
class. When $G$ is classical, we compute the intertwining between any
two such $K$types.


468  TwoWeight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type Edmunds, D. E.; Kokilashvili, V.; Meskhi, A.
Twoweight inequalities of strong and weak type are obtained in the
context of spaces of homogeneous type. Various applications are
given, in particular to Cauchy singular integrals on regular curves.


503  The Level 2 and 3 Modular Invariants for the Orthogonal Algebras Gannon, Terry
The `1loop partition function' of a rational conformal field theory
is a sesquilinear combination of characters, invariant under a natural
action of $\SL_2(\bbZ)$, and obeying an integrality condition.
Classifying these is a clearly defined mathematical problem, and at
least for the affine KacMoody algebras tends to have interesting
solutions. This paper finds for each affine algebra $B_r^{(1)}$ and
$D_r^{(1)}$ all of these at level $k\le 3$. Previously, only those at
level 1 were classified. An extraordinary number of exceptionals
appear at level 2the $B_r^{(1)}$, $D_r^{(1)}$ level 2
classification is easily the most anomalous one known and this
uniqueness is the primary motivation for this paper. The only level 3
exceptionals occur for $B_2^{(1)} \cong C_2^{(1)}$ and $D_7^{(1)}$.
The $B_{2,3}$ and $D_{7,3}$ exceptionals are cousins of the ${\cal
E}_6$exceptional and $\E_8$exceptional, respectively, in the
ADE classification for $A_1^{(1)}$, while the level 2 exceptionals
are related to the lattice invariants of affine~$u(1)$.


522  On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems Gui, Changfeng; Wei, Juncheng
We consider the problem
\begin{equation*}
\begin{cases}
\varepsilon^2 \Delta u  u + f(u) = 0, u > 0 & \mbox{in } \Omega\\
\frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega,
\end{cases}
\end{equation*}
where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small
parameter and $f$ is a superlinear, subcritical nonlinearity. It is
known that this equation possesses multiple boundary spike solutions
that concentrate, as $\epsilon$ approaches zero, at multiple critical
points of the mean curvature function $H(P)$, $P \in \partial \Omega$.
It is also proved that this equation has multiple interior spike solutions
which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$.
In this paper, we prove the existence of solutions with multiple spikes
{\it both\/} on the boundary and in the interior. The main difficulty
lies in the fact that the boundary spikes and the interior spikes usually
have different scales of error estimation. We have to choose a special set
of boundary spikes to match the scale of the interior spikes in a
variational approach.


539  On SquareIntegrable Representations of Classical $p$adic Groups Jantzen, Chris
In this paper, we use Jacquet module methods to study the problem
of classifying discrete series for the classical $p$adic groups
$\Sp(2n,F)$ and $\SO(2n+1,F)$.


582  Symplectic Geometry of the Moduli Space of Flat Connections on a Riemann Surface: Inductive Decompositions and Vanishing Theorems Jeffrey, Lisa C.; Weitsman, Jonathan
This paper treats the moduli space ${\cal M}_{g,1}(\Lambda)$ of
representations of the fundamental group of a Riemann surface of
genus $g$ with one boundary component which send the loop around
the boundary to an element conjugate to $\exp \Lambda$, where
$\Lambda$ is in the fundamental alcove of a Lie algebra. We
construct natural line bundles over ${\cal M}_{g,1} (\Lambda)$ and
exhibit natural homology cycles representing the Poincar\'e dual of
the first Chern class. We use these cycles to prove differential
equations satisfied by the symplectic volumes of these spaces.
Finally we give a bound on the degree of a nonvanishing element of
a particular subring of the cohomology of the moduli space of
stable bundles of coprime rank $k$ and degree $d$.


613  Small Solutions of $\phi_1 x_1^2 + \cdots + \phi_n x_n^2 = 0$ Ou, Zhiming M.; Williams, Kenneth S.
Let $\phi_1,\dots,\phi_n$ $(n\geq 2)$ be nonzero integers such that
the equation
$$
\sum_{i=1}^n \phi_i x_i^2 = 0
$$
is solvable in integers $x_1,\dots,x_n$ not all zero. It is shown
that there exists a solution satisfying
$$
0 < \sum_{i=1}^n \phi_i x_i^2 \leq 2 \phi_1 \cdots \phi_n,
$$
and that the constant 2 is best possible.


633  Chern Characters of Fourier Modules Walters, Samuel G.
Let $A_\theta$ denote the rotation algebrathe universal $C^\ast$algebra
generated by unitaries $U,V$ satisfying $VU=e^{2\pi i\theta}UV$, where
$\theta$ is a fixed real number. Let $\sigma$ denote the Fourier
automorphism of $A_\theta$ defined by $U\mapsto V$, $V\mapsto U^{1}$,
and let $B_\theta = A_\theta \rtimes_\sigma \mathbb{Z}/4\mathbb{Z}$ denote
the associated $C^\ast$crossed product. It is shown that there is a
canonical inclusion $\mathbb{Z}^9 \hookrightarrow K_0(B_\theta)$ for each
$\theta$ given by nine canonical modules. The unbounded trace functionals
of $B_\theta$ (yielding the Chern characters here) are calculated to obtain
the cyclic cohomology group of order zero $\HC^0(B_\theta)$ when
$\theta$ is irrational. The Chern characters of the nine modulesand more
importantly, the Fourier moduleare computed and shown to involve techniques
from the theory of Jacobi's theta functions. Also derived are explicit
equations connecting unbounded traces across strong Morita equivalence, which
turn out to be noncommutative extensions of certain theta function equations.
These results provide the basis for showing that for a dense $G_\delta$ set
of values of $\theta$ one has $K_0(B_\theta)\cong\mathbb{Z}^9$ and is
generated by the nine classes constructed here.


673  Sums of Two Squares in Short Intervals Balog, Antal; Wooley, Trevor D.
Let $\calS$ denote the set of integers representable as a sum of two
squares. Since $\calS$ can be described as the unsifted elements of a
sieving process of positive dimension, it is to be expected that
$\calS$ has many properties in common with the set of prime numbers.
In this paper we exhibit ``unexpected irregularities'' in the
distribution of sums of two squares in short intervals, a phenomenon
analogous to that discovered by Maier, over a decade ago, in the
distribution of prime numbers. To be precise, we show that there are
infinitely many short intervals containing considerably more elements
of $\calS$ than expected, and infinitely many intervals containing
considerably fewer than expected.


695  Correspondences, von Neumann Algebras and Holomorphic $L^2$ Torsion Carey, A.; Farber, M.; Mathai, V.
Given a holomorphic Hilbertian bundle on a compact complex manifold, we
introduce the notion of holomorphic $L^2$ torsion, which lies in the
determinant line of the twisted $L^2$ Dolbeault cohomology and
represents a volume element there. Here we utilise the theory of
determinant lines of Hilbertian modules over finite von~Neumann
algebras as developed in \cite{CFM}. This specialises to the
RaySingerQuillen holomorphic torsion in the finite dimensional case.
We compute a metric variation formula for the holomorphic $L^2$
torsion, which shows that it is {\it not\/} in general independent of
the choice of Hermitian metrics on the complex manifold and on the
holomorphic Hilbertian bundle, which are needed to define it. We
therefore initiate the theory of correspondences of determinant lines,
that enables us to define a relative holomorphic $L^2$ torsion for a
pair of flat Hilbertian bundles, which we prove is independent of the
choice of Hermitian metrics on the complex manifold and on the flat
Hilbertian bundles.


737  An Automorphic Theta Module for Quaternionic Exceptional Groups Gan, Wee Teck
We construct an automorphic realization of the global minimal
representation of quaternionic exceptional groups, using the theory
of Eisenstein series, and use this for the study of theta
correspondences.


757  Le problème de Neumann pour certaines équations du type de MongeAmpère sur une variété riemannienne Hanani, Abdellah
Let $(M_n,g)$ be a strictly convex riemannian manifold with
$C^{\infty}$ boundary. We prove the existence\break
of classical solution for the nonlinear elliptic partial
differential equation of MongeAmp\`ere:\break
$\det (u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a
Neumann condition on the boundary of the form $\frac{\partial
u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM
\times \bbR)$ is an everywhere strictly positive function
satisfying some assumptions, $\nu$ stands for the unit normal
vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$
is a nondecreasing function in $u$.


789  The DunfordPettis Property for Symmetric Spaces Kamińska, Anna; Mastyło, Mieczysław
A complete description of symmetric spaces on a separable measure
space with the DunfordPettis property is given. It is shown that
$\ell^1$, $c_0$ and $\ell^{\infty}$ are the only symmetric sequence
spaces with the DunfordPettis property, and that in the class of
symmetric spaces on $(0, \alpha)$, $0 < \alpha \leq \infty$, the only
spaces with the DunfordPettis property are $L^1$, $L^{\infty}$, $L^1
\cap L^{\infty}$, $L^1 + L^{\infty}$, $(L^{\infty})^\circ$ and $(L^1 +
L^{\infty})^\circ$, where $X^\circ$ denotes the norm closure of $L^1
\cap L^{\infty}$ in $X$. It is also proved that all Banach dual
spaces of $L^1 \cap L^{\infty}$ and $L^1 + L^{\infty}$ have the
DunfordPettis property. New examples of Banach spaces showing that
the DunfordPettis property is not a threespace property are also
presented. As applications we obtain that the spaces $(L^1 +
L^{\infty})^\circ$ and $(L^{\infty})^\circ$ have a unique symmetric
structure, and we get a characterization of the DunfordPettis
property of some K\"otheBochner spaces.


804  The Distributions in the Invariant Trace Formula Are Supported on Characters Kottwitz, Robert E.; Rogawski, Jonathan D.
J.~Arthur put the trace formula in invariant form for all connected
reductive groups and certain disconnected ones. However his work was
written so as to apply to the general disconnected case, modulo two
missing ingredients. This paper supplies one of those missing
ingredients, namely an argument in Galois cohomology of a kind first
used by D.~Kazhdan in the connected case.


815  On the Maximum and Minimum Modulus of Rational Functions Lubinsky, D. S.
We show that if $m$, $n\geq 0$, $\lambda >1$, and $R$ is a rational function
with numerator, denominator of degree $\leq m$, $n$, respectively, then there
exists a set $\mathcal{S}\subset [0,1] $ of linear measure $\geq
\frac{1}{4}\exp (\frac{13}{\log \lambda })$ such that for $r\in
\mathcal{S}$,
\[
\max_{z =r} R(z) / \min_{z =r}  R(z) \leq \lambda ^{m+n}.
\]
Here, one may not replace $\frac{1}{4}\exp ( \frac{13}{\log \lambda })$
by $\exp (\frac{2\varepsilon }{\log \lambda })$, for any $\varepsilon >0$.
As our motivating application, we prove a convergence result for diagonal
Pad\'{e} approximants for functions meromorphic in the unit ball.


833  WGroups under Quadratic Extensions of Fields Mináč, Ján; Smith, Tara L.
To each field $F$ of characteristic not $2$, one can associate a
certain Galois group $\G_F$, the socalled Wgroup of $F$, which
carries essentially the same information as the Witt ring $W(F)$ of
$F$. In this paper we investigate the connection between $\wg$ and
$\G_{F(\sqrt{a})}$, where $F(\sqrt{a})$ is a proper quadratic
extension of $F$. We obtain a precise description in the case when
$F$ is a pythagorean formally real field and $a = 1$, and show that
the Wgroup of a proper field extension $K/F$ is a subgroup of the
Wgroup of $F$ if and only if $F$ is a formally real pythagorean field
and $K = F(\sqrt{1})$. This theorem can be viewed as an analogue of
the classical ArtinSchreier's theorem describing fields fixed by
finite subgroups of absolute Galois groups. We also obtain precise
results in the case when $a$ is a doublerigid element in $F$. Some
of these results carry over to the general setting.


849  Operator Estimates for Fredholm Modules Sukochev, F. A.
We study estimates of the type
$$
\Vert \phi(D)  \phi(D_0) \Vert_{\emt} \leq C \cdot \Vert D  D_0
\Vert^{\alpha}, \quad \alpha = \frac12, 1
$$
where $\phi(t) = t(1 + t^2)^{1/2}$, $D_0 = D_0^*$ is an unbounded
linear operator affiliated with a semifinite von Neumann algebra
$\calM$, $D  D_0$ is a bounded selfadjoint linear operator from
$\calM$ and $(1 + D_0^2)^{1/2} \in \emt$, where $\emt$ is a symmetric
operator space associated with $\calM$. In particular, we prove that
$\phi(D)  \phi(D_0)$ belongs to the noncommutative $L_p$space for
some $p \in (1,\infty)$, provided $(1 + D_0^2)^{1/2}$ belongs to the
noncommutative weak $L_r$space for some $r \in [1,p)$. In the case
$\calM = \calB (\calH)$ and $1 \leq p \leq 2$, we show that this
result continues to hold under the weaker assumption $(1 +
D_0^2)^{1/2} \in \calC_p$. This may be regarded as an odd
counterpart of A.~Connes' result for the case of even Fredholm
modules.


897  Higher Order Scattering on Asymptotically Euclidean Manifolds Christiansen, T. J.; Joshi, M. S.
We develop a scattering theory for perturbations of powers of the
Laplacian on asymptotically Euclidean manifolds. The (absolute)
scattering matrix is shown to be a Fourier integral operator
associated to the geodesic flow at time $\pi$ on the boundary.
Furthermore, it is shown that on $\Real^n$ the asymptotics of certain
shortrange perturbations of $\Delta^k$ can be recovered from the
scattering matrix at a finite number of energies.


920  Real Interpolation with Logarithmic Functors and Reiteration Evans, W. D.; Opic, B.
We present ``reiteration theorems'' with limiting values
$\theta=0$ and $\theta = 1$ for a real interpolation method
involving brokenlogarithmic functors. The resulting spaces
lie outside of the original scale of spaces and to describe them
new interpolation functors are introduced. For an ordered couple
of (quasi) Banach spaces similar results were presented without
proofs by Doktorskii in [D].


961  Algebraic Evaluations of Some Euler Integrals, Duplication Formulae for Appell's Hypergeometric Function $F_1$, and Brownian Variations Ismail, Mourad E. H.; Pitman, Jim
Explicit evaluations of the symmetric Euler integral $\int_0^1
u^{\alpha} (1u)^{\alpha} f(u) \,du$ are obtained for some particular
functions $f$. These evaluations are related to duplication formulae
for Appell's hypergeometric function $F_1$ which give reductions of
$F_1 (\alpha, \beta, \beta, 2 \alpha, y, z)$ in terms of more
elementary functions for arbitrary $\beta$ with $z = y/(y1)$ and for
$\beta = \alpha + \half$ with arbitrary $y$, $z$. These duplication
formulae generalize the evaluations of some symmetric Euler integrals
implied by the following result: if a standard Brownian bridge is
sampled at time $0$, time $1$, and at $n$ independent random times
with uniform distribution on $[0,1]$, then the broken line
approximation to the bridge obtained from these $n+2$ values has a
total variation whose mean square is $n(n+1)/(2n+1)$.


982  Holomorphic Functions of Slow Growth on Nested Covering Spaces of Compact Manifolds Lárusson, Finnur
Let $Y$ be an infinite covering space of a projective manifold
$M$ in $\P^N$ of dimension $n\geq 2$. Let $C$ be the intersection with
$M$ of at most $n1$ generic hypersurfaces of degree $d$ in $\mathbb{P}^N$.
The preimage $X$ of $C$ in $Y$ is a connected submanifold. Let $\phi$
be the smoothed distance from a fixed point in $Y$ in a metric pulled up
from $M$. Let $\O_\phi(X)$ be the Hilbert space of holomorphic
functions $f$ on $X$ such that $f^2 e^{\phi}$ is integrable on $X$, and
define $\O_\phi(Y)$ similarly. Our main result is that (under more
general hypotheses than described here) the restriction $\O_\phi(Y)
\to \O_\phi(X)$ is an isomorphism for $d$ large enough.


999  Compact Groups of Operators on Subproportional Quotients of $l^m_1$ Mankiewicz, Piotr
It is proved that a ``typical'' $n$dimensional quotient $X_n$ of
$l^m_1$ with $n = m^{\sigma}$, $0 < \sigma < 1$, has the property
$$
\Average \int_G \Tx\_{X_n} \,dh_G(T) \geq
\frac{c}{\sqrt{n\log^3 n}} \biggl( n  \int_G \tr T \,dh_G (T)
\biggr),
$$
for every compact group $G$ of operators acting on $X_n$, where
$d_G(T)$ stands for the normalized Haar measure on $G$ and the average
is taken over all extreme points of the unit ball of $X_n$. Several
consequences of this estimate are presented.


1018  Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$Varieties Reichstein, Zinovy; Youssin, Boris
Let $G$ be an algebraic group and let $X$ be a generically free $G$variety.
We show that $X$ can be transformed, by a sequence of blowups with smooth
$G$equivariant centers, into a $G$variety $X'$ with the following
property the stabilizer of every point of $X'$ is isomorphic to a
semidirect product $U \sdp A$ of a unipotent group $U$ and a
diagonalizable group $A$.
As an application of this result, we prove new lower bounds on essential
dimensions of some algebraic groups. We also show that certain
polynomials in one variable cannot be simplified by a Tschirnhaus
transformation.


1057  The Spectrum of an Infinite Graph Urakawa, Hajime
In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.


1085  Complex MongeAmpère Measures of Plurisubharmonic Functions with Bounded Values Near the Boundary Xing, Yang
We give a characterization of bounded plurisubharmonic functions by
using their complex MongeAmp\`ere measures. This implies a both necessary
and sufficient condition for a positive measure to be complex
MongeAmp\`ere measure of some bounded plurisubharmonic function.


1101  Discrete Series of Classical Groups Zhang, Yuanli
Let $G_n$ be the split classical groups $\Sp(2n)$, $\SO(2n+1)$ and
$\SO(2n)$ defined over a $p$adic field F or the quasisplit
classical groups $U(n,n)$ and $U(n+1,n)$ with respect to a
quadratic extension $E/F$. We prove the selfduality of unitary
supercuspidal data of standard Levi subgroups of $G_n(F)$ which
give discrete series representations of $G_n(F)$.


1121  Ramanujan Type Buildings Ballantine, Cristina M.
We will construct a finite union of finite quotients of the affine
building of the group $\GL_3$ over the field of $p$adic numbers
$\mathbb{Q}_p$. We will view this object as a hypergraph and estimate
the spectrum of its underlying graph.


1149  Canonical Resolution of a Quasiordinary Surface Singularity Ban, Chunsheng; McEwan, Lee J.
We describe the embedded resolution of an irreducible quasiordinary
surface singularity $(V,p)$ which results from applying the canonical
resolution of BierstoneMilman to $(V,p)$. We show that this process
depends solely on the characteristic pairs of $(V,p)$, as predicted
by Lipman. We describe the process explicitly enough that a resolution
graph for $f$ could in principle be obtained by computer using only
the characteristic pairs.


1164  Perforated Ordered $\K_0$Groups Elliott, George A.; Villadsen, Jesper
A simple $\C^*$algebra is constructed for which the Murrayvon
Neumann equivalence classes of projections, with the usual
additioninduced by addition of orthogonal projectionsform the
additive semigroup
$$
\{0,2,3,\dots\}.
$$
(This is a particularly simple instance of the phenomenon of
perforation of the ordered $\K_0$group, which has long been known in
the commutative casefor instance, in the case of the
foursphereand was recently observed by the second author in the
case of a simple $\C^*$algebra.)


1192  Orbital Integrals on $p$Adic Lie Algebras Herb, Rebecca A.
Let $G$ be a connected reductive $p$adic group and let $\frakg$ be its
Lie algebra. Let $\calO$ be any $G$orbit in $\frakg$. Then the orbital
integral $\mu_{\calO}$ corresponding to $\calO$ is an invariant distribution
on $\frakg $, and HarishChandra proved that its Fourier transform $\hat
\mu_{\calO}$ is a locally constant function on the set $\frakg'$ of regular
semisimple elements of $\frakg$. If $\frakh$ is a Cartan subalgebra of
$\frakg$, and $\omega$ is a compact subset of $\frakh\cap\frakg'$, we give
a formula for $\hat \mu_{\calO}(tH)$ for $H\in\omega$ and $t\in F^{\times}$
sufficiently large. In the case that $\calO$ is a regular semisimple orbit,
the formula is already known by work of Waldspurger. In the case that
$\calO$ is a nilpotent orbit, the behavior of $\hat\mu_{\calO}$ at
infinity is already known because of its homogeneity properties. The
general case combines aspects of these two extreme cases. The formula
for $\hat\mu _{\calO}$ at infinity can be used to formulate a ``theory
of the constant term'' for the space of distributions spanned by the
Fourier transforms of orbital integrals. It can also be used to show
that the Fourier transforms of orbital integrals are ``linearly
independent at infinity.''


1221  Nest Representations of TAF Algebras Hopenwasser, Alan; Peters, Justin R.; Power, Stephen C.
A nest representation of a strongly maximal TAF algebra $A$ with
diagonal $D$ is a representation $\pi$ for which $\lat \pi(A)$ is
totally ordered. We prove that $\ker \pi$ is a meet irreducible ideal
if the spectrum of $A$ is totally ordered or if (after an appropriate
similarity) the von Neumann algebra $\pi(D)''$ contains an atom.


1235  Representations with Weighted Frames and Framed Parabolic Bundles Hurtubise, J. C.; Jeffrey, L. C.
There is a wellknown correspondence (due to Mehta and Seshadri in
the unitary case, and extended by Bhosle and Ramanathan to other
groups), between the symplectic variety $M_h$ of representations of
the fundamental group of a punctured Riemann surface into a compact
connected Lie group~$G$, with fixed conjugacy classes $h$ at the
punctures, and a complex variety ${\cal M}_h$ of holomorphic bundles
on the unpunctured surface with a parabolic structure at the puncture
points. For $G = \SU(2)$, we build a symplectic variety $P$ of pairs
(representations of the fundamental group into $G$, ``weighted frame''
at the puncture points), and a corresponding complex variety ${\cal
P}$ of moduli of ``framed parabolic bundles'', which encompass
respectively all of the spaces $M_h$, ${\cal M}_h$, in the sense that
one can obtain $M_h$ from $P$ by symplectic reduction, and ${\cal
M}_h$ from ${\cal P}$ by a complex quotient. This allows us to
explain certain features of the toric geometry of the $\SU(2)$ moduli
spaces discussed by Jeffrey and Weitsman, by giving the actual toric
variety associated with their integrable system.


1269  Well Ramified Extensions of Complete Discrete Valuation Fields with Applications to the Kato Conductor Spriano, Luca
We study extensions $L/K$ of complete discrete valuation fields $K$
with residue field $\oK$ of characteristic $p > 0$, which we do not
assume to be perfect. Our work concerns ramification theory for such
extensions, in particular we show that all classical properties which
are true under the hypothesis {\it ``the residue field extension
$\oL/\oK$ is separable''} are still valid under the more general
hypothesis that the valuation ring extension is monogenic. We also
show that conversely, if classical ramification properties hold true
for an extension $L/K$, then the extension of valuation rings is
monogenic. These are the ``{\it well ramified}'' extensions. We show
that there are only three possible types of well ramified extensions
and we give examples. In the last part of the paper we consider, for
the three types, Kato's generalization of the conductor, which we show
how to bound in certain cases.


1310  On the Homology of $\GL_n$ and Higher PreBloch Groups Yagunov, Serge
For every integer $n>1$ and infinite field $F$ we construct a spectral
sequence converging to the homology of $\GL_n(F)$ relative to the
group of monomial matrices $\GM_n(F)$. Some entries in $E^2$terms of
these spectral sequences may be interpreted as a natural
generalization of the Bloch group to higher dimensions. These groups
may be characterized as homology of $\GL_n$ relatively to $\GL_{n1}$
and $\GM_n$. We apply the machinery developed to the investigation of
stabilization maps in homology of General Linear Groups.


1339  Author Index  Index des auteurs 2000, for 2000  pour
No abstract.

