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3  Subgroups of the adjoint group of a radical ring Amberg, B.; Dickenschied, O.; Sysak, Ya. P.
It is shown that the adjoint group $R^\circ$ of an arbitrary
radical ring $R$ has a series with abelian factors and that its finite
subgroups are nilpotent. Moreover, some criteria for subgroups of
$R^\circ$ to be locally nilpotent are given.


16  Asymptotic shape of finite packings Böröczky, Károly Jr.; Schnell, Uwe
Let $K$ be a convex body in $\ed$ and denote by $\cn$
the set of centroids of $n$ nonoverlapping translates
of $K$. For $\varrho>0$, assume that the parallel body
$\cocn+\varrho K$ of $\cocn$ has minimal volume.
The notion of parametric density (see~\cite{Wil93})
provides a bridge between finite and infinite
packings (see~\cite{BHW94} or~\cite{Hen}).
It is known that
there exists a maximal $\varrho_s(K)\geq 1/(32d^2)$ such that
$\cocn$ is a segment for $\varrho<\varrho_s$ (see~\cite{BHW95}).
We prove the existence of a minimal $\varrho_c(K)\leq d+1$ such that
if $\varrho>\varrho_c$ and $n$ is large then
the shape of $\cocn$ can not be too far from the shape of $K$.
For $d=2$, we verify that $\varrho_s=\varrho_c$.
For $d\geq 3$, we present the first example of a convex
body with known $\varrho_s$ and $\varrho_c$; namely, we have
$\varrho_s=\varrho_c=1$ for the
parallelotope.


29  Weighted norm inequalities for fractional integral operators with rough kernel Ding, Yong; Lu, Shanzhen
Given function $\Omega$ on ${\Bbb R^n}$, we define the fractional
maximal operator and the fractional integral operator by
$$
M_{\Omega,\alpha}\,f(x)=\sup_{r>0}\frac 1{r^{n\alpha}}
\int_{\,y<r} \Omega(\,y)\,\,f(xy)\,dy
$$
and
$$
T_{\Omega,\alpha}\,f(x)=\int_{\Bbb R^n}\frac {\Omega(\,y)}{y^{n\alpha}}
\,f(xy)\,dy
$$
respectively, where $0<\alpha<n$.
In this paper we study the weighted norm inequalities of $ M_{\Omega,
\alpha}$ and $T_{\Omega,\alpha}$ for appropriate $\alpha,s$ and $A(\,p,q)$
weights in the case that $\Omega\in L^s(S^{n1})(s>1)$, homogeneous of
degree zero.


40  Green's functions for powers of the invariant Laplacian Engliš, Miroslav; Peetre, Jaak
The aim of the present paper is the computation of Green's functions
for the powers $\DDelta^m$ of the invariant Laplace operator on rankone
Hermitian symmetric spaces. Starting with the noncompact case, the
unit ball in $\CC^d$, we obtain a complete result for $m=1,2$ in
all dimensions. For $m\ge3$ the formulas grow quite complicated so
we restrict ourselves to the case of the unit disc ($d=1$) where
we develop a method, possibly applicable also in other situations,
for reducing the number of integrations by half, and use it to give
a description of the boundary behaviour of these Green functions
and to obtain their (multivalued) analytic continuation to the
entire complex plane. Next we discuss the type of special functions
that turn up (hyperlogarithms of Kummer). Finally we treat also
the compact case of the complex projective space $\Bbb P^d$ (for
$d=1$, the Riemann sphere) and, as an application of our results,
use eigenfunction expansions to obtain some new identities involving
sums of Legendre ($d=1$) or Jacobi ($d>1$) polynomials and the
polylogarithm function. The case of Green's functions of powers of
weighted (no longer invariant, but only covariant) Laplacians is
also briefly discussed.


74  Elementary proof of the fundamental lemma for a unitary group Flicker, Yuval Z.
The fundamental lemma in the theory of automorphic forms is proven
for the (quasisplit) unitary group $U(3)$ in three variables
associated with a quadratic extension of $p$adic fields, and its
endoscopic group $U(2)$, by means of a new, elementary technique.
This lemma is a prerequisite for an application of the trace
formula to classify the automorphic and admissible representations
of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$.
It compares the (unstable) orbital integral of the characteristic
function of the standard maximal compact subgroup $K$ of $U(3)$ at
a regular element (whose centralizer $T$ is a torus), with an
analogous (stable) orbital integral on the endoscopic group $U(2)$.
The technique is based on computing the sum over the double coset
space $T\bs G/K$ which describes the integral, by means of an
intermediate double coset space $H\bs G/K$ for a subgroup $H$ of
$G=U(3)$ containing $T$. Such an argument originates from
Weissauer's work on the symplectic group. The lemma is proven for
both ramified and unramified regular elements, for which endoscopy
occurs (the stable conjugacy class is not a single orbit).


99  $A_\phi$invariant subspaces on the torus Izuchi, Keiji; Matsugu, Yasuo
Generalizing the notion of invariant subspaces on
the 2dimensional torus $T^2$, we study the structure
of $A_\phi$invariant subspaces of $L^2(T^2)$. A
complete description is given of $A_\phi$invariant
subspaces that satisfy conditions similar to those
studied by Mandrekar, Nakazi, and Takahashi.


134  On critical level sets of some two degrees of freedom integrable Hamiltonian systems Médan, Christine
We prove that all Liouville's tori generic bifurcations of a
large class of two degrees of freedom integrable Hamiltonian
systems (the so called JacobiMoserMumford systems) are
nondegenerate in the sense of Bott. Thus, for such systems,
Fomenko's theory~\cite{fom} can be applied (we give the example
of Gel'fandDikii's system). We also check the Bott property
for two interesting systems: the Lagrange top and the geodesic
flow on an ellipsoid.


152  Inequalities for rational functions with prescribed poles Min, G.
This paper considers the rational system ${\cal P}_n
(a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (xa_k)},
P\in {\cal P}_n\bigr\}$ with nonreal elements in
$\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [1,1]$ paired by complex
conjugation. It gives a sharp (to constant) Markovtype inequality
for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$.
The corresponding Markovtype inequality for high derivatives
is established, as well as Nikolskiitype inequalities. Some
sharp Markov and Bernsteintype inequalities with curved majorants
for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are
obtained, which generalize some results for the classical
polynomials. A sharp Schurtype inequality is also proved and
plays a key role in the proofs of our main results.


167  MurnaghanNakayama rules for characters of IwahoriHecke algebras of the complex reflection groups $G(r,p,n)$ Halverson, Tom; Ram, Arun
IwahoriHecke algebras for the infinite series of complex
reflection groups $G(r,p,n)$ were constructed recently in
the work of Ariki and Koike~\cite{AK}, Brou\'e and Malle
\cite{BM}, and Ariki~\cite{Ari}. In this paper we give
MurnaghanNakayama type formulas for computing the irreducible
characters of these algebras. Our method is a generalization
of that in our earlier paper ~\cite{HR} in which we derived
MurnaghanNakayama rules for the characters of the
IwahoriHecke algebras of the classical Weyl groups.
In both papers we have been
motivated by C. Greene~\cite{Gre}, who gave a new derivation
of the MurnaghanNakayama formula for irreducible symmetric
group characters by summing diagonal matrix entries in Young's
seminormal representations. We use the analogous representations
of the IwahoriHecke algebra of $G(r,p,n)$ given by Ariki and
Koike~\cite{AK} and Ariki ~\cite{Ari}.


193  Intertwining operator and $h$harmonics associated with reflection groups Xu, Yuan
We study the intertwining operator and $h$harmonics in
Dunkl's theory on $h$harmonics associated with reflection groups. Based
on a biorthogonality between the ordinary harmonics and the action of the
intertwining operator $V$ on the harmonics, the main result provides a
method to compute the action of the intertwining operator $V$ on polynomials
and to construct an orthonormal basis for the space of $h$harmonics.


210  Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$ Zhao, Kaiming
In this paper, we determine when two simple generalized Cartan
type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss
the relationship between the Jacobian conjecture and the generalized
Cartan type $W$ Lie algebras.


225  Derivations and invariant forms of Lie algebras graded by finite root systems Benkart, Georgia
Lie algebras graded by finite reduced root systems have been
classified up to isomorphism. In this paper we describe the derivation
algebras of these Lie algebras and determine when they possess invariant
bilinear forms. The results which we develop to do this are much more
general and apply to Lie algebras that are completely reducible with
respect to the adjoint action of a finitedimensional subalgebra.


242  Intégration du sousdifférentiel proximal: un contre exemple Benoist, Joël
Etant donn\'ee une partie $D$ d\'enombrable et dense de
${\R}$, nous construisons une infinit\'e de fonctions
Lipschitziennes d\'efinies sur ${\R}$, s'annulant
en z\'ero, dont le sousdiff\'erentiel proximal est \'egal
\`a $]1, 1[$ en tout point de $D$ et est vide en tout point
du compl\'ementaire de $D$. Nous d\'eduisons que deux
fonctions dont la diff\'erence n'est pas constante peuvent
avoir les m\^emes sousdiff\'erentiels.


266  The torsion free Pieri formula Britten, D. J.; Lemire, F. W.
Central to the study of simple infinite dimensional
$g\ell(n, \Bbb C)$modules having finite dimensional weight spaces are the
torsion free modules. All degree $1$ torsion free modules are known.
Torsion free modules of arbitrary degree can be constructed by tensoring
torsion free modules of degree $1$ with finite dimensional simple modules.
In this paper, the central characters of such a tensor product module are
shown to be given by a Pierilike formula, complete reducibility is
established when these central characters are distinct and an example
is presented illustrating the existence of a nonsimple indecomposable
submodule when these characters are not distinct.


290  Noncommutative disc algebras for semigroups Davidson, Kenneth R.; Popescu, Gelu
We study noncommutative disc algebras associated to the free
product of discrete subsemigroups of $\bbR^+$. These algebras are
associated to generalized Cuntz algebras, which are shown to be
simple and purely infinite. The nonselfadjoint subalgebras
determine the semigroup up to isomorphism. Moreover, we establish
a dilation theorem for contractive representations of these
semigroups which yields a variant of the von Neumann inequality.
These methods are applied to establish a solution to the truncated
moment problem in this context.


312  Units in group rings of free products of prime cyclic groups Dokuchaev, Michael A.; Singer, Maria Lucia Sobral
Let $G$ be a free product of cyclic groups of prime order. The
structure of the unit group ${\cal U}(\Q G)$ of the rational group
ring $\Q G$ is given in terms of free products and amalgamated free
products of groups. As an application, all finite subgroups of
${\cal U}(\Q G)$, up to conjugacy, are described and the
Zassenhaus Conjecture for finite subgroups in $\Z G$ is proved. A
strong version of the Tits Alternative for ${\cal U}(\Q G)$ is
obtained as a corollary of the structural result.


323  Purely infinite, simple $C^\ast$algebras arising from free product constructions Dykema, Kenneth J.; Rørdam, Mikael
Examples of simple, separable, unital, purely infinite
$C^\ast$algebras are constructed, including:
\item{(1)} some that are not approximately divisible;
\item{(2)} those that arise as crossed products of any of a certain class of
$C^\ast$algebras by any of a certain class of nonunital endomorphisms;
\item{(3)} those that arise as reduced free products of pairs of
$C^\ast$algebras with respect to any from a certain class of states.


342  Shape fibrations, multivalued maps and shape groups Giraldo, Antonio
The notion of shape fibration with the near lifting of near
multivalued paths property is studied. The relation of these
mapswhich agree with shape fibrations having totally disconnected
fiberswith Hurewicz fibrations with the unique path lifting
property is completely settled. Some results concerning homotopy and
shape groups are presented for shape fibrations with the near lifting
of near multivalued paths property. It is shown that for this class of
shape fibrations the existence of liftings of a fine multivalued map,
is equivalent to an algebraic problem relative to the homotopy, shape
or strong shape groups associated.


356  Some norms on universal enveloping algebras Gross, Leonard
The universal enveloping algebra, $U(\frak g)$, of a Lie algebra $\frak g$
supports some norms and seminorms that have arisen naturally in the
context of heat kernel analysis on Lie groups. These norms and seminorms
are investigated here from an algebraic viewpoint. It is shown
that the norms corresponding to heat kernels on the associated Lie
groups decompose as product norms under the natural isomorphism
$U(\frak g_1 \oplus \frak g_2) \cong U(\frak g_1) \otimes U(\frak
g_2)$. The seminorms corresponding to Green's functions are
examined at a purely Lie algebra level for $\rmsl(2,\Bbb C)$. It
is also shown that the algebraic dual space $U'$ is spanned by its
finite rank elements if and only if $\frak g$ is nilpotent.


378  Equivariant polynomial automorphism of $\Theta$representations Kurth, Alexandre
We show that every equivariant polynomial automorphism of a
$\Theta$repre\sen\ta\tion and of the reduction of an irreducible
$\Theta$representation is a multiple of the identity.


401  The hypercentre and the $n$centre of the unit group of an integral group ring Li, Yuanlin
In this paper, we first show that the central height of the unit group of
the integral group ring of a periodic group is at most $2$. We then
give a complete characterization of the $n$centre of that unit group.
The $n$centre of the unit group is either the centre or the second
centre (for $n \geq 2$).


412  Asymptotic transformations of $q$series McIntosh, Richard J.
For the $q$series $\sum_{n=0}^\infty a^nq^{bn^2+cn}/(q)_n$
we construct a companion $q$series such that the asymptotic
expansions of their logarithms as $q\to 1^{\scriptscriptstyle }$
differ only in the dominant few terms. The asymptotic expansion
of their quotient then has a simple closed form; this gives rise
to a new $q$hypergeometric identity. We give an asymptotic
expansion of a general class of $q$series containing some of
Ramanujan's mock theta functions and Selberg's identities.


426  The groups of the regular starpolytopes McMullen, Peter
No abstract.


449  $Q_p$ spaces on Riemann surfaces Aulaskari, Rauno; He, Yuzan; Ristioja, Juha; Zhao, Ruhan 

465  Six primes and an almost prime in four linear equations Balog, Antal
There are infinitely many triplets of primes $p,q,r$ such that the
arithmetic means of any two of them, ${p+q\over2}$, ${p+r\over2}$,
${q+r\over2}$ are also primes. We give an asymptotic formula for
the number of such triplets up to a limit. The more involved
problem of asking that in addition to the above the arithmetic mean
of all three of them, ${p+q+r\over3}$ is also prime seems to be out
of reach. We show by combining the HardyLittlewood method with the
sieve method that there are quite a few triplets for which six of
the seven entries are primes and the last is almost prime.}


487  On the Liouville property for divergence form operators Barlow, Martin T.
In this paper we construct a bounded strictly positive
function $\sigma$ such that the Liouville property fails for the
divergence form operator $L=\nabla (\sigma^2 \nabla)$. Since in
addition $\Delta \sigma/\sigma$ is bounded, this example also gives a
negative answer to a problem of Berestycki, Caffarelli and Nirenberg
concerning linear Schr\"odinger operators.


497  Morse index of approximating periodic solutions for the billiard problem. Application to existence results Bolle, Philippe
This paper deals with periodic solutions for the billiard problem in a
bounded open set of $\hbox{\Bbbvii R}^N$ which are limits of regular
solutions of Lagrangian systems with a potential well. We give a
precise link between the Morse index of approximate solutions
(regarded as critical points of Lagrangian functionals) and the
properties of the bounce trajectory to which they converge.


525  Nilpotent orbit varieties and the atomic decomposition of the $q$Kostka polynomials Brockman, William; Haiman, Mark
We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of
schemetheoretic
intersections of nilpotent orbit closures with the diagonal matrices.
Here $\mu'$ gives the Jordan block structure of the nilpotent matrix.
de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of
Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology
rings of the varieties constructed by
Springer~\cite{Springer76,Springer78}. The famous $q$Kostka
polynomial~$\Klmt(q)$ is the Hilbert series for the
multiplicity of the irreducible symmetric group representation indexed
by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$.
\LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition
of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with
nonnegative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobeniussplitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a directsum decomposition of
the ideals of nilpotent orbit closures, arising from the inclusions of
the corresponding sets. We carry out the restriction to the diagonal
using a recent theorem of Broer~\cite{Broer}. This gives a directsum
decomposition of the ideals yielding the $k[\Cmubar\cap
\hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of
the $q$Kostka polynomials.


538  Upper bounds for the resonance counting function of Schrödinger operators in odd dimensions Froese, Richard
The purpose of this note is to provide a simple proof of the sharp
polynomial upper bound for the resonance counting function of
a Schr\"odinger operator in odd dimensions. At the same time
we generalize the result to the class of superexponentially
decreasing potentials.


547  MittagLeffler theorems on Riemann surfaces and Riemannian manifolds Gauthier, Paul M.
Cauchy and Poisson integrals over {\it unbounded\/} sets are employed to
prove MittagLeffler type theorems with massive singularities as well as
approximation theorems for holomorphic and harmonic functions.


563  Primes in short segments of arithmetic progressions Goldston, D. A.; Yildirim, C. Y.
Consider the variance for the number of primes that are both in the
interval $[y,y+h]$ for $y \in [x,2x]$ and in an arithmetic
progression of modulus $q$. We study the total variance
obtained by adding these variances over all the reduced residue
classes modulo $q$. Assuming a strong form of the twin prime
conjecture and the Riemann Hypothesis one can obtain an asymptotic
formula for the total variance in the range when $1 \leq h/q \leq
x^{1/2\epsilon}$, for any $\epsilon >0$. We show that one can still
obtain some weaker asymptotic results assuming the Generalized Riemann
Hypothesis (GRH) in place of the twin prime conjecture. In their
simplest form, our results are that on GRH the same asymptotic formula
obtained with the twin prime conjecture is true for ``almost all'' $q$
in the range $1 \leq h/q \leq h^{1/4\epsilon}$, that on averaging
over $q$ one obtains an asymptotic formula in the extended range $1
\leq h/q \leq h^{1/2\epsilon}$, and that there are lower bounds with
the correct order of magnitude for all $q$ in the range $1 \leq h/q
\leq x^{1/3\epsilon}$.


581  The homology of singular polygon spaces Kamiyama, Yasuhiko
Let $M_n$ be the variety of spatial polygons $P= (a_1, a_2, \dots,
a_n)$ whose sides are vectors $a_i \in \text{\bf R}^3$ of length
$\vert a_i \vert=1 \; (1 \leq i \leq n),$ up to motion in
$\text{\bf R}^3.$ It is known that for odd $n$, $M_n$ is a
smooth manifold, while for even $n$, $M_n$ has conelike singular
points. For odd $n$, the rational homology of $M_n$ was determined
by Kirwan and Klyachko [6], [9]. The purpose of this paper is to
determine the rational homology of $M_n$ for even $n$. For even
$n$, let ${\tilde M}_n$ be the manifold obtained from $M_n$ by the
resolution of the singularities. Then we also determine the
integral homology of ${\tilde M}_n$.


595  Multipliers of fractional Cauchy transforms and smoothness conditions Luo, Donghan; MacGregor, Thomas
This paper studies conditions on an analytic function that imply it
belongs to ${\cal M}_\alpha$, the set of multipliers of the family of
functions given by $f(z) = \int_{\zeta=1} {1 \over
(1\overline\zeta z)^\alpha} \,d\mu (\zeta)$ $(z<1)$ where $\mu$ is a
complex Borel measure on the unit circle and $\alpha >0$. There are
two main theorems. The first asserts that if $0<\alpha<1$ and
$\sup_{\zeta=1} \int^1_0 f'(r\zeta) (1r)^{\alpha1} \,dr<\infty$
then $f \in {\cal M}_\alpha$. The second asserts that if $0<\alpha
\leq 1$, $f \in H^\infty$ and $\sup_t \int^\pi_0 {f(e^{i(t+s)}) 
2f(e^{it}) + f(e^{i(ts)}) \over s^{2\alpha}} \, ds < \infty$ then
$f \in {\cal M}_\alpha$. The conditions in these theorems are shown
to relate to a number of smoothness conditions on the unit circle for
a function analytic in the open unit disk and continuous in its closure.


605  Hardy spaces of conjugate systems of temperatures GuzmánPartida, Martha; PérezEsteva, Salvador
We define Hardy spaces of conjugate systems of temperature functions on
${\bbd R}_{+}^{n+1}$. We show that their boundary distributions are the same
as the boundary distributions of the usual Hardy spaces of conjugate systems
of harmonic functions.


620  The Eichler trace of $\bbd Z_p$ actions on Riemann surfaces Sjerve, Denis; Yang, Qing Jie
We study $\hbox{\Bbbvii Z}_p$ actions on compact connected Riemann
surfaces via their associated Eichler traces. We determine the set
of possible Eichler traces and determine the relationship between 2
actions if they have the same trace.


638  Fractals in the large Strichartz, Robert S.
A {\it reverse iterated function system} (r.i.f.s.) is defined to be a
set of expansive maps
$\{T_1,\ldots,T_m\}$ on a discrete metric space $M$. An invariant set
$F$ is defined to be a set satisfying
$F = \bigcup^m_{j=1} T_jF$, and an invariant measure $\mu$ is
defined to be a solution of
$\mu = \sum^m_{j=1} p_j\mu\circ T_j^{1}$ for positive weights
$p_j$. The structure and basic properties of such invariant sets
and measures is described, and some examples are given.
A {\it blowup} $\cal F$ of a selfsimilar set $F$ in
$\Bbb R^n$ is defined to be the union of an increasing sequence of
sets, each similar to $F$. We give a general construction of
blowups, and show that under certain hypotheses a blowup is the sum set of
$F$ with an invariant set for a r.i.f.s. Some examples of blowups of
familiar fractals are described. If $\mu$ is an invariant measure
on $\Bbb Z^+$ for a linear r.i.f.s., we describe the behavior of its
{\it analytic} transform, the power series
$\sum^\infty_{n=0} \mu(n)z^n$ on the unit disc.


658  Hankel operators on pseudoconvex domains of finite type in ${\Bbb C}^2$ Symesak, Frédéric
The aim of this paper is to study small Hankel operators $h$ on the
Hardy space or on weighted Bergman spaces, where $\Omega$ is a
finite type domain in ${\Bbbvii C}^2$ or a strictly pseudoconvex
domain in ${\Bbbvii C}^n$. We give a sufficient condition on the
symbol $f$ so that $h$ belongs to the Schatten class ${\cal S}_p$,
$1\le p<+\infty$.


673  Fredholm modules and spectral flow Carey, Alan; Phillips, John
An {\it odd unbounded\/} (respectively, $p${\it summable})
{\it Fredholm module\/} for a unital Banach $\ast$algebra, $A$, is a pair $(H,D)$
where $A$ is represented on the Hilbert space, $H$, and $D$ is an unbounded
selfadjoint operator on $H$ satisfying:
\item{(1)} $(1+D^2)^{1}$ is compact (respectively, $\Trace\bigl((1+D^2)^{(p/2)}\bigr)
<\infty$), and
\item{(2)} $\{a\in A\mid [D,a]$ is bounded$\}$ is a dense
$\ast$subalgebra of $A$.


719  Indecomposable almost free modulesthe local case Göbel, Rüdiger; Shelah, Saharon
Let $R$ be a countable, principal ideal domain which is not a field and
$A$ be a countable $R$algebra which is free as an $R$module. Then we
will construct an $\aleph_1$free $R$module $G$ of rank $\aleph_1$
with endomorphism algebra End$_RG = A$. Clearly the result does not
hold for fields. Recall that an $R$module is $\aleph_1$free if all
its countable submodules are free, a condition closely related to
Pontryagin's theorem. This result has many consequences, depending on
the algebra $A$ in use. For instance, if we choose $A = R$, then
clearly $G$ is an indecomposable `almost free' module. The existence of
such modules was unknown for rings with only finitely many primes like
$R = \hbox{\Bbbvii Z}_{(p)}$, the integers localized at some prime $p$. The result
complements a classical realization theorem of Corner's showing that
any such algebra is an endomorphism algebra of some torsionfree,
reduced $R$module $G$ of countable rank. Its proof is based on new
combinatorialalgebraic techniques related with what we call {\it rigid
treeelements\/} coming from a module generated over a forest of trees.


739  Eigenpolytopes of distance regular graphs Godsil, C. D.
Let $X$ be a graph with vertex set $V$ and let $A$ be
its adjacency matrix. If $E$ is the matrix representing orthogonal
projection onto an eigenspace of $A$ with dimension $m$, then $E$ is
positive semidefinite. Hence it is the Gram matrix of a set of $V$
vectors in $\re^m$. We call the convex hull of a such a set of vectors
an eigenpolytope of $X$. The connection between the properties of this
polytope and the graph is strongest when $X$ is distance regular and,
in this case, it is most natural to consider the eigenpolytope
associated to the second largest eigenvalue of $A$. The main result
of this paper is the characterisation of those distance regular graphs
$X$ for which the $1$skeleton of this eigenpolytope is isomorphic to
$X$.


756  Estimates on renormalization group transformations Brydges, D.; Dimock, J.; Hurd, T. R.
We consider a specific realization of the renormalization group (RG)
transformation acting on functional measures for scalar quantum
fields which are expressible as a polymer expansion times an
ultraviolet cutoff Gaussian measure. The new and improved
definitions and estimates we present are sufficiently general and
powerful to allow iteration of the transformation, hence the
analysis of complete renormalization group flows, and hence the
construction of a variety of scalar quantum field theories.


794  Upper bounds on $L(1,\chi)$ and applications Louboutin, Stéphane
We give upper bounds on the modulus of the values at $s=1$ of
Artin $L$functions of abelian extensions unramified at all
the infinite places. We also explain how we can compute better
upper bounds and explain how useful such computed bounds are
when dealing with class number problems for $\CM$fields. For
example, we will reduce the determination of all the
nonabelian normal $\CM$fields of degree $24$ with Galois
group $\SL_2(F_3)$ (the special linear group over the finite
field with three elements) which have class number one to the
computation of the class numbers of $23$ such $\CM$fields.


816  Tableaux realization of generalized Verma modules Mazorchuk, Volodymyr
We construct the tableaux realization of generalized Verma modules
over the Lie algebra $\sl(3,{\bbd C})$. By the same procedure we
construct and investigate the structure of a new family of
generalized Verma modules over $\sl(n,{\bbd C})$.


829  Conjugacy classes and nilpotent variety of a reductive monoid Putcha, Mohan S.
We continue in this paper our study of conjugacy classes
of a reductive monoid $M$. The main theorems establish a strong connection
with the BruhatRenner decomposition of $M$. We use our results to decompose
the variety $M_{\nil}$ of nilpotent elements of $M$ into irreducible components.
We also identify a class of nilpotent elements that we call standard and prove
that the number of conjugacy classes of standard nilpotent elements is always
finite.


845  LusternikSchnirelmann category and algebraic $R$local homotopy theory Scheerer, H.; Tanré, D.
In this paper, we define the notion of $R_{\ast}$$\LS$ category
associated to an increasing system of subrings of $\Q$ and we relate
it to the usual $\LS$category. We also relate it to the invariant
introduced by F\'elix and Lemaire in tame homotopy theory, in which
case we give a description in terms of Lie algebras and of cocommutative
coalgebras, extending results of LemaireSigrist and F\'elixHalperin.


863  Smooth formal embeddings and the residue complex Yekutieli, Amnon
Let $\pi\colon X \ar S$ be a finite type morphism of noetherian schemes.
A {\it smooth formal embedding\/} of $X$ (over $S$) is a bijective closed
immersion $X \subset \mfrak{X}$, where $\mfrak{X}$ is a noetherian
formal scheme, formally smooth over $S$. An example of such an embedding
is the formal completion $\mfrak{X} = Y_{/ X}$ where $X \subset Y$
is an algebraic embedding. Smooth formal embeddings can be used to
calculate algebraic De~Rham (co)homology.
Our main application is an explicit construction of the Grothendieck
residue complex when $S$ is a regular scheme. By definition the residue
complex is the Cousin complex of $\pi^{!} \mcal{O}_{S}$, as in \cite{RD}.
We start with IC.~Huang's theory of pseudofunctors on modules with
$0$dimensional support, which provides a graded sheaf $\bigoplus_{q}
\mcal{K}^{q}_{\,X / S}$. We then use smooth formal embeddings to obtain
the coboundary operator $\delta \colon\mcal{K}^{q}_{X / S} \ar
\mcal{K}^{q + 1}_{\,X / S}$. We exhibit a canonical isomorphism between
the complex $(\mcal{K}^{\bdot}_{\,X / S}, \delta)$ and the residue complex
of \cite{RD}. When $\pi$ is equidimensional of dimension $n$ and
generically smooth we show that $\mrm{H}^{n} \mcal{K}^{\bdot}_{\,X/S}$
is canonically isomorphic to to the sheaf of regular differentials of
KunzWaldi \cite{KW}.
Another issue we discuss is Grothendieck Duality on a noetherian formal
scheme $\mfrak{X}$. Our results on duality are used in the construction
of $\mcal{K}^{\bdot}_{\,X / S}$.


897  Fourier multipliers for local hardy spaces on ChébliTrimèche hypergroups Bloom, Walter R.; Xu, Zengfu
In this paper we consider Fourier multipliers on local
Hardy spaces $\qin$ $(0<p \leq 1)$ for Ch\'ebliTrim\`eche hypergroups.
The molecular characterization is investigated which allows us to prove
a version of H\"ormander's multiplier theorem.


929  Decomposition varieties in semisimple Lie algebras Broer, Abraham
The notion of decompositon class in a semisimple Lie algebra is a
common generalization of nilpotent orbits and the set of
regular semisimple elements. We prove that the closure of a
decomposition class has many properties in common with nilpotent
varieties, \eg, its normalization has rational singularities.


972  Trace class elements and crosssections in KacMoody groups Brüchert, Gerd
Let $G$ be an affine KacMoody group, $\pi_0,\dots,\pi_r,\pi_{\delta}$
its fundamental irreducible representations and $\chi_0, \dots,
\chi_r, \chi_{\delta}$ their characters. We determine the set of all
group elements $x$ such that all $\pi_i(x)$ act as trace class
operators, \ie, such that $\chi_i(x)$ exists, then prove that the
$\chi_i$ are class functions. Thus, $\chi:=(\chi_0, \dots, \chi_r,
\chi_{\delta})$ factors to an adjoint quotient $\bar{\chi}$ for $G$.
In a second part, following Steinberg, we define a crosssection $C$
for the potential regular classes in $G$. We prove that the
restriction $\chi_C$ behaves well algebraically. Moreover, we obtain
an action of $\hbox{\Bbbvii C}^{\times}$ on $C$, which leads to a
functional identity for $\chi_C$ which shows that $\chi_C$ is
quasihomogeneous.


1007  Galois module structure of ambiguous ideals in biquadratic extensions Elder, G. Griffith
Let $N/K$ be a biquadratic extension of algebraic number fields, and
$G=\Gal (N/K)$. Under a weak restriction on the ramification filtration
associated with each prime of $K$ above $2$, we explicitly describe the
$\bZ[G]$module structure of each ambiguous ideal of $N$. We find under
this restriction that in the representation of each ambiguous ideal as a
$\bZ[G]$module, the exponent (or multiplicity) of each indecomposable
module is determined by the invariants of ramification, alone.
For a given group, $G$, define ${\cal S}_G$ to be the set of
indecomposable $\bZ[G]$modules, ${\cal M}$, such that there
is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$
is a $\bZ[G]$module summand of an ambiguous ideal of $N$. Can
${\cal S}_G$ ever be infinite? In this paper we answer this
question of Chinburg in the affirmative.


1048  Localization theories for simplicial presheaves Goerss, P. G.; Jardine, J. F.
Most extant localization theories for spaces, spectra and diagrams
of such can be derived from a simple list of axioms which are verified
in broad generality. Several new theories are introduced, including
localizations for simplicial presheaves and presheaves of spectra at
homology theories represented by presheaves of spectra, and a theory
of localization along a geometric topos morphism. The
$f$localization concept has an analog for simplicial presheaves, and
specializes to the $\hbox{\Bbbvii A}^1$local theory of
MorelVoevodsky. This theory answers a question of Soul\'e concerning
integral homology localizations for diagrams of spaces.


1090  Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes Lohoué, Noël; Mustapha, Sami
Let $\Delta$ be a left invariant subLaplacian on a Lie group $G$
and let $\nabla$ be the associated gradient. In this paper we
investigate the boundness of the Riesz transform
$\nabla\Delta^{1/2}$ on Lie groups $G$ which are amenable and with
exponential volume growth and on certain homogenous spaces.


1105  Tempered representations and the theta correspondence Roberts, Brooks
Let $V$ be an even dimensional nondegenerate symmetric bilinear
space over a nonarchimedean local field $F$ of characteristic zero,
and let $n$ be a nonnegative integer. Suppose that $\sigma \in
\Irr \bigl(\OO (V)\bigr)$ and $\pi \in \Irr \bigl(\Sp (n,F)\bigr)$
correspond under the theta correspondence. Assuming that $\sigma$
is tempered, we investigate the problem of determining the
Langlands quotient data for $\pi$.


1119  Ward's solitons II: exact solutions Anand, Christopher Kumar
In a previous paper, we gave a correspondence between certain exact
solutions to a \((2+1)\)dimensional integrable Chiral Model and
holomorphic bundles on a compact surface. In this paper, we use
algebraic geometry to derive a closedform expression for those
solutions and show by way of examples how the algebraic data which
parametrise the solution space dictates the behaviour of the
solutions.


1138  Compound invariants and mixed $F$, $\DF$power spaces Chalov, P. A.; Terzioğlu, T.; Zahariuta, V. P.
The problems on isomorphic classification and quasiequivalence of bases
are studied for the class of mixed $F$, $\DF$power series spaces,
{\it i.e.} the spaces of the following kind
$$
G(\la,a)=\lim_{p \to \infty} \proj \biggl(\lim_{q \to \infty}\ind
\Bigl(\ell_1\bigl(a_i (p,q)\bigr)\Bigr)\biggr),
$$
where $a_i (p,q)=\exp\bigl((p\la_i q)a_i\bigr)$, $p,q \in \N$, and
$\la =( \la_i)_{i \in \N}$, $a=(a_i)_{i \in \N}$ are
some sequences of positive numbers. These spaces, up to isomorphisms,
are basis subspaces of tensor products of power series spaces of $F$ and
$\DF$types, respectively. The $m$rectangle characteristic
$\mu_m^{\lambda,a}(\delta,\varepsilon; \tau,t)$, $m \in \N$ of the
space $G(\la,a)$ is defined as the number of members of the sequence
$(\la_i, a_i)_{i \in \N}$ which are contained in the union of $m$
rectangles $P_k = (\delta_k, \varepsilon_k] \times (\tau_k, t_k]$,
$k = 1,2, \ldots, m$. It is shown that each $m$rectangle characteristic
is an invariant on the considered class under some proper definition of an
equivalency relation. The main tool are new compound invariants, which
combine some version of the classical approximative dimensions (Kolmogorov,
Pe{\l}czynski) with appropriate geometrical and interpolational operations
under neighborhoods of the origin (taken from a given basis).


1163  Gradient estimates for harmonic Functions on manifolds with Lipschitz metrics Chen, Jingyi; Hsu, Elton P.
We introduce a distributional Ricci curvature on complete smooth
manifolds with Lipschitz continuous metrics. Under an assumption
on the volume growth of geodesics balls, we obtain a gradient
estimate for weakly harmonic functions if the distributional Ricci
curvature is bounded below.


1176  Isomorphism problem for metacirculant graphs of order a product of distinct primes Dobson, Edward
In this paper, we solve the isomorphism problem for metacirculant
graphs of order $pq$ that are not circulant. To solve this problem,
we first extend Babai's characterization of the CIproperty to
nonCayley vertextransitive hypergraphs. Additionally, we find a
simple characterization of metacirculant Cayley graphs of order $pq$,
and exactly determine the full isomorphism classes of circulant graphs
of order $pq$.


1189  Totally real rigid elements and Galois theory Engler, Antonio José
Abelian closed subgroups of the Galois group of the pythagorean closure
of a formally real field are described by means of the inertia group of
suitable valuation rings.


1209  A lower bound for $K_X L$ of quasipolarized surfaces $(X,L)$ with nonnegative Kodaira dimension Fukuma, Yoshiaki
Let $X$ be a smooth projective surface over the complex
number field and let $L$ be a nefbig divisor on $X$. Here we consider
the following conjecture; If the Kodaira dimension $\kappa(X)\geq 0$,
then $K_{X}L\geq 2q(X)4$, where $q(X)$ is the irregularity of $X$. In
this paper, we prove that this conjecture is true if (1) the case in which
$\kappa(X)=0$ or $1$, (2) the case in which $\kappa(X)=2$ and $h^{0}(L)\geq
2$, or (3) the case in which $\kappa(X)=2$, $X$ is minimal, $h^{0}(L)=1$,
and $L$ satisfies some conditions.


1236  The behaviour of Legendre and ultraspherical polynomials in $L_p$spaces Kalton, N. J.; Tzafriri, L.
We consider the analogue of the $\Lambda(p)$problem for
subsets of the Legendre polynomials or more general ultraspherical
polynomials. We obtain the ``best possible'' result that if $2<p<4$
then a random subset of $N$ Legendre polynomials of size
$N^{4/p1}$ spans an Hilbertian subspace. We also answer a question of
K\"onig concerning the structure of the space of polynomials of degree
$n$ in various weighted $L_p$spaces.


1253  Integral representation of $p$class groups in ${\Bbb Z}_p$extensions and the Jacobian variety LópezBautista, Pedro Ricardo; VillaSalvador, Gabriel Daniel
For an arbitrary finite Galois $p$extension $L/K$ of
$\zp$cyclotomic number fields of $\CM$type with Galois group $G =
\Gal(L/K)$ such that the Iwasawa invariants $\mu_K^$, $ \mu_L^$
are zero, we obtain unconditionally and explicitly the Galois
module structure of $\clases$, the minus part of the $p$subgroup
of the class group of $L$. For an arbitrary finite Galois
$p$extension $L/K$ of algebraic function fields of one variable
over an algebraically closed field $k$ of characteristic $p$ as its
exact field of constants with Galois group $G = \Gal(L/K)$ we
obtain unconditionally and explicitly the Galois module structure
of the $p$torsion part of the Jacobian variety $J_L(p)$ associated
to $L/k$.


1273  Mean convergence of Lagrange interpolation for exponential weights on $[1,1]$ Lubinsky, D. S.
We obtain necessary and sufficient conditions for mean convergence of
Lagrange interpolation at zeros of orthogonal polynomials for weights on
$[1,1]$, such as
\[
w(x)=\exp \bigl((1x^{2})^{\alpha }\bigr),\quad \alpha >0
\]
or
\[
w(x)=\exp \bigl(\exp _{k}(1x^{2})^{\alpha }\bigr),\quad k\geq 1,
\ \alpha >0,
\]
where $\exp_{k}=\exp \Bigl(\exp \bigl(\cdots\exp (\ )\cdots\bigr)\Bigr)$
denotes the $k$th iterated exponential.


1298  Imprimitively generated Liealgebraic Hamiltonians and separation of variables Milson, Robert
Turbiner's conjecture posits that a Liealgebraic Hamiltonian
operator whose domain is a subset of the Euclidean plane admits a
separation of variables. A proof of this conjecture is given in
those cases where the generating Liealgebra acts imprimitively.
The general form of the conjecture is false. A counterexample is
given based on the trigonometric OlshanetskyPerelomov potential
corresponding to the $A_2$ root system.


1323  L'invariant de HasseWitt de la forme de Killing Morales, Jorge
Nous montrons que l'invariant de HasseWitt de la forme de Killing
d'une alg{\`e}bre de Lie semisimple $L$ s'exprime {\`a} l'aide de
l'invariant de Tits de la repr{\'e}sentation irr{\'e}ductible de
$L$ de poids dominant $\rho=\frac{1}{2}$ (somme des racines
positives), et des invariants associ{\'e}s au groupe des
sym{\'e}tries du diagramme de Dynkin de $L$.


1337  Author Index  Index des auteurs 1998, for 1998  pour
No abstract.
