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3  On a Conjecture of Goresky, Kottwitz and MacPherson Allday, C.; Puppe, V.
We settle a conjecture of Goresky, Kottwitz and MacPherson related
to Koszul duality, \ie, to the correspondence between differential
graded modules over the exterior algebra and those over the
symmetric algebra.


10  Tractable Fields Chacron, M.; Tignol, J.P.; Wadsworth, A. R.
A field $F$ is said to be tractable when a condition
described below on the simultaneous representation of
quaternion algebras holds over $F$. It is shown
that a global field $F$ is tractable i{f}f $F$ has
at most one dyadic place. Several other examples
of tractable and nontractable fields are given.


26  Separable Reduction and Supporting Properties of FréchetLike Normals in Banach Spaces Fabian, Marián; Mordukhovich, Boris S.
We develop a method of separable reduction for Fr\'{e}chetlike
normals and $\epsilon$normals to arbitrary sets in general Banach
spaces. This method allows us to reduce certain problems involving
such normals in nonseparable spaces to the separable case. It is
particularly helpful in Asplund spaces where every separable subspace
admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable
reduction method in Asplund spaces, we provide a new direct proof of a
nonconvex extension of the celebrated BishopPhelps density theorem.
Moreover, in this way we establish new characterizations of Asplund
spaces in terms of $\epsilon$normals.


49  Algèbres quasicommutatives et carrés de Steenrod Ndombol, Bitjong; El haouari, M.
Soit $k$ un corps de caract\'eristique $p$ quelconque. Nous
d\'efinissons la cat\'egorie des $k$alg\`ebres de cocha\^{\i}nes
fortement quasicommutatives et nous donnons une condition
n\'ecessaire et suffisante pour que l'alg\`ebre de cohomologie \`a
coefficients dans $\mathbb{Z}_2$ d'un objet de cette cat\'egorie soit
un module instable sur l'alg\`ebre de Steenrod \`a coefficients dans
$\mathbb{Z}_2$.


69  On a Theorem of Hermite and Joubert Reichstein, Zinovy
A classical theorem of Hermite and Joubert asserts that any field
extension of degree $n=5$ or $6$ is generated by an element whose
minimal polynomial is of the form $\lambda^n + c_1 \lambda^{n1} +
\cdots + c_{n1} \lambda + c_n$ with $c_1=c_3=0$. We show that this
theorem fails for $n=3^m$ or $3^m + 3^l$ (and more generally, for $n =
p^m$ or $p^m + p^l$, if 3 is replaced by another prime $p$), where $m
> l \geq 0$. We also prove a similar result for division algebras and
use it to study the structure of the universal division algebra $\UD
(n)$.


96  Partial Characters and Signed Quotient Hypergroups Rösler, Margit; Voit, Michael
If $G$ is a closed subgroup of a commutative hypergroup $K$, then the
coset space $K/G$ carries a quotient hypergroup structure. In this
paper, we study related convolution structures on $K/G$ coming from
deformations of the quotient hypergroup structure by certain functions
on $K$ which we call partial characters with respect to $G$. They are
usually not probabilitypreserving, but lead to socalled signed
hypergroups on $K/G$. A first example is provided by the Laguerre
convolution on $\left[ 0,\infty \right[$, which is interpreted as a
signed quotient hypergroup convolution derived from the Heisenberg
group. Moreover, signed hypergroups associated with the Gelfand pair
$\bigl( U(n,1), U(n) \bigr)$ are discussed.


117  Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation Sauer, A.
We construct meromorphic functions with asymptotic power series
expansion in $z^{1}$ at $\infty$ on an Arakelyan set $A$ having
prescribed zeros and poles outside $A$. We use our results to prove
approximation theorems where the approximating function fulfills
interpolation restrictions outside the set of approximation.


130  The Dual Pair $G_2 \times \PU_3 (D)$ ($p$Adic Case) Savin, Gordan; Gan, Wee Teck
We study the correspondence of representations arising by
restricting the minimal representation of the linear group of type
$E_7$ and relative rank $4$. The main tool is computations of the
Jacquet modules of the minimal representation with respect to
maximal parabolic subgroups of $G_2$ and $\PU_3(D)$.


147  Homeomorphic Analytic Maps into the Maximal Ideal Space of $H^\infty$ Suárez, Daniel
Let $m$ be a point of the maximal ideal space of $\papa$ with
nontrivial Gleason part $P(m)$. If $L_m \colon \disc \rr P(m)$ is the
Hoffman map, we show that $\papa \circ L_m$ is a closed subalgebra
of $\papa$. We characterize the points $m$ for which $L_m$ is a
homeomorphism in terms of interpolating sequences, and we show that in
this case $\papa \circ L_m$ coincides with $\papa$. Also, if
$I_m$ is the ideal of functions in $\papa$ that identically vanish
on $P(m)$, we estimate the distance of any $f\in \papa$ to $I_m$.


164  Poles of Siegel Eisenstein Series on $U(n,n)$ Tan, Victor
Let $U(n,n)$ be the rank $n$ quasisplit unitary group over a
number field. We show that the normalized Siegel Eisenstein series
of $U(n,n)$ has at most simple poles at the integers or half
integers in certain strip of the complex plane.


176  Values of the Dedekind Eta Function at Quadratic Irrationalities van der Poorten, Alfred; Williams, Kenneth S.
Let $d$ be the discriminant of an imaginary quadratic field. Let
$a$, $b$, $c$ be integers such that
$$
b^2  4ac = d, \quad a > 0, \quad \gcd (a,b,c) = 1.
$$
The value of $\bigl\eta \bigl( (b + \sqrt{d})/2a \bigr) \bigr$ is
determined explicitly, where $\eta(z)$ is Dedekind's eta function
$$
\eta (z) = e^{\pi iz/12} \prod^\ty_{m=1} (1  e^{2\pi imz})
\qquad \bigl( \im(z) > 0 \bigr). %\eqno({\rm im}(z)>0).
$$


225  Asymptotic Formulae for the Lattice Point Enumerator Betke, U.; Böröczky, K. Jr.
Let $M$ be a convex body such that the boundary has positive
curvature. Then by a well developed theory dating back to Landau and
Hlawka for large $\lambda$ the number of lattice points in $\lambda M$
is given by $G(\lambda M) =V(\lambda M) + O(\lambda^{d1\varepsilon
(d)})$ for some positive $\varepsilon(d)$. Here we give for general
convex bodies the weaker estimate
\[
\left G(\lambda M) V(\lambda M) \right 
\le \frac{1}{2} S_{\Z^d}(M) \lambda^{d1}+o(\lambda^{d1})
\]
where $S_{\Z^d}(M)$ denotes the lattice surface area of $M$. The term
$S_{\Z^d}(M)$ is optimal for all convex bodies and $o(\lambda^{d1})$
cannot be improved in general. We prove that the same estimate even
holds if we allow small deformations of $M$.


250  Convergence of Subdifferentials of Convexly Composite Functions Combari, C.; Poliquin, R.; Thibault, L.
In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'eKuratowski
convergence of the graphs of the subdifferentials of convexly
composite functions. We also provide applications to the
convergence of multipliers of families of constrained optimization
problems and to the generalized secondorder derivability of
convexly composite functions.


266  Spectral Estimates for Towers of Noncompact Quotients Deitmar, Anton; Hoffman, Werner
We prove a uniform upper estimate on the number of cuspidal
eigenvalues of the $\Ga$automorphic Laplacian below a given bound
when $\Ga$ varies in a family of congruence subgroups of a given
reductive linear algebraic group. Each $\Ga$ in the family is assumed
to contain a principal congruence subgroup whose index in $\Ga$ does
not exceed a fixed number. The bound we prove depends linearly on the
covolume of $\Ga$ and is deduced from the analogous result about the
cutoff Laplacian. The proof generalizes the heatkernel method which
has been applied by Donnelly in the case of a fixed lattice~$\Ga$.


294  A Homotopy of Quiver Morphisms with Applications to Representations Enochs, Edgar E.; Herzog, Ivo
It is shown that a morphism of quivers having a certain path
lifting property has a decomposition that mimics the decomposition
of maps of topological spaces into homotopy equivalences composed
with fibrations. Such a decomposition enables one to describe the
right adjoint of the restriction of the representation functor
along a morphism of quivers having this path lifting property.
These right adjoint functors are used to construct injective
representations of quivers. As an application, the injective
representations of the cyclic quivers are classified when the base
ring is left noetherian. In particular, the indecomposable
injective representations are described in terms of the injective
indecomposable $R$modules and the injective indecomposable
$R[x,x^{1}]$modules.


309  Symmetric sequence subspaces of $C(\alpha)$, II Leung, Denny H.; Tang, WeeKee
If $\alpha$ is an ordinal, then the space of all ordinals less than or
equal to $\alpha$ is a compact Hausdorff space when endowed with the
order topology. Let $C(\alpha)$ be the space of all continuous
realvalued functions defined on the ordinal interval $[0,
\alpha]$. We characterize the symmetric sequence spaces which embed
into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy
$(E_\alpha)$ of symmetric sequence spaces is constructed so that, for
each countable ordinal $\alpha$, $E_\alpha$ embeds into
$C(\omega^{\omega^\alpha})$, but does not embed into
$C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.


326  Association Schemes for Ordered Orthogonal Arrays and $(T,M,S)$Nets Martin, W. J.; Stinson, D. R.
In an earlier paper~\cite{stinmar}, we studied a generalized Rao bound
for ordered orthogonal arrays and $(T,M,S)$nets. In this paper,
we extend this to a codingtheoretic approach to ordered orthogonal
arrays. Using a certain association
scheme, we prove a MacWilliamstype theorem for linear ordered orthogonal
arrays and linear ordered codes as well as a linear programming bound
for the general case. We include some tables which compare this
bound against two previously known bounds for ordered orthogonal arrays.
Finally we show that, for even strength, the LP bound is always at
least as strong as the generalized Rao bound.


347  Exceptional Moufang Quadrangles of Type $\mathsf{F}_4$ Mühlherr, Bernhard; Van Maldeghem, Hendrik
In this paper, we present a geometric construction of the Moufang
quadrangles discovered by Richard Weiss (see Tits \& Weiss
\cite{TitWei:97} or Van Maldeghem \cite{Mal:97}). The construction
uses fixed point free involutions in certain mixed quadrangles, which
are then extended to involutions of certain buildings of type
$\ssF_4$. The fixed flags of each such involution constitute a
generalized quadrangle. This way, not only the new exceptional
quadrangles can be constructed, but also some special type of mixed
quadrangles.


372  Uniqueness for a Competing Species Model Mytnik, Leonid
We show that a martingale problem associated with a competing
species model has a unique solution. The proof of uniqueness of the
solution for the martingale problem is based on duality
technique. It requires the construction of dual probability
measures.


449  A BrunnMinkowski Type Theorem on the Minkowski Spacetime Bahn, Hyoungsick; Ehrlich, Paul
In this article, we derive a BrunnMinkowski type theorem
for sets bearing some relation to the causal structure
on the Minkowski spacetime $\mathbb{L}^{n+1}$. We also
present an isoperimetric inequality in the Minkowski
spacetime $\mathbb{L}^{n+1}$ as a consequence of this
BrunnMinkowski type theorem.


470  Exterior Univalent Harmonic Mappings With Finite Blaschke Dilatations Bshouty, D.; Hengartner, W.
In this article we characterize the univalent harmonic mappings from
the exterior of the unit disk, $\Delta$, onto a simply connected
domain $\Omega$ containing infinity and which are solutions of the system
of elliptic partial differential equations $\fzbb = a(z)f_z(z)$
where the second dilatation function $a(z)$ is a finite Blaschke
product. At the end of this article, we apply our results to
nonparametric minimal surfaces having the property that the image
of its Gauss map is the upper halfsphere covered once or twice.


488  Homological Aspects of Semigroup Gradings on Rings and Algebras Burgess, W. D.; Saorín, Manuel
This article studies algebras $R$ over a simple artinian ring $A$,
presented by a quiver and relations and graded by a semigroup $\Sigma$.
Suitable semigroups often arise from a presentation of $R$.
Throughout, the algebras need not be finite dimensional. The graded
$K_0$, along with the $\Sigma$graded Cartan endomorphisms and Cartan
matrices, is examined. It is used to study homological properties.
A test is found for finiteness of the global dimension of a
monomial algebra in terms of the invertibility of the Hilbert
$\Sigma$series in the associated path incidence ring.
The rationality of the $\Sigma$Euler characteristic, the Hilbert
$\Sigma$series and the Poincar\'eBetti $\Sigma$series is studied
when $\Sigma$ is torsionfree commutative and $A$ is a division ring.
These results are then applied to the classical series. Finally, we
find new finite dimensional algebras for which the strong no loops
conjecture holds.


506  On Polynomial Invariants of Exceptional Simple Algebraic Groups Elduque, A.; Iltyakov, A. V.
We study polynomial invariants of systems of vectors with respect
to exceptional simple algebraic groups in their minimal linear
representations. For each type we prove that the algebra of
invariants is integral over the subalgebra of trace polynomials
for a suitable algebraic system (\cf\ \cite{Schw1},
\cite{Schw2}, \cite{Ilt}).


523  Representations of VirasoroHeisenberg Algebras and VirasoroToroidal Algebras Fabbri, Marc A.; Okoh, Frank
Virasorotoroidal algebras, $\tilde\mathcal{T}_{[n]}$, are
semidirect products of toroidal algebras $\mathcal{T}_{[n]}$ and
the Virasoro algebra. The toroidal algebras are, in turn,
multiloop versions of affine KacMoody algebras. Let $\Gamma$ be
an extension of a simply laced lattice $\dot{Q}$ by a hyperbolic
lattice of rank two. There is a Fock space $V(\Gamma)$
corresponding to $\Gamma$ with a decomposition as a complex vector
space: $V(\Gamma) = \coprod_{m \in \mathbf{Z}}K(m)$. Fabbri and
Moody have shown that when $m \neq 0$, $K(m)$ is an irreducible
representation of $\tilde\mathcal{T}_{[2]}$. In this paper we
produce a filtration of $\tilde\mathcal{T}_{[2]}$submodules of
$K(0)$. When $L$ is an arbitrary geometric lattice and $n$ is a
positive integer, we construct a VirasoroHeisenberg algebra
$\tilde\mathcal{H}(L,n)$. Let $Q$ be an extension of $\dot{Q}$ by
a degenerate rank one lattice. We determine the components of
$V(\Gamma)$ that are irreducible $\tilde\mathcal{H}(Q,1)$modules
and we show that the reducible components have a filtration of
$\tilde\mathcal{H}(Q,1)$submodules with completely reducible
quotients. Analogous results are obtained for $\tilde\mathcal{H}
(\dot{Q},2)$. These results complement and extend results of
Fabbri and Moody.


546  Strong Converse Inequalities for Averages in Weighted $L^p$ Spaces on $[1,1]$ Felten, M.
Averages in weighted spaces $L^p_\phi[1,1]$ defined by additions
on $[1,1]$ will be shown to satisfy strong converse inequalities
of type A and B with appropriate $K$functionals. Results for
higher levels of smoothness are achieved by combinations of
averages. This yields, in particular, strong converse inequalities
of type D between $K$functionals and suitable difference operators.


566  Quotient Hereditarily Indecomposable Banach Spaces Ferenczi, V.
A Banach space $X$ is said to be {\it quotient hereditarily
indecomposable\/} if no infinite dimensional quotient of a subspace
of $X$ is decomposable. We provide an example of a quotient
hereditarily indecomposable space, namely the space $X_{\GM}$
constructed by W.~T.~Gowers and B.~Maurey in \cite{GM}. Then we
provide an example of a reflexive hereditarily indecomposable space
$\hat{X}$ whose dual is not hereditarily indecomposable; so
$\hat{X}$ is not quotient hereditarily indecomposable. We also
show that every operator on $\hat{X}^*$ is a strictly singular
perturbation of an homothetic map.


585  Smooth Finite Dimensional Embeddings Mansfield, R.; MovahediLankarani, H.; Wells, R.
We give necessary and sufficient conditions for a normcompact subset
of a Hilbert space to admit a $C^1$ embedding into a finite dimensional
Euclidean space. Using quasibundles, we prove a structure theorem
saying that the stratum of $n$dimensional points is contained in an
$n$dimensional $C^1$ submanifold of the ambient Hilbert space. This
work sharpens and extends earlier results of G.~Glaeser on paratingents.
As byproducts we obtain smoothing theorems for compact subsets of
Hilbert space and disjunction theorems for locally compact subsets
of Euclidean space.


616  Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory Panyushev, Dmitri I.
Let $L$ be a simple algebraic group and $P$ a parabolic subgroup
with Abelian unipotent radical $P^u$. Many familiar varieties
(determinantal varieties, their symmetric and skewsymmetric
analogues) arise as closures of $P$orbits in $P^u$. We give a
unified invarianttheoretic treatment of various properties of
these orbit closures. We also describe the closures of the
conormal bundles of these orbits as the irreducible components of
some commuting variety and show that the polynomial algebra
$k[P^u]$ is a free module over the algebra of covariants.


636  First Occurrence for the Dual Pairs $\bigl(U(p,q),U(r,s)\bigr)$ Paul, Annegret
We prove a conjecture of Kudla and Rallis about the first occurrence
in the theta correspondence, for dual pairs of the form
$\bigl(U(p,q),U(r,s)\bigr)$ and most representations.


658  Nilpotency of Some Lie Algebras Associated with $p$Groups Shumyatsky, Pavel
Let $ L=L_0+L_1$ be a $\mathbb{Z}_2$graded Lie algebra over a
commutative ring with unity in which $2$ is invertible. Suppose
that $L_0$ is abelian and $L$ is generated by finitely many
homogeneous elements $a_1,\dots,a_k$ such that every commutator in
$a_1,\dots,a_k$ is adnilpotent. We prove that $L$ is nilpotent.
This implies that any periodic residually finite $2'$group $G$
admitting an involutory automorphism $\phi$ with $C_G(\phi)$
abelian is locally finite.


673  Brownian Motion and Harmonic Analysis on Sierpinski Carpets Barlow, Martin T.; Bass, Richard F.
We consider a class of fractal subsets of $\R^d$ formed in a manner
analogous to the construction of the Sierpinski carpet. We prove a
uniform Harnack inequality for positive harmonic functions; study
the heat equation, and obtain upper and lower bounds on the heat
kernel which are, up to constants, the best possible; construct a
locally isotropic diffusion $X$ and determine its basic properties;
and extend some classical Sobolev and Poincar\'e inequalities to
this setting.


745  Induced Coactions of Discrete Groups on $C^*$Algebras Echterhoff, Siegfried; Quigg, John
Using the close relationship between coactions of discrete groups and
Fell bundles, we introduce a procedure for inducing a $C^*$coaction
$\delta\colon D\to D\otimes C^*(G/N)$ of a quotient group $G/N$ of a
discrete group $G$ to a $C^*$coaction $\Ind\delta\colon\Ind D\to \Ind
D\otimes C^*(G)$ of $G$. We show that induced coactions behave in many
respects similarly to induced actions. In particular, as an analogue of
the well known imprimitivity theorem for induced actions we prove that
the crossed products $\Ind D\times_{\Ind\delta}G$ and $D\times_{\delta}G/N$
are always Morita equivalent. We also obtain nonabelian analogues of a
theorem of Olesen and Pedersen which show that there is a duality between
induced coactions and twisted actions in the sense of Green. We further
investigate amenability of Fell bundles corresponding to induced coactions.


771  Stable BiPeriod Summation Formula and Transfer Factors Flicker, Yuval Z.
This paper starts by introducing a biperiodic summation formula
for automorphic forms on a group $G(E)$, with periods by a subgroup
$G(F)$, where $E/F$ is a quadratic extension of number fields. The
split case, where $E = F \oplus F$, is that of the standard trace
formula. Then it introduces a notion of stable biconjugacy, and
stabilizes the geometric side of the biperiod summation formula.
Thus weighted sums in the stable biconjugacy class are expressed
in terms of stable biorbital integrals. These stable integrals
are on the same endoscopic groups $H$ which occur in the case of
standard conjugacy.
The spectral side of the biperiod summation formula involves
periods, namely integrals over the group of $F$adele points of
$G$, of cusp forms on the group of $E$adele points on the group
$G$. Our stabilization suggests that such cusp formswith non
vanishing periodsand the resulting biperiod distributions
associated to ``periodic'' automorphic forms, are related to
analogous biperiod distributions associated to ``periodic''
automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$.
This offers a sharpening of the theory of liftings, where periods
play a key role.
The stabilization depends on the ``fundamental lemma'', which
conjectures that the unit elements of the Hecke algebras on $G$ and
$H$ have matching orbital integrals. Even in stating this
conjecture, one needs to introduce a ``transfer factor''. A
generalization of the standard transfer factor to the biperiodic
case is introduced. The generalization depends on a new definition
of the factors even in the standard case.
Finally, the fundamental lemma is verified for $\SL(2)$.


792  Tensor Products and Transferability of Semilattices Grätzer, G.; Wehrung, F.
In general, the tensor product, $A \otimes B$, of the lattices $A$ and
$B$ with zero is not a lattice (it is only a joinsemilattice with
zero). If $A\otimes B$ is a {\it capped\/} tensor product, then
$A\otimes B$ is a lattice (the converse is not known). In this paper, we
investigate lattices $A$ with zero enjoying the property that $A\otimes
B$ is a capped tensor product, for {\it every\/} lattice $B$ with zero;
we shall call such lattices {\it amenable}.
The first author introduced in 1966 the concept of a {\it sharply
transferable lattice}. In 1972, H.~Gaskill defined,
similarly, sharply transferable semilattices, and characterized them
by a very effective condition (T).
We prove that {\it a finite lattice $A$ is\/} amenable {\it if{}f it is\/}
sharply transferable {\it as a joinsemilattice}.
For a general lattice $A$ with zero, we obtain the result: {\it $A$ is
amenable if{}f $A$ is locally finite and every finite sublattice of $A$
is transferable as a joinsemilattice}.
This yields, for example, that a finite lattice $A$ is amenable
if{}f $A\otimes\FL(3)$ is a lattice if{}f $A$ satisfies (T), with
respect to join. In particular, $M_3\otimes\FL(3)$ is not a lattice.
This solves a problem raised by R.~W.~Quackenbush in 1985 whether
the tensor product of lattices with zero is always a lattice.


816  A New Form of the SegalBargmann Transform for Lie Groups of Compact Type Hall, Brian C.
I consider a twoparameter family $B_{s,t}$ of unitary transforms
mapping an $L^{2}$space over a Lie group of compact type onto a
holomorphic $L^{2}$space over the complexified group. These were
studied using infinitedimensional analysis in joint work with
B.~Driver, but are treated here by finitedimensional means. These
transforms interpolate between two previously known transforms, and
all should be thought of as generalizations of the classical
SegalBargmann transform. I consider also the limiting cases $s
\rightarrow \infty$ and $s \rightarrow t/2$.


835  LanglandsShahidi Method and Poles of Automorphic $L$Functions: Application to Exterior Square $L$Functions Kim, Henry H.
In this paper we use LanglandsShahidi method and the result of
Langlands which says that non selfconjugate maximal parabolic
subgroups do not contribute to the residual spectrum, to prove the
holomorphy of several completed automorphic $L$functions on the
whole complex plane which appear in constant terms of the Eisenstein
series. They include the exterior square $L$functions of $\GL_n$, $n$
odd, the RankinSelberg $L$functions of $\GL_n\times \GL_m$, $n\ne m$,
and $L$functions $L(s,\sigma,r)$, where $\sigma$ is a generic
cuspidal representation of $\SO_{10}$ and $r$ is the halfspin
representation of $\GSpin(10, \mathbb{C})$. The main part is
proving the holomorphy and nonvanishing of the local normalized
intertwining operators by reducing them to natural conjectures in
harmonic analysis, such as standard module conjecture.


850  Tensor Algebras, Induced Representations, and the Wold Decomposition Muhly, Paul S.; Solel, Baruch
Our objective in this sequel to \cite{MSp96a} is to develop extensions,
to representations of tensor algebras over $C^{*}$correspondences, of
two fundamental facts about isometries on Hilbert space: The Wold
decomposition theorem and Beurling's theorem, and to apply these to
the analysis of the invariant subspace structure of certain subalgebras
of CuntzKrieger algebras.


881  The Representation Ring and the Centre of a Hopf Algebra Witherspoon, Sarah J.
When $H$ is a finite dimensional, semisimple, almost cocommutative
Hopf algebra, we examine a table of characters which extends the
notion of the character table for a finite group. We obtain a
formula for the structure constants of the representation ring in
terms of values in the character table, and give the example of the
quantum double of a finite group. We give a basis of the centre of
$H$ which generalizes the conjugacy class sums of a finite group,
and express the class equation of $H$ in terms of this basis. We
show that the representation ring and the centre of $H$ are dual
character algebras (or signed hypergroups).


897  Cohomology of Complex Projective Stiefel Manifolds Astey, L.; Gitler, S.; Micha, E.; Pastor, G.
The cohomology algebra mod $p$ of the complex projective Stiefel
manifolds is determined for all primes $p$. When $p=2$ we also
determine the action of the Steenrod algebra and apply this to the
problem of existence of trivial subbundles of multiples of the
canonical line bundle over a lens space with $2$torsion, obtaining
optimal results in many cases.


915  Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers Balogh, Zoltán M.; Leuenberger, Christoph
Consider the polynomial hull of a smoothly varying family of
strictly convex smooth domains fibered over the unit circle. It is
wellknown that the boundary of the hull is foliated by graphs of
analytic discs. We prove that this foliation is smooth, and we
show that it induces a complex flow of contactomorphisms. These
mappings are quasiconformal in the sense of Kor\'anyi and Reimann.
A similar bound on their quasiconformal distortion holds as in the
onedimensional case of holomorphic motions. The special case when
the fibers are rotations of a fixed domain in $\C^2$ is studied in
details.


936  Galois Representations with NonSurjective Traces David, Chantal; Kisilevsky, Hershy; Pappalardi, Francesco
Let $E$ be an elliptic curve over $\q$, and let $r$ be an integer.
According to the LangTrotter conjecture, the number of primes $p$
such that $a_p(E) = r$ is either finite, or is asymptotic to
$C_{E,r} {\sqrt{x}} / {\log{x}}$ where $C_{E,r}$ is a nonzero
constant. A typical example of the former is the case of rational
$\ell$torsion, where $a_p(E) = r$ is impossible if $r \equiv 1
\pmod{\ell}$. We prove in this paper that, when $E$ has a rational
$\ell$isogeny and $\ell \neq 11$, the number of primes $p$ such
that $a_p(E) \equiv r \pmod{\ell}$ is finite (for some $r$ modulo
$\ell$) if and only if $E$ has rational $\ell$torsion over the
cyclotomic field $\q(\zeta_\ell)$. The case $\ell=11$ is special,
and is also treated in the paper. We also classify all those
occurences.


952  On Limit Multiplicities for Spaces of Automorphic Forms Deitmar, Anton; Hoffmann, Werner
Let $\Gamma$ be a rankone arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
squareintegrable $\Gamma$automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorgeWallach
and Delorme.


977  Extreme PickNevanlinna Interpolants Fisher, Stephen D.; Khavinson, Dmitry
Following the investigations of B.~Abrahamse [1], F.~Forelli [11],
M.~Heins [14] and others, we continue the study of the
PickNevanlinna interpolation problem in multiplyconnected planar
domains. One major focus is on the problem of characterizing the
extreme points of the convex set of interpolants of a fixed data
set. Several other related problems are discussed.


996  Counting in Ergodic Theory Jones, Roger L.; Rosenblatt, Joseph M.; Wierdl, Máté
Many aspects of the behavior of averages in ergodic theory
are a matter of counting the number of times a particular
event occurs. This theme is pursued in this article where
we consider large deviations, square functions, jump
inequalities and related topics.


1020  On Functions Satisfying Modular Equations for Infinitely Many Primes Kozlov, Dmitry N.
In this paper we study properties of the functions which satisfy
modular equations for infinitely many primes. The two main results
are:
\begin{enumerate}
\item[1)] every such function is analytic in the upper half plane;


1035  The Homology of Abelian Covers of Knotted Graphs Litherland, R. A.
Let $\tilde M$ be a regular branched cover of a homology 3sphere
$M$ with deck group $G\cong \zt^d$ and branch set a trivalent graph
$\Gamma$; such a cover is determined by a coloring of the edges of
$\Gamma$ with elements of $G$. For each index2 subgroup $H$ of
$G$, $M_H = \tilde M/H$ is a double branched cover of $M$. Sakuma
has proved that $H_1(\tilde M)$ is isomorphic, modulo 2torsion, to
$\bigoplus_H H_1(M_H)$, and has shown that $H_1(\tilde M)$ is
determined up to isomorphism by $\bigoplus_H H_1(M_H)$ in certain
cases; specifically, when $d=2$ and the coloring is such that the
branch set of each cover $M_H\to M$ is connected, and when $d=3$
and $\Gamma$ is the complete graph $K_4$. We prove this for a
larger class of coverings: when $d=2$, for any coloring of a
connected graph; when $d=3$ or $4$, for an infinite class of
colored graphs; and when $d=5$, for a single coloring of the
Petersen graph.


1073  The Hausdorff and Packing Dimensions of Some Sets Related to Sierpi\'nski Carpets Nielsen, Ole A.
The Sierpi\'nski carpets first considered by C.~McMullen and later
studied by Y.~Peres are modified by insisting that the allowed
digits in the expansions occur with prescribed frequencies. This
paper (i)~~calculates the Hausdorff, box (or Minkowski), and
packing dimensions of the modified Sierpi\'nski carpets and
(ii)~~shows that for these sets the Hausdorff and packing measures
in their dimension are never zero and gives necessary and
sufficient conditions for these measures to be infinite.


1089  The Characteristic Numbers of Quartic Plane Curves Vakil, Ravi
The characteristic numbers of smooth plane quartics are computed
using intersection theory on a component of the moduli space of
stable maps. This completes the verification of Zeuthen's
prediction of characteristic numbers of smooth plane curves. A
short sketch of a computation of the characteristic numbers of
plane cubics is also given as an illustration.


1123  First Steps of Local Contact Algebra Arnold, V. I.
We consider germs of mappings of a line to contact space and
classify the first simple singularities up to the action of
contactomorphisms in the target space and diffeomorphisms of the
line. Even in these first cases there arises a new interesting
interaction of local commutative algebra with contact structure.


1135  Endoscopic $L$Functions and a Combinatorial Identity Arthur, James
The trace formula contains terms on the spectral side that are
constructed from unramified automorphic $L$functions. We shall
establish an identify that relates these terms with corresponding
terms attached to endoscopic groups of $G$. In the process, we
shall show that the $L$functions of $G$ that come from automorphic
representations of endoscopic groups have meromorphic continuation.


1149  Linear Groups Generated by Reflection Tori Cohen, A. M.; Cuypers, H.; Sterk, H.
A reflection is an invertible linear transformation of a vector
space fixing a given hyperplane, its axis, vectorwise and a given
complement to this hyperplane, its center, setwise. A reflection
torus is a onedimensional group generated by all reflections with
fixed axis and center.


1175  Reflection Subquotients of Unitary Reflection Groups Lehrer, G. I.; Springer, T. A.
Let $G$ be a finite group generated by (pseudo) reflections in a
complex vector space and let $g$ be any linear transformation which
normalises $G$. In an earlier paper, the authors showed how to
associate with any maximal eigenspace of an element of the coset
$gG$, a subquotient of $G$ which acts as a reflection group on the
eigenspace. In this work, we address the questions of
irreducibility and the coexponents of this subquotient, as well as
centralisers in $G$ of certain elements of the coset. A criterion
is also given in terms of the invariant degrees of $G$ for an
integer to be regular for $G$. A key tool is the investigation of
extensions of invariant vector fields on the eigenspace, which
leads to some results and questions concerning the geometry of
intersections of invariant hypersurfaces.


1194  Subregular Nilpotent Elements and Bases in $K$Theory Lusztig, G.
In this paper we describe a canonical basis for the equivariant
$K$theory (with respect to a $\bc^*$action) of the variety of
Borel subalgebras containing a subregular nilpotent element of a
simple complex Lie algebra of type $D$ or~$E$.


1226  SemiAffine CoxeterDynkin Graphs and $G \subseteq \SU_2(C)$ McKay, John
The semiaffine CoxeterDynkin graph is introduced, generalizing
both the affine and the finite types.


1230  Symmetric Tessellations on Euclidean SpaceForms Hartley, Michael I.; McMullen, Peter; Schulte, Egon
It is shown here that, for $n \geq 2$, the $n$torus is the only
$n$dimensional compact euclidean spaceform which can admit a
regular or chiral tessellation. Further, such a tessellation can
only be chiral if $n = 2$.


1240  Realizations of Regular Toroidal Maps Monson, B.; Weiss, A. Ivić
We determine and completely describe all pure realizations of the
finite regular toroidal polyhedra of types $\{3,6\}$ and $\{6,3\}$.


1258  Similarity Submodules and Root Systems in Four Dimensions Baake, Michael; Moody, Robert V.
Lattices and $\ZZ$modules in Euclidean space possess an infinitude
of subsets that are images of the original set under similarity
transformation. We classify such selfsimilar images according to
their indices for certain 4D examples that are related to 4D root
systems, both crystallographic and noncrystallographic. We
encapsulate their statistics in terms of Dirichlet series
generating functions and derive some of their asymptotic properties.


1277  Isomorphism Invariants for Projective Configurations Shephard, G. C.
An isomorphism invariant is an expression, defined for a
configuration in the projective plane, which takes the same value
for all isomorphic configurations. Examples are given as well as a
general method (Nehring sequences) for constructing such
invariants.


1300  On the Existence of Similar Sublattices Conway, J. H.; Rains, E. M.; Sloane, N. J. A.
Partial answers are given to two questions. When does a lattice
$\Lambda$ contain a sublattice $\Lambda'$ of index $N$ that is
geometrically similar to $\Lambda$? When is the sublattice
``clean'', in the sense that the boundaries of the Voronoi cells
for $\Lambda'$ do not intersect $\Lambda$?


1307  Quadratic Integers and Coxeter Groups Johnson, Norman W.; Weiss, Asia Ivić
Matrices whose entries belong to certain rings of algebraic
integers can be associated with discrete groups of transformations
of inversive $n$space or hyperbolic $(n+1)$space
$\mbox{H}^{n+1}$. For small $n$, these may be Coxeter groups,
generated by reflections, or certain subgroups whose generators
include direct isometries of $\mbox{H}^{n+1}$. We show how linear
fractional transformations over rings of rational and (real or
imaginary) quadratic integers are related to the symmetry groups of
regular tilings of the hyperbolic plane or 3space. New light is
shed on the properties of the rational modular group $\PSL_2
(\bbZ)$, the Gaussian modular (Picard) group $\PSL_2 (\bbZ[{\it
i}])$, and the Eisenstein modular group $\PSL_2 (\bbZ[\omega ])$.


1337  Author Index  Index des auteurs 1999, for 1999  pour
No abstract.

