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3  Holomorphic Generation of Continuous Inverse Algebras Biller, Harald
We study complex commutative Banach algebras
(and, more generally, continuous
inverse algebras) in which the holomorphic functions of a fixed $n$tuple
of elements are dense. In particular, we characterize the compact subsets
of~$\C^n$ which appear as joint spectra of such $n$tuples. The
characterization is compared with several established notions of holomorphic
convexity by means of approximation
conditions.


36  Classification of Ding's Schubert Varieties: Finer Rook Equivalence Develin, Mike; Martin, Jeremy L.; Reiner, Victor
K.~Ding studied a class of Schubert varieties $X_\lambda$
in type A partial
flag manifolds, indexed by
integer partitions $\lambda$ and in bijection
with dominant permutations. He observed that the
Schubert cell structure of $X_\lambda$ is indexed by maximal rook
placements on the Ferrers board $B_\lambda$, and that the
integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are
additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$
satisfy the combinatorial condition of rookequivalence.


63  Some Results on the SchroederBernstein Property for Separable Banach Spaces Ferenczi, Valentin; Galego, Elói Medina
We construct a continuum of mutually
nonisomorphic
separable Banach spaces which are complemented in each other.
Consequently, the SchroederBernstein Index of any of these spaces is
$2^{\aleph_0}$. Our
construction is based on a Banach space introduced by W. T. Gowers
and
B. Maurey in 1997.
We also use classical descriptive set theory methods, as in some
work of the first author and C. Rosendal, to improve some results
of P. G. Casazza and
of N. J. Kalton on the
SchroederBernstein Property for
spaces with an unconditional finitedimensional Schauder
decomposition.


85  On the Convergence of a Class of Nearly Alternating Series Foster, J. H.; Serbinowska, Monika
Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.


109  On Fiber Cones of $\m$Primary Ideals Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.
Two formulas for the multiplicity of the fiber cone
$F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$primary ideal of
a $d$dimensional CohenMacaulay local ring $(R,\m)$ are derived in
terms of the mixed multiplicity $e_{d1}(\m  I)$, the multiplicity
$e(I)$, and superficial elements. As a consequence, the
CohenMacaulay property of $F(I)$ when $I$ has minimal mixed
multiplicity or almost minimal mixed multiplicity is characterized
in terms of the reduction number of $I$ and lengths of certain ideals.
We also characterize the CohenMacaulay and Gorenstein properties of
fiber cones of $\m$primary ideals with a $d$generated minimal
reduction $J$ satisfying $\ell(I^2/JI)=1$ or
$\ell(I\m/J\m)=1.$


127  Smooth Values of the Iterates of the Euler PhiFunction Lamzouri, Youness
Let $\phi(n)$ be the Euler phifunction, define
$\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all
$k\geq 0$. We will determine an asymptotic formula for the set of
integers $n$ less than $x$ for which $\phi_k(n)$ is $y$smooth,
conditionally on a weak form of the ElliottHalberstam conjecture.


148  On Certain Classes of Unitary Representations for Split Classical Groups Muić, Goran
In this paper we prove the unitarity of duals of tempered
representations supported on minimal parabolic subgroups for split
classical $p$adic groups. We also construct a family of unitary
spherical representations for real and complex classical groups


186  Endomorphism Algebras of Kronecker Modules Regulated by Quadratic Function Fields Okoh, F.; Zorzitto, F.
Purely simple Kronecker modules ${\mathcal M}$, built from an algebraically closed field $K$,
arise from a triplet $(m,h,\alpha)$ where $m$ is a positive integer,
$h\colon\ktil\ar \{\infty,0,1,2,3,\dots\}$ is a height function, and
$\alpha$ is a $K$linear functional on the space $\krx$ of rational
functions in one variable $X$. Every pair $(h,\alpha)$ comes with a
polynomial $f$ in $K(X)[Y]$ called the regulator. When the module
${\mathcal M}$ admits nontrivial endomorphisms, $f$ must be linear or
quadratic in $Y$. In that case ${\mathcal M}$ is purely simple if and
only if $f$ is an irreducible quadratic. Then the $K$algebra
$\edm\cm$ embeds in the quadratic function field $\krx[Y]/(f)$. For
some height functions $h$ of infinite support $I$, the search for a
functional $\alpha$ for which $(h,\alpha)$ has regulator $0$ comes
down to having functions $\eta\colon I\ar K$ such that no planar curve
intersects the graph of $\eta$ on a cofinite subset. If $K$ has
characterictic not $2$, and the triplet $(m,h,\alpha)$ gives a
purelysimple Kronecker module ${\mathcal M}$ having nontrivial
endomorphisms, then $h$ attains the value $\infty$ at least once on
$\ktil$ and $h$ is finitevalued at least twice on
$\ktil$. Conversely all these $h$ form part of such triplets. The
proof of this result hinges on the fact that a rational function $r$
is a perfect square in $\krx$ if and only if $r$ is a perfect square
in the completions of $\krx$ with respect to all of its valuations.


211  On Two Exponents of Approximation Related to a Real Number and Its Square Roy, Damien
For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the
supremum of all real numbers $\lambda$ such that, for each
sufficiently large $X$, the inequalities $x_0 \le X$,
$x_0\xix_1 \le X^{\lambda}$ and $x_0\xi^2x_2 \le
X^{\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$
not all zero, and let $\omegahat_2(\xi)$ denote the supremum of
all real numbers $\omega$ such that, for each sufficiently large
$X$, the dual inequalities $x_0+x_1\xi+x_2\xi^2 \le
X^{\omega}$, $x_1 \le X$ and $x_2 \le X$ admit a solution in
integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a
question of Y.~Bugeaud and M.~Laurent, we show that the exponents
$\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers
with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2,
(\sqrt{5}1)/2]$ while, for the same values of $\xi$, the dual
exponents $\omegahat_2(\xi)$ form a dense subset of $[2,
(\sqrt{5}+3)/2]$. Part of the proof rests on a result of
V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) =
1\omegahat_2(\xi)^{1}$ for any real number $\xi$ with
$[\bQ(\xi)\wcol\bQ]>2$.


225  Harmonic Analysis on Metrized Graphs Baker, Matt; Rumely, Robert
This paper studies the Laplacian operator on a metrized graph, and its
spectral theory.


276  Weighted Inequalities for HardySteklov Operators Bernardis, A. L.; MartínReyes, F. J.; Salvador, P. Ortega
We characterize the pairs of weights $(v,w)$ for which the
operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$
increasing and continuous functions is of strong type
$(p,q)$ or weak type $(p,q)$ with respect to the pair
$(v,w)$ in the case $0<q<p$ and $1<p<\infty$. The result
for the weak type is new while the characterizations for
the strong type improve the ones given by H.~P. Heinig and
G. Sinnamon. In particular, we do not assume
differentiability properties on $s$ and $h$ and we obtain
that the strong type inequality $(p,q)$, $q<p$, is
characterized by the fact that the function
$$\Phi(x)=\sup
\Bigl(\int_c^dg^qw\Bigr)^{1/p}
\Bigl(\int_{s(d)}^{h(c)}v^{1p'}\Bigr)^{1/p'}$$
belongs to $L^{r}(g^qw)$, where $1/r=1/q1/p$ and the
supremum is taken over all $c$ and $d$
such that $c\le x\le d$ and $s(d)\leq h(c)$.


296  Bol Loops of Nilpotence Class Two Chein, Orin; Goodaire, Edgar G.
Call a nonMoufang Bol loop minimally nonMoufang
if every proper subloop is Moufang and
minimally nonassociative if every proper subloop is
associative. We prove that these concepts are
the same for Bol loops which are nilpotent of
class two and in which certain associators square to $1$.
In the process, we derive many commutator and associator identities
which hold in such loops.


311  Growth and Zeros of the Zeta Function for Hyperbolic Rational Maps Christianson, Hans
This paper describes new results on the growth and zeros of the Ruelle
zeta function for the Julia set of a hyperbolic rational map. It is
shown that the zeta function is bounded by $\exp(C_K s^{\delta})$ in
strips $\Real s \leq K$, where $\delta$ is the dimension of the
Julia set. This leads to bounds on the number of zeros in strips
(interpreted as the PollicottRuelle resonances of this dynamical
system). An upper bound on the number of zeros in polynomial regions
$\{\Real s  \leq \Imag s^\alpha\}$ is given, followed by weaker
lower bound estimates in strips $\{\Real s > C, \Imag s\leq r\}$,
and logarithmic neighbourhoods
$\{ \Real s  \leq \rho \log \Imag s \}$.
Recent numerical work of StrainZworski suggests the upper
bounds in strips are optimal.


332  Endomorphism Rings of Finite Global Dimension Leuschke, Graham J.
For a commutative local ring $R$, consider (noncommutative)
$R$algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$
where $M$ is a reflexive $R$module with nonzero free direct summand.
Such algebras $\Lambda$ of finite global dimension can be viewed as
potential substitutes for, or analogues of, a resolution of
singularities of $\operatorname{Spec} R$. For example, Van den Bergh
has shown that a threedimensional Gorenstein normal
$\mathbb{C}$algebra with isolated terminal singularities has a
crepant resolution of singularities if and only if it has such an
algebra $\Lambda$ with finite global dimension and which is maximal
CohenMacaulay over $R$ (a ``noncommutative crepant resolution of
singularities''). We produce algebras
$\Lambda=\operatorname{End}_R(M)$ having finite global dimension in
two contexts: when $R$ is a reduced onedimensional complete local
ring, or when $R$ is a CohenMacaulay local ring of finite
CohenMacaulay type. If in the latter case $R$ is Gorenstein, then
the construction gives a noncommutative crepant resolution of
singularities in the sense of Van den Bergh.


343  Weak Semiprojectivity in Purely Infinite Simple $C^*$Algebras Lin, Huaxin
Let $A$ be a separable amenable purely infinite simple \CA which
satisfies the Universal Coefficient Theorem. We prove that $A$ is
weakly semiprojective if and only if $K_i(A)$ is a countable
direct sum of finitely generated groups ($i=0,1$). Therefore, if
$A$ is such a \CA, for any $\ep>0$ and any finite subset ${\mathcal
F}\subset A$ there exist $\dt>0$ and a finite subset ${\mathcal
G}\subset A$ satisfying the following: for any contractive
positive linear map $L: A\to B$ (for any \CA $B$) with $
\L(ab)L(a)L(b)\<\dt$ for $a, b\in {\mathcal G}$
there exists a homomorphism $h\from A\to B$ such that
$ \h(a)L(a)\<\ep$ for $a\in {\mathcal F}$.


372  Zeta Functions of Supersingular Curves of Genus 2 Maisner, Daniel; Nart, Enric
We determine which isogeny classes of supersingular abelian
surfaces over a finite field $k$ of characteristic $2$ contain
jacobians. We deal with this problem in a direct way by computing
explicitly the zeta function of all supersingular curves of genus
$2$. Our procedure is constructive, so that we are able to exhibit
curves with prescribed zeta function and find formulas for the
number of curves, up to $k$isomorphism, leading to the same zeta
function.


393  Le splitting pour l'opérateur de KleinGordon: une approche heuristique et numérique Servat, E.
Dans cet article on \'etudie la diff\'erence entre les deux
premi\`eres valeurs propres, le splitting, d'un op\'erateur de
KleinGordon semiclassique unidimensionnel, dans le cas d'un
potentiel sym\'etrique pr\'esentant un double puits. Dans le cas d'une
petite barri\`ere de potentiel, B. Helffer et B. Parisse ont obtenu
des r\'esultats analogues \`a ceux existant pour l'op\'erateur de
Schr\"odinger. Dans le cas d'une grande barri\`ere de potentiel, on
obtient ici des estimations des tranform\'ees de Fourier des fonctions
propres qui conduisent \`a une conjecture du splitting. Des calculs
num\'eriques viennent appuyer cette conjecture.


418  On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials Stoimenow, A.
It is known that the BrandtLickorishMillettHo polynomial $Q$
contains Casson's knot invariant. Whether there are (essentially)
other Vassiliev knot invariants obtainable from $Q$ is an open
problem. We show that this is not so up to degree $9$. We also
give the (apparently) first examples of knots not distinguished
by 2cable HOMFLY polynomials which are not mutants. Our calculations
provide evidence of a negative answer to the question whether Vassiliev
knot invariants of degree $d \le 10$ are determined by the HOMFLY and
Kauffman polynomials and their 2cables, and for the existence of
algebras of such Vassiliev invariants not isomorphic to the algebras
of their weight systems.


449  $\SL_n$, Orthogonality Relations and Transfer Badulescu, Alexandru Ioan
Let $\pi$ be a square integrable representation of
$G'=\SL_n(D)$, with $D$ a central division algebra of finite dimension
over a local field $F$ of nonzero characteristic. We prove
that, on the elliptic set, the character of $\pi$ equals the complex
conjugate of the orbital integral of one of the pseudocoefficients
of~$\pi$. We prove also the orthogonality relations for characters of
square integrable representations of $G'$. We prove the stable
transfer of orbital integrals between $\SL_n(F)$ and its inner forms.


465  Searching for Absolute $\mathcal{CR}$Epic Spaces Barr, Michael; Kennison, John F.; Raphael, R.
In previous papers, Barr and Raphael investigated the situation of a
topological space $Y$ and a subspace $X$ such that the induced map
$C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative
rings (with units). We call such an embedding a $\CR$epic embedding
and we say that $X$ is absolute $\CR$epic if every embedding of $X$
is $\CR$epic. We continue this investigation. Our most notable
result shows that a Lindel\"of space $X$ is absolute $\CR$epic if a
countable intersection of $\beta X$neighbourhoods of $X$ is a $\beta
X$neighbourhood of $X$. This condition is stable under countable
sums, the formation of closed subspaces, cozerosubspaces, and being
the domain or codomain of a perfect map. A strengthening of the
Lindel\"of property leads to a new class with the same closure
properties that is also closed under finite products. Moreover, all
\scompact spaces and all Lindel\"of $P$spaces satisfy this stronger
condition. We get some results in the nonLindel\"of case that are
sufficient to show that the Dieudonn\'e plank and some closely related
spaces are absolute $\CR$epic.


488  Osculating Varieties of Veronese Varieties and Their Higher Secant Varieties Bernardi, A.; Catalisano, M. V.; Gimigliano, A.; Idà, M.
We consider the $k$osculating varieties
$O_{k,n.d}$ to the (Veronese) $d$uple embeddings of $\PP^n$. We
study the dimension of their higher secant varieties via inverse
systems (apolarity). By associating certain 0dimensional schemes
$Y\subset \PP^n$ to $O^s_{k,n,d}$ and by studying their Hilbert
functions, we are able, in several cases, to determine whether those
secant varieties are defective or not.


503  Cyclic Groups and the Three Distance Theorem Chevallier, Nicolas
We give a two dimensional extension of the three distance Theorem. Let
$\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a
triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$translations,
whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose
number of different triangles, up to translations, is bounded above by a
constant which does not depend on $\theta$ and $q$.


553  Computations of Elliptic Units for Real Quadratic Fields Dasgupta, Samit
Let $K$ be a real quadratic field, and $p$ a rational prime which is
inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an
earlier joint article with Henri Darmon, we presented the definition
of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$
and each $\tau \in K$. We conjectured that the $p$adic number
$u(\alpha, \tau)$ lies in a specific ring class extension of $K$
depending on $\tau$, and proposed a ``Shimura reciprocity law"
describing the permutation action of Galois on the set of $u(\alpha,
\tau)$. This article provides computational evidence for these
conjectures. We present an efficient algorithm for computing
$u(\alpha, \tau)$, and implement this algorithm with the modular unit
$\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$
and $11$, and all real quadratic fields $K$ with discriminant $D <
500$ such that $2$ splits in $K$ and $K$ contains no unit of negative
norm, we obtain results supporting our conjectures. One of the
theoretical results in this paper is that a certain measure used to
define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$valued rather
than only $\mathbf{Z}_p \cap \mathbf{Q}$valued; this is an
improvement over our previous result and allows for a precise
definition of $u(\alpha, \tau)$, instead of only up to a root of
unity.


575  Cardinal Invariants of Analytic $P$Ideals HernándezHernández, Fernando; Hrušák, Michael
We study the cardinal invariants of analytic $P$ideals, concentrating on the
ideal $\mathcal{Z}$ of asymptotic density zero. Among other results we prove
$ \min\{ \mathfrak{b},\cov\ (\mathcal{N})
\} \leq\cov^{\ast}(\mathcal{Z}) \leq\max\{
\mathfrak{b},\non(\mathcal{N}) \right\}.
$


596  Eigenvalues, $K$theory and Minimal Flows ItzáOrtiz, Benjamín A.
Let $(Y,T)$ be a minimal suspension flow built over a dynamical
system $(X,S)$ and with (strictly positive, continuous) ceiling
function $f\colon X\to\R$. We show that the eigenvalues of
$(Y,T)$ are contained in the range of a trace on the $K_0$group
of $(X,S)$. Moreover, a trace gives an order isomorphism of a
subgroup of $K_0(\cprod{C(X)}{S})$ with the group of
eigenvalues of $(Y,T)$. Using this result, we relate the values of
$t$ for which the time$t$ map on the minimal suspension flow is
minimal with the $K$theory of the base of this suspension.


614  Preduals and Nuclear Operators Associated with Bounded, $p$Convex, $p$Concave and Positive $p$Summing Operators Labuschagne, C. C. A.
We use Krivine's form of the Grothendieck inequality
to renorm the space of bounded linear maps acting between Banach
lattices. We
construct preduals and describe the nuclear operators
associated with these preduals for this renormed space
of bounded operators as well as for
the spaces of $p$convex,
$p$concave and positive $p$summing operators acting
between Banach lattices and Banach spaces.
The nuclear operators obtained are described in
terms of factorizations through
classical Banach spaces via positive operators.


638  Distance from Idempotents to Nilpotents MacDonald, Gordon W.
We give bounds on the distance from a nonzero idempotent to the
set of nilpotents in the set of $n\times n$ matrices in terms of
the norm of the idempotent. We construct explicit idempotents and
nilpotents which achieve these distances, and determine exact
distances in some special cases.


658  Division Algebras of Prime Degree and Maximal Galois $p$Extensions Mináč, J.; Wadsworth, A.
Let $p$ be an odd prime number, and let $F$
be a field of characteristic not $p$ and not containing
the group $\mu_p$ of $p$th roots of unity.
We consider cyclic $p$algebras over $F$ by descent from
$L = F(\mu_p)$. We generalize a theorem of Albert by
showing that if $\mu_{p^n} \subseteq L$, then a division
algebra $D$ of degree $p^n$ over $F$ is a cyclic
algebra if and only if there is $d\in D$ with $d^{p^n}\in
F  F^p$. Let $F(p)$ be the maximal $p$extension
of $F$. We show that $F(p)$ has a noncyclic algebra
of degree $p$ if and only if a certain eigencomponent of the
$p$torsion of $\Br(F(p)(\mu_p))$ is nontrivial.
To get a better understanding of $F(p)$, we consider
the valuations on $F(p)$ with residue characteristic
not $p$, and determine what residue fields and value
groups can occur. Our results support the conjecture
that the $p$ torsion in $\Br(F(p))$ is always trivial.


673  Hecke $L$Functions and the Distribution of Totally Positive Integers Ash, Avner; Friedberg, Solomon
Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.


696  Algèbres de Lie d'homotopie associées à une protobigèbre de Lie Bangoura, Momo
On associe \`a toute structure de protobig\`ebre de Lie sur un espace
vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie
d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre
ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres,
tous les crochets $n$aires sont nuls pour $n \geq 4$ du fait qu'ils
proviennent d'une structure de protobig\`ebre de Lie. Plus
g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de
l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une
collection d'applications multilin\'eaires antisym\'etriques sur
l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les
identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres
de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour
le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de
$F$ et de~$F^*$.


712  Jet Modules Billig, Yuly
In this paper we classify indecomposable modules for the Lie algebra
of vector fields on a torus that admit a compatible action of the algebra
of functions. An important family of such modules is given by spaces of jets
of tensor fields.


730  Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials Erdélyi, T.; Lubinsky, D. S.
We investigate large sieve inequalities such as
\[
\frac{1}{m}\sum_{j=1}^{m}\psi ( \log  P( e^{i\tau
_{j}})  ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left(
\log [ e P( e^{i\tau })  ] \right) \,d\tau
,
\]
where $\psi $ is convex and increasing, $P$ is a polynomial or an
exponential of a potential, and the constant $C$ depends on the degree of $P$,
and the distribution of the points $0\leq \tau _{1}<\tau _{2}<\dots<\tau
_{m}\leq 2\pi $. The method allows greater generality and is in some ways
simpler than earlier ones. We apply our results to estimate the Mahler
measure of Fekete polynomials.


742  Geometry and Spectra of Closed Extensions of Elliptic Cone Operators Gil, Juan B.; Krainer, Thomas; Mendoza, Gerardo A.
We study the geometry of the set of closed extensions of index $0$ of
an elliptic differential cone operator and its model operator in
connection with the spectra of the extensions, and we give a necessary
and sufficient condition for the existence of rays of minimal growth
for such operators.


795  The ChoquetDeny Equation in a Banach Space Jaworski, Wojciech; Neufang, Matthias
Let $G$ be a locally compact group and $\pi$ a representation of
$G$ by weakly$^*$ continuous isometries acting in a dual Banach space $E$.
Given a
probability measure $\mu$ on $G$, we study the ChoquetDeny equation
$\pi(\mu)x=x$, $x\in E$. We prove that the solutions of this equation
form the range of a projection of norm $1$ and can be represented by means of a
``Poisson formula'' on the same boundary space that is used to represent the
bounded harmonic functions of the random walk of law $\mu$. The relation
between the space of solutions of the ChoquetDeny equation in $E$ and the
space of bounded harmonic functions can be understood in terms of a
construction resembling the $W^*$crossed product and coinciding precisely
with the crossed product in the special case of the ChoquetDeny equation in
the space $E=B(L^2(G))$ of bounded linear operators on $L^2(G)$. Other
general properties of the ChoquetDeny equation in a Banach space are also
discussed.


828  NonBacktracking Random Walks and Cogrowth of Graphs Ortner, Ronald; Woess, Wolfgang
Let $X$ be a locally finite, connected graph without vertices of
degree $1$. Nonbacktracking random walk moves at each step with equal
probability to one of the ``forward'' neighbours of the actual state,
i.e., it does not go back along
the preceding edge to the preceding
state. This is not a Markov chain, but can be turned into a Markov
chain whose state space is the set of oriented edges of $X$. Thus we
obtain for infinite $X$ that the $n$step nonbacktracking transition
probabilities tend to zero, and we can also compute their limit when
$X$ is finite. This provides a short proof of old results concerning
cogrowth of groups, and makes the extension of that result to
arbitrary regular graphs rigorous. Even when $X$ is nonregular, but
small cycles are dense in $X$, we show that the graph $X$ is
nonamenable if and only if the nonbacktracking $n$step transition
probabilities decay exponentially fast. This is a partial
generalization of the cogrowth criterion for regular graphs which
comprises the original cogrowth criterion for finitely generated
groups of Grigorchuk and Cohen.


845  Representations of the Fundamental Group of an $L$Punctured Sphere Generated by Products of Lagrangian Involutions Schaffhauser, Florent
In this paper, we characterize unitary representations of $\pi:=\piS$ whose
generators $u_1, \dots, u_l$ (lying in conjugacy classes fixed initially)
can be decomposed as products of two Lagrangian involutions
$u_j=\s_j\s_{j+1}$ with $\s_{l+1}=\s_1$. Our main result is that such
representations are exactly the elements of the fixedpoint set of an
antisymplectic involution defined on the moduli space
$\Mod:=\Hom_{\mathcal C}(\pi,U(n))/U(n)$. Consequently, as this fixedpoint set is
nonempty, it is a Lagrangian submanifold of $\Mod$. To prove this, we use
the quasiHamiltonian description of the symplectic structure of $\Mod$ and
give conditions on an involution defined on a quasiHamiltonian $U$space
$(M, \w, \mu\from M \to U)$ for it to induce an antisymplectic involution on
the reduced space $M/\!/U := \mu^{1}(\{1\})/U$.


880  Radical Ideals in Valuation Domains van, John E. den
An ideal $I$ of a ring $R$ is called a radical ideal if
$I={\mathcalR}(R)$ where ${\mathcal R}$ is a radical in the sense of
KuroshAmitsur. The main theorem of this paper asserts that if $R$
is a valuation domain, then a proper ideal $I$ of $R$ is a radical
ideal if and only if $I$ is a distinguished ideal of $R$ (the
latter property means that if $J$ and $K$ are ideals of $R$ such
that $J\subset I\subset K$ then we cannot have $I/J\cong K/I$ as
rings) and that such an ideal is necessarily prime. Examples are
exhibited which show that, unlike prime ideals, distinguished
ideals are not characterizable in terms of a property of the
underlying value group of the valuation domain.


897  The Ground State Problem for a Quantum Hamiltonian Model Describing Friction Bruneau, Laurent
In this paper, we consider the quantum version of a Hamiltonian model
describing friction.
This model consists of
a particle which interacts with a bosonic reservoir representing a
homogeneous medium through which the particle moves. We show that if
the particle is confined, then the Hamiltonian admits a ground state
if and only if a suitable infrared condition is satisfied. The latter
is violated in the case of linear friction, but satisfied when the
friction force is proportional to a higher power of the particle
speed.


917  Admissibility for a Class of Quasiregular Representations Currey, Bradley N.
Given a semidirect product $G = N \rtimes H$ where $N$ is%%
nilpotent, connected, simply connected and normal in $G$ and where
$H$ is a vector group for which $\ad(\h)$ is completely reducible and
$\mathbf R$split, let $\tau$ denote the quasiregular representation of
$G$ in $L^2(N)$. An element $\psi \in L^2(N)$ is said to be admissible
if the wavelet transform $f \mapsto \langle f, \tau(\cdot)\psi\rangle$
defines an isometry from $L^2(N)$ into $L^2(G)$. In this paper we give
an explicit construction of admissible vectors in the case where $G$
is not unimodular and the stabilizers in $H$ of its action on $\hat N$
are almost everywhere trivial. In this situation we prove
orthogonality relations and we construct an explicit decomposition of
$L^2(G)$ into $G$invariant, multiplicityfree subspaces each of which
is the image of a wavelet transform . We also show that, with the
assumption of (almosteverywhere) trivial stabilizers,
nonunimodularity is necessary for the existence of admissible
vectors.


943  A Weighted $L^2$Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds Finster, Felix; Kraus, Margarita
We derive a weighted $L^2$estimate of the Witten spinor in
a complete Riemannian spin manifold~$(M^n, g)$ of nonnegative scalar curvature
which is asymptotically Schwarzschild.
The interior geometry of~$M$ enters this estimate only
via the lowest eigenvalue of the square of the Dirac
operator on a conformal compactification of $M$.


966  Operator Amenability of the Fourier Algebra in the $\cb$Multiplier Norm Forrest, Brian E.; Runde, Volker; Spronk, Nico
Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the
closure of $A(G)$, the Fourier algebra of $G$, in the space of completely
bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group
such that $\cstar(G)$ is residually finitedimensional, we show that
$A_{\cb}(G)$ is operator amenable. In particular,
$A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free
group in two generators, is not an amenable group. Moreover, we show that
if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable,
a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$
if and only if it has an approximate identity bounded in the $\cb$multiplier
norm.


981  The ChenRuan Cohomology of Weighted Projective Spaces Jiang, Yunfeng
In this paper we study the ChenRuan cohomology ring of weighted
projective spaces. Given a weighted projective space ${\bf
P}^{n}_{q_{0}, \dots, q_{n}}$, we determine all of its twisted
sectors and the corresponding degree shifting numbers. The main
result of this paper is that the obstruction bundle over any
3\nobreakdashmulti\sector is a direct sum of line bundles which we use to
compute the orbifold cup product. Finally we compute the
ChenRuan cohomology ring of weighted projective space ${\bf
P}^{5}_{1,2,2,3,3,3}$.


1008  Ideas from Zariski Topology in the Study of Cubical Homology Kaczynski, Tomasz; Mrozek, Marian; Trahan, Anik
Cubical sets and their homology have been
used in dynamical systems as well as in digital imaging. We take a
fresh look at this topic, following Zariski ideas from
algebraic geometry. The cubical topology is defined to be a
topology in $\R^d$ in which a set is closed if and only if it is
cubical. This concept is a convenient frame for describing a
variety of important features of cubical sets. Separation axioms
which, in general, are not satisfied here, characterize exactly
those pairs of points which we want to distinguish. The noetherian
property guarantees the correctness of the algorithms. Moreover, maps
between cubical sets which are continuous and closed with respect
to the cubical topology are precisely those for whom the homology
map can be defined and computed without grid subdivisions. A
combinatorial version of the VietorisBegle theorem is derived. This theorem
plays the central role in an algorithm computing homology
of maps which are continuous
with respect to the Euclidean topology.


1029  The Geometry of $L_0$ Kalton, N. J.; Koldobsky, A.; Yaskin, V.; Yaskina, M.
Suppose that we have the unit Euclidean ball in
$\R^n$ and construct new bodies using three operations  linear
transformations, closure in the radial metric, and multiplicative
summation defined by $\x\_{K+_0L} = \sqrt{\x\_K\x\_L}.$ We prove
that in dimension $3$ this procedure gives all originsymmetric convex
bodies, while this is no longer true in dimensions $4$ and higher. We
introduce the concept of embedding of a normed space in $L_0$ that
naturally extends the corresponding properties of $L_p$spaces with
$p\ne0$, and show that the procedure described above gives exactly the
unit balls of subspaces of $L_0$ in every dimension. We provide
Fourier analytic and geometric characterizations of spaces embedding
in $L_0$, and prove several facts confirming the place of $L_0$ in the
scale of $L_p$spaces.


1050  On the Restriction to $\D^* \times \D^*$ of Representations of $p$Adic $\GL_2(\D)$ Raghuram, A.
Let $\mathcal{D}$ be a division algebra
over a nonarchimedean local field. Given
an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we
describe its restriction to the diagonal subgroup $\mathcal{D}^* \times
\mathcal{D}^*$. The description is in terms of the structure of the
twisted Jacquet module of the representation $\pi$. The proof
involves Kirillov theory that we have developed earlier in joint work
with Dipendra Prasad. The main result on restriction also shows that
$\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$distinguished if and only if
$\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$
is a quaternion division algebra then the twisted Jacquet module
is multiplicityfree by proving an appropriate theorem on invariant
distributions; this then proves a multiplicityone theorem on the
restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic
case.


1069  Quotients jacobiens : une approche algébrique Reydy, Carine
Le diagramme d'Eisenbud et Neumann d'un germe est un arbre qui
repr\'esente ce germe et permet d'en calculer les invariants. On donne
une d\'emonstration alg\'ebrique d'un r\'esultat caract\'erisant
l'ensemble des quotients jacobiens d'un germe d'application $(f,g)$
\`a partir du diagramme d'Eisenbud et Neumann de $fg$.


1098  Ruled Exceptional Surfaces and the Poles of Motivic Zeta Functions Rodrigues, B.
In this paper we study ruled surfaces which appear as an exceptional
surface in a succession of blowingups. In particular we prove
that the $e$invariant of such a ruled exceptional surface $E$ is
strictly positive whenever its intersection with the other
exceptional surfaces does not contain a fiber (of $E$). This fact
immediately enables us to resolve an open problem concerning an
intersection configuration on such a ruled exceptional surface
consisting of three nonintersecting sections. In the second part
of the paper we apply the nonvanishing of $e$ to the study of the
poles of the wellknown topological, Hodge and motivic zeta
functions.


1121  Meromorphic Continuation of Spherical Cuspidal Data Eisenstein Series Alayont, Feryâl
Meromorphic continuation of the Eisenstein series induced from spherical,
cuspidal data on parabolic subgroups is achieved via reworking
Bernstein's adaptation of Selberg's third proof of meromorphic
continuation.


1135  Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari
We study when characteristic and H\"older continuous functions
are traces of Sobolev functions on doubling metric measure spaces.
We provide analytic and geometric conditions sufficient for extending
characteristic and H\"older continuous functions into globally defined
Sobolev functions.


1154  $k(n)$TorsionFree $H$Spaces and $P(n)$Cohomology Boardman, J. Michael; Wilson, W. Stephen
The $H$space that represents BrownPeterson cohomology
$\BP^k ()$ was split by the second author into indecomposable
factors, which all have torsionfree homotopy and homology.
Here, we do the same for the related spectrum $P(n)$, by constructing
idempotent operations in $P(n)$cohomology $P(n)^k ($$)$ in the style
of BoardmanJohnsonWilson; this relies heavily on the
RavenelWilson determination of the relevant Hopf ring.
The resulting $(i 1)$connected $H$spaces $Y_i$ have
free connective Morava $\K$homology $k(n)_* (Y_i)$, and may be
built from the spaces in the $\Omega$spectrum for $k(n)$
using only $v_n$torsion invariants.


1207  $H^p$Maximal Regularity and Operator Valued Multipliers on Hardy Spaces Bu, Shangquan; Le, Christian Merdy
We consider maximal regularity in the $H^p$ sense for the Cauchy
problem $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed
operator on a Banach space $X$ and $f$ is an $X$valued function
defined on $\R$. We prove that if $X$ is an AUMD Banach space,
then $A$ satisfies $H^p$maximal regularity if and only if $A$ is
Rademacher sectorial of type $<\frac{\pi}{2}$. Moreover we find an
operator $A$ with $H^p$maximal regularity that does not have the
classical $L^p$maximal regularity. We prove a related Mikhlin
type theorem for operator valued Fourier multipliers on Hardy
spaces $H^p(\R;X)$, in the case when $X$ is an AUMD Banach space.


1223  CalderónZygmund Operators Associated to Ultraspherical Expansions Buraczewski, Dariusz; Martinez, Teresa; Torrea, José L.
We define the higher order Riesz transforms and the LittlewoodPaley
$g$function
associated to the differential operator $L_\l f(\theta)=f''(\theta)2\l\cot\theta
f'(\theta)+\l^2f(\theta)$. We prove that these operators are
Calder\'{o}nZygmund operators in the homogeneous type space
$((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted,
$H^1L^1$ and $L^\inftyBMO$ inequalities are obtained.


1245  On Gap Properties and Instabilities of $p$YangMills Fields Chen, Qun; Zhou, ZhenRong
We consider the
$p$YangMills functional
$(p\geq 2)$
defined as
$\YM_p(\nabla):=\frac 1 p \int_M \\rn\^p$.
We call critical points of $\YM_p(\cdot)$ the $p$YangMills
connections, and the associated curvature $\rn$ the $p$YangMills
fields. In this paper, we prove gap properties and instability theorems for $p$YangMills
fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$.


1260  Generic Extensions and Canonical Bases for Cyclic Quivers Deng, Bangming; Du, Jie; Xiao, Jie
We use the monomial basis theory developed by Deng and Du to
present an elementary algebraic construction of the canonical
bases for both the RingelHall algebra of a cyclic quiver and the
positive part $\bU^+$ of the quantum affine $\frak{sl}_n$. This
construction relies on analysis of quiver representations and the
introduction of a new integral PBWlike basis for the Lusztig
$\mathbb Z[v,v^{1}]$form of~$\bU^+$.


1284  On Effective Witt Decomposition and the CartanDieudonn{é Theorem Fukshansky, Lenny
Let $K$ be a number field, and let $F$ be a symmetric bilinear form in
$2N$ variables over $K$. Let $Z$ be a subspace of $K^N$. A classical
theorem of Witt states that the bilinear space $(Z,F)$ can be
decomposed into an orthogonal sum of hyperbolic planes and singular and
anisotropic components. We prove the existence of such a decomposition
of small height, where all bounds on height are explicit in terms of
heights of $F$ and $Z$. We also prove a special version of Siegel's
lemma for a bilinear space, which provides a smallheight orthogonal
decomposition into onedimensional subspaces. Finally, we prove an
effective version of the CartanDieudonn{\'e} theorem. Namely, we show
that every isometry $\sigma$ of a regular bilinear space $(Z,F)$ can
be represented as a product of reflections of bounded heights with an
explicit bound on heights in terms of heights of $F$, $Z$, and
$\sigma$.


1301  Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group Furioli, Giulia; Melzi, Camillo; Veneruso, Alessandro
We prove dispersive and Strichartz inequalities for the solution of the wave
equation related to the full
Laplacian on the Heisenberg group, by means of Besov spaces defined by a
LittlewoodPaley
decomposition related to the spectral resolution of the full Laplacian.
This requires a careful
analysis due also to the nonhomogeneous nature of the full Laplacian.
This result has to be compared to a previous one by Bahouri, G\'erard
and Xu concerning the solution of the wave equation related to
the Kohn Laplacian.


1323  On a Conjecture of Jacquet, Lai, and Rallis: Some Exceptional Cases Ginzburg, David; Lapid, Erez
We prove two spectral identities. The first one relates the relative
trace formula for the spherical variety $\GSpin(4,3)/G_2$ with a
weighted trace formula for $\GL_2$. The second relates a spherical
variety pertaining to $F_4$ to one of $\GSp(6)$. These identities are
in accordance with a conjecture made by Jacquet, Lai, and Rallis,
and are obtained without an appeal to a geometric comparison.


1341  Author Index  Index des auteurs 2007, for 2007  pour
No abstract.

