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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.

