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1201  Chern Classes of Splayed Intersections Aluffi, Paolo; Faber, Eleonore
We generalize the Chern class relation for the transversal intersection
of two nonsingular
varieties to a relation for possibly singular varieties, under
a splayedness assumption.
We show that the relation for the ChernSchwartzMacPherson classes
holds for two splayed hypersurfaces in a nonsingular variety,
and under a `strong splayedness' assumption for more
general subschemes. Moreover, the relation is shown to hold for
the ChernFulton classes
of any two splayed subschemes.
The main tool is a formula for Segre classes of splayed
subschemes. We also discuss the Chern class relation under the
assumption that one of the
varieties is a general very ample divisor.


1219  $p$adic and Motivic Measure on Artin $n$stacks Balwe, Chetan
We define a notion of $p$adic measure on Artin $n$stacks which are
of strongly finite type over the ring of $p$adic integers. $p$adic
measure on schemes can be evaluated by counting points on the
reduction of the scheme modulo $p^n$. We show that an analogous
construction works in the case of Artin stacks as well if we count the
points using the counting measure defined by Toën. As a consequence,
we obtain the result that the Poincaré and Serre series of such
stacks are rational functions, thus extending Denef's result for
varieties. Finally, using motivic integration we show that as $p$
varies, the rationality of the Serre series of an Artin stack defined
over the integers is uniform with respect to $p$.


1247  Lyapunov Stability and Attraction Under Equivariant Maps Barros, Carlos Braga; Rocha, Victor; Souza, Josiney
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed
with
admissible families of open coverings. Assume that $\mathcal{S}$ is a
semigroup acting on both $M$ and $N$. In this paper we study the behavior of
limit sets, prolongations, prolongational limit sets, attracting sets,
attractors and Lyapunov stable sets (all concepts defined for the action of
the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations
from $M$ to $N$.


1270  Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems Carcamo, Cristian; Vidal, Claudio
In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.


1290  On Twofaced Families of Noncommutative Random Variables Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul
We demonstrate that the notions of bifree independence and combinatorialbifree
independence of twofaced families are equivalent using a diagrammatic
view of binoncrossing partitions.
These diagrams produce an operator model on a Fock space suitable
for representing any twofaced family of noncommutative random
variables.
Furthermore, using a Kreweras complement on binoncrossing partitions
we establish the expected formulas for the multiplicative convolution
of a bifree pair of twofaced families.


1326  The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank Cojocaru, Alina Carmen; Shulman, Andrew Michael
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study
the first elementary divisor
$d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies.
In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$
and prove that, on average, it has a large norm. Our work is motivated by the study of J.P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.


1358  On the Rate of Convergence of Empirical Measures in $\infty$transportation Distance Garcia Trillos, Nicolas; Slepcev, Dejan
We consider random i.i.d. samples of absolutely continuous measures
on bounded connected domains.
We prove an upper bound on the $\infty$transportation distance
between the measure and the empirical measure of the sample.
The bound is optimal in terms of scaling with the number of sample
points.


1384  Strong Logarithmic Sobolev Inequalities for LogSubharmonic Functions Graczyk, Piotr; Kemp, Todd; Loeb, JeanJacques
We prove an intrinsic equivalence between strong
hypercontractivity and a strong logarithmic Sobolev
inequality for the cone of logarithmically subharmonic
(LSH) functions. We introduce a new large class of measures,
Euclidean regular and exponential type, in addition to all compactlysupported
measures, for which this equivalence holds. We prove a Sobolev
density theorem through LSH functions and use it to prove
the equivalence of strong
hypercontractivity and the strong logarithmic Sobolev
inequality for such logsubharmonic
functions.


1411  Functiontheoretic Properties for the Gauss Maps of Various Classes of Surfaces Kawakami, Yu
We elucidate the geometric background of functiontheoretic properties
for the Gauss maps of
several classes of immersed surfaces in threedimensional space
forms, for example, minimal surfaces in Euclidean threespace, improper affine spheres in the affine threespace, and constant
mean curvature one surfaces and flat surfaces in hyperbolic threespace. To achieve this purpose, we prove an optimal curvature bound
for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for
the Gauss maps of these classes of surfaces.

