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961  Densities of Short Uniform Random Walks Borwein, Jonathan M.; Straub, Armin; Wan, James; Zudilin, Wadim
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic
to expect similar results for more than five steps. New results are also
presented concerning the moments of uniform random walks and, in particular,
their derivatives. Relations with Mahler measures are discussed.


991  Poisson Brackets with Prescribed Casimirs Damianou, Pantelis A.; Petalidou, Fani
We consider the problem of constructing Poisson brackets on smooth
manifolds $M$ with prescribed Casimir functions. If $M$ is of even
dimension, we achieve our construction by considering a suitable
almost symplectic structure on $M$, while, in the case where $M$ is
of odd dimension, our objective is achieved by using a convenient
almost cosymplectic structure. Several examples and applications are
presented.


1019  On a Theorem of Bombieri, Friedlander, and Iwaniec Fiorilli, Daniel
In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a BombieriVinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.


1036  Harmonic Analysis Related to Homogeneous Varieties in Three Dimensional Vector Spaces over Finite Fields Koh, Doowon; Shen, ChunYen
In this paper we study the extension problem, the
averaging problem, and the generalized ErdősFalconer distance
problem associated with arbitrary homogeneous varieties in three
dimensional vector spaces over finite fields. In the case when the
varieties do not contain any plane passing through the origin, we
obtain the best possible results on the aforementioned three problems. In
particular, our result on the extension problem modestly generalizes
the result by Mockenhaupt and Tao who studied the particular conical
extension problem. In addition, investigating the Fourier decay on
homogeneous varieties enables us to give complete mapping properties
of averaging operators. Moreover, we improve the size condition on a
set such that the cardinality of its distance set is nontrivial.


1058  Optimal Roughening of Convex Bodies Plakhov, Alexander
A body moves in a rarefied medium composed of point particles at
rest. The particles make elastic reflections when colliding with the
body surface, and do not interact with each other. We consider a
generalization of Newton's minimal resistance problem: given two
bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2
\subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the
resistance in the class of connected bodies $B$ such that $C_1 \subset
B \subset C_2$. We prove that the infimum of resistance is zero; that
is, there exist "almost perfectly streamlined" bodies.


1075  A Stochastic Difference Equation with Stationary Noise on Groups Raja, Chandiraraj Robinson Edward
We consider the stochastic difference equation $$\eta _k = \xi _k
\phi (\eta _{k1}), \quad k \in \mathbb Z $$ on a locally compact group $G$
where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$valued
random variables and $\eta _k$ are unknown $G$valued random variables.
This equation was considered by Tsirelson and Yor on
onedimensional torus. We consider the case when $\xi _k$ have a
common law $\mu$ and prove that if $G$ is a distal group and $\phi$
is a distal automorphism of $G$ and if the equation has a solution,
then extremal solutions of the equation are in oneone
correspondence with points on the coset space $K\backslash G$ for
some compact subgroup $K$ of $G$ such that $\mu$ is supported on
$Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide
a necessary and sufficient condition for the existence of solutions
to the equation.


1090  Classic and Mirabolic RobinsonSchenstedKnuth Correspondence for Partial Flags Rosso, Daniele
In this paper we first generalize to the case of
partial flags a result proved both by Spaltenstein and by Steinberg
that relates the relative position of two complete flags and the
irreducible components of the flag variety in which they lie, using
the RobinsonSchenstedKnuth correspondence. Then we use this result
to generalize the mirabolic RobinsonSchenstedKnuth correspondence
defined by Travkin, to the case of two partial flags and a line.


1122  $p$adic $L$functions and the Rationality of Darmon Cycles Seveso, Marco Adamo
Darmon cycles are a higher weight analogue of StarkHeegner points. They
yield local cohomology classes in the Deligne representation associated with a
cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$.
They are conjectured to be the restriction of global cohomology classes in
the BlochKato Selmer group defined over narrow ring class fields attached
to a real quadratic field. We show that suitable linear combinations of them
obtained by genus characters satisfy these conjectures. We also prove $p$adic GrossZagier type formulas, relating the derivatives of $p$adic $L$functions of the weight variable attached to imaginary (resp. real)
quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express
the second derivative of the MazurKitagawa $p$adic $L$function of the
weight variable in terms of a global cycle defined over a quadratic
extension of $\mathbb{Q}$.


1182  PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$ Tall, Franklin D.
Extending the work of Larson and Todorcevic,
we show there
is a model of set theory in which normal spaces are collectionwise
Hausdorff if they are either first countable or locally compact, and
yet there are no first countable $L$spaces or compact
$S$spaces. The model is one of the form PFA$(S)[S]$, where $S$
is a coherent Souslin tree.
