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1201  Application of the Strong Artin Conjecture to the Class Number Problem Cho, Peter J.; Kim, Henry H.
We construct unconditionally several families of number fields with
the largest possible class numbers. They are number fields of degree 4
and 5 whose Galois closures have the Galois group $A_4, S_4$ and
$S_5$. We first construct families of number fields with smallest
regulators, and by using the strong Artin conjecture and applying zero
density result of KowalskiMichel, we choose subfamilies of
$L$functions which are zero free close to 1.
For these subfamilies, the $L$functions have the extremal value at
$s=1$, and by the class number formula, we obtain the extreme class
numbers.


1217  Beltrami Equation with Coefficient in Sobolev and Besov Spaces Cruz, Victor; Mateu, Joan; Orobitg, Joan
Our goal in this work is to present some function spaces on the
complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of
the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f
(z)$, have first derivatives locally in $X(\mathbb C)$, provided that the
Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.


1236  Higher Connectedness Properties of Support Points and Functionals of Convex Sets De Bernardi, Carlo Alberto
We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinitedimensional Banach space $X$ is $\mathrm{AR(}\sigma$$\mathrm{compact)}$ and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph and the range of the subdifferential map of a proper convex l.s.c. function on $X$.


1255  Variations of Integrals in Diffeology IglesiasZemmour, Patrick
We establish the formula for the variation of
integrals of differential forms on cubic chains, in the
context of diffeological spaces. Then, we establish the diffeological version of Stoke's
theorem, and we apply that to get the diffeological variant of the
CartanLie formula. Still in the context of CartanDeRham calculus
in diffeology, we
construct a ChainHomotopy Operator $\mathbf K$ we apply it here to
get the homotopic invariance of De Rham cohomology for
diffeological spaces. This is the ChainHomotopy Operator which used in
symplectic diffeology to construct the Moment Map.


1287  $K$theory of Furstenberg Transformation Group $C^*$algebras Reihani, Kamran
The paper studies the $K$theoretic invariants of the crossed product
$C^{*}$algebras associated with an important family of homeomorphisms
of the tori $\mathbb{T}^{n}$ called Furstenberg transformations.
Using the PimsnerVoiculescu theorem, we prove that given $n$, the
$K$groups of those crossed products, whose corresponding $n\times n$
integer matrices are unipotent of maximal degree, always have the same
rank $a_{n}$. We show using the theory developed here that a claim
made in the literature about the torsion subgroups of these $K$groups
is false. Using the representation theory of the simple Lie algebra
$\frak{sl}(2,\mathbb{C})$, we show that, remarkably, $a_{n}$ has a
combinatorial significance. For example, every $a_{2n+1}$ is just the
number of ways that $0$ can be represented as a sum of integers
between $n$ and $n$ (with no repetitions). By adapting an argument
of van Lint (in which he answered a question of Erdős), a simple,
explicit formula for the asymptotic behavior of the sequence
$\{a_{n}\}$ is given. Finally, we describe the order structure of the
$K_{0}$groups of an important class of Furstenberg crossed products,
obtaining their complete Elliott invariant using classification
results of H. Lin and N. C. Phillips.


1320  Orbital $L$functions for the Space of Binary Cubic Forms Taniguchi, Takashi; Thorne, Frank
We introduce the notion of orbital $L$functions
for the space of binary cubic forms
and investigate their analytic properties.
We study their functional equations and residue formulas in some detail.
Aside from their intrinsic interest,
the results from this paper are used to
prove the existence of secondary terms in counting
functions for cubic fields.
This is worked out in a companion paper.


1384  Estimates of Hausdorff Dimension for Nonwandering Sets of Higher Dimensional Open Billiards Wright, Paul
This article concerns a class of open billiards consisting of a finite
number of strictly convex, noneclipsing obstacles $K$. The
nonwandering set $M_0$ of the billiard ball map is a topological
Cantor set and its Hausdorff dimension has been previously estimated
for billiards in $\mathbb{R}^2$, using wellknown techniques. We
extend these estimates to billiards in $\mathbb{R}^n$, and make
various refinements to the estimates. These refinements also allow
improvements to other results. We also show that in many cases, the
nonwandering set is confined to a particular subset of $\mathbb{R}^n$
formed by the convex hull of points determined by period 2
orbits. This allows more accurate bounds on the constants used in
estimating Hausdorff dimension.


1401  A Universal Volume Comparison Theorem for Finsler Manifolds and Related Results Zhao, Wei; Shen, Yibing
In this paper, we establish a universal volume comparison theorem
for Finsler manifolds and give the BergerKazdan inequality and
Santaló's formula in Finsler geometry. Being based on these, we
derive a BergerKazdan type comparison theorem and a Croke type
isoperimetric inequality for Finsler manifolds.

