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481  Uniform Distribution of Fractional Parts Related to Pseudoprimes Banks, William D.; Garaev, Moubariz Z.; Luca, Florian; Shparlinski, Igor E.
We estimate exponential sums with the Fermatlike quotients
$$
f_g(n) = \frac{g^{n1}  1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n1}1}{P(n)},
$$
where $g$ and $n$ are positive integers, $n$ is composite, and
$P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$
and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base
$g$, and if $n$ is a Carmichael number, this is true for all $g$
coprime to $n$. Nevertheless, our bounds imply that the fractional
parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on
average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We
also obtain similar results with the functions ${\widetilde f}_g(n)
= gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.


503  Subspaces of de~Branges Spaces Generated by Majorants Baranov, Anton; Woracek, Harald
For a given de~Branges space $\mc H(E)$ we investigate
de~Branges subspaces defined in terms of majorants
on the real axis. If $\omega$ is a nonnegative function on $\mathbb R$,
we consider the subspace
\[
\mc R_\omega(E)=\clos_{\mc H(E)} \big\{F\in\mc H(E):
\text{ there exists } C>0:
E^{1} F\leq C\omega \mbox{ on }{\mathbb R}\big\}
.
\]
We show that $\mc R_\omega(E)$ is a de~Branges subspace and
describe all subspaces of this form. Moreover,
we give a criterion for the existence of positive minimal majorants.


518  Global Units Modulo Circular Units: Descent Without Iwasawa's Main Conjecture Belliard, JeanRobert
Iwasawa's classical asymptotical formula relates the orders of the $p$parts $X_n$ of the ideal
class groups along a $\mathbb{Z}_p$extension $F_\infty/F$ of a number
field $F$ to Iwasawa structural invariants $\la$ and $\mu$
attached to the inverse limit $X_\infty=\varprojlim X_n$.
It relies on ``good" descent properties satisfied by
$X_n$. If $F$ is abelian and $F_\infty$ is cyclotomic, it is known
that the $p$parts of the orders of the global units modulo
circular units $U_n/C_n$ are asymptotically equivalent to the
$p$parts of the ideal class numbers. This suggests that these
quotients $U_n/C_n$, so to speak unit class groups, also satisfy
good descent properties. We show this directly, i.e., without using Iwasawa's Main Conjecture.


534  Girsanov Transformations for NonSymmetric Diffusions Chen, ChuanZhong; Sun, Wei
Let $X$ be a diffusion process, which is assumed to be
associated with a (nonsymmetric) strongly local Dirichlet form
$(\mathcal{E},\mathcal{D}(\mathcal{E}))$ on $L^2(E;m)$. For
$u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichlet
space, we investigate some properties of the Girsanov transformed
process $Y$ of $X$. First, let $\widehat{X}$ be the dual process of
$X$ and $\widehat{Y}$ the Girsanov transformed process of $\widehat{X}$.
We give a necessary and sufficient condition for $(Y,\widehat{Y})$ to
be in duality with respect to the measure $e^{2u}m$. We also
construct a counterexample, which shows that this condition may
not be satisfied and hence $(Y,\widehat{Y})$ may not be dual
processes. Then we present a sufficient condition under which $Y$
is associated with a semiDirichlet form. Moreover, we give an
explicit representation of the semiDirichlet form.


548  Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces Girouard, Alexandre
We study the effect of two types of degeneration of a Riemannian
metric on the first eigenvalue of the Laplace operator on
surfaces. In both cases we prove that the first eigenvalue of the
round sphere is an optimal asymptotic upper bound. The first type of
degeneration is concentration of the density to a point within a
conformal class. The second is degeneration of the
conformal class to the boundary of the moduli space on the torus and
on the Klein bottle. In the latter, we follow the outline proposed
by N. Nadirashvili in 1996.


566  Convex Subordination Chains in Several Complex Variables Graham, Ian; Hamada, Hidetaka; Kohr, Gabriela; Pfaltzgraff, John A.
In this paper we study the notion of a convex subordination chain in several
complex variables. We obtain certain necessary and sufficient conditions for a
mapping to be a convex subordination chain, and we give various examples of
convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We
also obtain a sufficient condition for injectivity of $f(z/\z\,\z\)$
on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain
over $(0,1)$.


583  Algebraic Properties of a Family of Generalized Laguerre Polynomials Hajir, Farshid
We study the algebraic properties of Generalized Laguerre Polynomials
for negative integral values of the parameter. For integers $r,n\geq
0$, we conjecture that $L_n^{(1nr)}(x) = \sum_{j=0}^n
\binom{nj+r}{nj}x^j/j!$ is a $\Q$irreducible polynomial whose
Galois group contains the alternating group on $n$ letters. That this
is so for $r=n$ was conjectured in the 1950's by Grosswald and proven
recently by Filaseta and Trifonov. It follows from recent work of
Hajir and Wong that the conjecture is true when $r$ is large with
respect to $n\geq 5$. Here we verify it in three situations: i) when
$n$ is large with respect to $r$, ii) when $r \leq 8$, and iii) when
$n\leq 4$. The main tool is the theory of $p$adic Newton Polygons.


604  First Countable Continua and Proper Forcing Hart, Joan E.; Kunen, Kenneth
Assuming the Continuum Hypothesis,
there is a compact, first countable, connected space of weight $\aleph_1$
with no totally disconnected perfect subsets.
Each such space, however, may be destroyed by
some proper forcing order which does not add reals.


617  Square Integrable Representations and the Standard Module Conjecture for General Spin Groups Kim, Wook
In this paper we study square integrable representations and
$L$functions for quasisplit general spin groups over a $p$adic
field. In the first part, the holomorphy of $L$functions in a half
plane is proved by using a variant form of Casselman's square
integrability criterion and the LanglandsShahidi method. The
remaining part focuses on the proof of the standard module
conjecture. We generalize Mui\'c's idea via the LanglandsShahidi method
towards a proof of the conjecture. It is used in the work of M. Asgari
and F. Shahidi on generic transfer for general spin groups.


641  Characterization of Parallel Isometric Immersions of Space Forms into Space Forms in the Class of Isotropic Immersions Maeda, Sadahiro; Udagawa, Seiichi
For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form
$\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we
show that if the mean curvature vector of $M^n$ is parallel and the
sectional curvature $K$ of $M^n$ satisfies some inequality, then
the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is
parallel and our manifold $M^n$ is a space form.


656  Generalized Polynomials and Mild Mixing McCutcheon, Randall; Quas, Anthony
An unsettled conjecture of V. Bergelson and I. H\aa land proposes that
if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving
system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ are
generalized polynomials (functions built out of regular polynomials
via iterated use of the greatest integer or floor function) having the
property that no $p_i$, nor any $p_ip_j$, $i\neq j$, is constant on a
set of positive density, then for any measurable sets
$A_0,A_1,\dots
,A_k$, there exists a zerodensity set $E\subset \z$ such that
\[\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots
\cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).\] We formulate and prove a
faithful version of this conjecture for mildly mixing systems and
partially characterize, in the degree two case, the set of families
$\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem.


674  A Construction of Rigid Analytic Cohomology Classes for Congruence Subgroups of $\SL_3(\mathbb Z)$ Pollack, David; Pollack, Robert
We give a constructive proof, in the special case of ${\rm GL}_3$, of
a theorem of Ash and Stevens which compares overconvergent cohomology
to classical cohomology. Namely, we show that every ordinary
classical Heckeeigenclass can be lifted uniquely to a rigid analytic
eigenclass. Our basic method builds on the ideas of M. Greenberg; we
first form an arbitrary lift of the classical eigenclass to a
distributionvalued cochain. Then, by appropriately iterating the
$U_p$operator, we produce a cocycle whose image in cohomology is the
desired eigenclass. The constructive nature of this proof makes it
possible to perform computer computations to approximate these
interesting overconvergent eigenclasses.


691  Prehomogeneity on QuasiSplit Classical Groups and Poles of Intertwining Operators Yu, Xiaoxiang
Suppose that $P=MN$ is a maximal parabolic subgroup of a quasisplit,
connected, reductive classical group $G$ defined over a nonArchimedean
field and $A$ is the standard intertwining operator attached to a
tempered representation of $G$ induced from $M$. In this paper we
determine all the cases in which $\Lie(N)$ is
prehomogeneous under $\Ad(m)$ when $N$ is nonabelian, and give necessary
and sufficient conditions for $A$ to have a pole at $0$.


708  Regular Homeomorphisms of Finite Order on Countable Spaces Zelenyuk, Yevhen
We present a structure theorem for a broad class of homeomorphisms of
finite order on countable zero dimensional spaces. As applications we
show the following.
\begin{compactenum}[\rm(a)]
\item Every countable nondiscrete topological group not containing an
open Boolean subgroup can be partitioned into infinitely many dense
subsets.

