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3  How the Roots of a Polynomial Vary with Its Coefficients: A Local Quantitative Result Beauzamy, Bernard
A wellknown result, due to Ostrowski, states that if $\Vert PQ
\Vert_2< \varepsilon$, then the roots $(x_j)$ of $P$ and $(y_j)$ of
$Q$ satisfy $x_j y_j\le C n \varepsilon^{1/n}$, where $n$ is the
degree of $P$ and $Q$. Though there are cases where this estimate
is sharp, it can still be made more precise in general, in two
ways: first by using Bombieri's norm instead of the classical $l_1$
or $l_2$ norms, and second by taking into account the multiplicity
of each root. For instance, if $x$ is a simple root of $P$, we show
that $xy< C \varepsilon$ instead of $\varepsilon^{1/n}$. The
proof uses the properties of Bombieri's scalar product and Walsh
Contraction Principle.


13  Dow's Principle and $Q$Sets Brendle, Jörg
A $Q$set is a set of reals every subset of which is a relative
$G_\delta$. We investigate the combinatorics of $Q$sets and
discuss a question of Miller and Zhou on the size $\qq$ of the smallest
set of reals which is not a $Q$set. We show in particular that various
natural lower bounds for $\qq$ are consistently strictly smaller than
$\qq$.


25  On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.
Analogues of van der Waerden's theorem on arithmetic progressions
are considered where the family of all arithmetic progressions,
$\AP$, is replaced by some subfamily of $\AP$. Specifically, we
want to know for which sets $A$, of positive integers, the
following statement holds: for all positive integers $r$ and $k$,
there exists a positive integer $n= w'(k,r)$ such that for every
$r$coloring of $[1,n]$ there exists a monochromatic $k$term
arithmetic progression whose common difference belongs to $A$. We
will call any subset of the positive integers that has the above
property {\em large}. A set having this property for a specific
fixed $r$ will be called {\em $r$large}. We give some necessary
conditions for a set to be large, including the fact that every
large set must contain an infinite number of multiples of each
positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can
have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$.
Sufficient conditions for a set to be large are also given. We
show that any set containing $n$cubes for arbitrarily large $n$,
is a large set. Results involving the connection between the
notions of ``large'' and ``2large'' are given. Several open
questions and a conjecture are presented.


37  Operators with Closed Range, PseudoInverses, and Perturbation of Frames for a Subspace Christensen, Ole
Recent work of Ding and Huang shows that if we perturb a bounded
operator (between Hilbert spaces) which has closed range, then the
perturbed operator again has closed range. We extend this result by
introducing a weaker perturbation condition, and our result is then
used to prove a theorem about the stability of frames for a subspace.


46  Generic Partial TwoPoint Sets Are Extendable Dijkstra, Jan J.
It is shown that under $\ZFC$ almost all planar compacta that meet
every line in at most two points are subsets of sets that meet every
line in exactly two points. This result was previously obtained by the
author jointly with K.~Kunen and J.~van~Mill under the assumption that
Martin's Axiom is valid.


52  Embedding Coverings in Bundles Edmonds, Allan L.
If $V\to X$ is a vector bundle of fiber dimension $k$ and $Y\to X$
is a finite sheeted covering map of degree $d$, the implications
for the Euler class $e(V)$ in $H^k(X)$ of $V$ implied by the
existence of an embedding $Y\to V$ lifting the covering map are
explored. In particular it is proved that $dd'e(V)=0$ where $d'$
is a certain divisor of $d1$, and often $d'=1$.


56  On the Square of the First Zero of the Bessel Function $J_\nu(z)$ Elbert, Árpád; Siafarikas, Panayiotis D.
Let $j_{\nu,1}$ be the smallest (first) positive zero of the Bessel
function $J_{\nu}(z)$, $\nu>1$, which becomes zero when $\nu$
approaches $1$. Then $j_{\nu,1}^{2}$ can be continued
analytically to $2<\nu<1$, where it takes on negative values. We
show that $j_{\nu,1}^{2}$ is a convex function of $\nu$ in the
interval $2<\nu\leq 0$, as an addition to an old result
[\'A.~Elbert and A.~Laforgia, SIAM J. Math. Anal. {\bf 15}(1984),
206212], stating this convexity for $\nu>0$. Also the
monotonicity properties of the functions $\frac{j_{\nu,1}^{2}}{4
(\nu+1)}$, $\frac{j_{\nu,1}^{2}}{4(\nu+1)\sqrt{\nu+2}}$ are
determined. Our approach is based on the series expansion of
Bessel function $J_{\nu}(z)$ and it turned out to be effective,
especially when $2<\nu<1$.


68  The Moments of the SumOfDigits Function in Number Fields Gittenberger, Bernhard; Thuswaldner, Jörg M.
We consider the asymptotic behavior of the moments of the sumofdigits
function of canonical number systems in number fields. Using Delange's
method we obtain the main term and smaller order terms which contain
periodic fluctuations.


78  Fermat Jacobians of Prime Degree over Finite Fields González, Josep
We study the splitting of Fermat Jacobians of prime
degree $\ell$ over an algebraic closure of a finite field of
characteristic $p$ not equal to $\ell$. We prove that their
decomposition is determined by the residue degree of $p$ in the
cyclotomic field of the $\ell$th roots of unity. We provide a
numerical criterion that allows to compute the absolutely simple
subvarieties and their multiplicity in the Fermat Jacobian.


87  Some norm inequalities for operators Kittaneh, Fuad
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a
separable Hilbert space. It is shown that if $f$ and $g$ are
nonnegative continuous functions on $[0,\infty)$ which satisfy the
relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then
$$
\Biglvert \,\Bigl\sum^n_{i=1} A^*_i X_i B_i \Bigr^r \,\Bigrvert^2
\leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (X^*_i)^2 A_i \Bigr)^r
\Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (X_i)^2 B_i
\Bigr)^r \Bigrvert
$$
for every $r>0$ and for every unitarily invariant norm. This result
improves some known CauchySchwarz type inequalities. Norm
inequalities related to the arithmeticgeometric mean inequality and
the classical Heinz inequalities are also obtained.


97  On Analytic Functions of Bergman $\BMO$ in the Ball Kwon, E. G.
Let $B = B_n$ be the open unit ball of $\bbd C^n$ with
volume measure $\nu$, $U = B_1$ and ${\cal B}$ be the Bloch space on
$U$. ${\cal A}^{2, \alpha} (B)$, $1 \leq \alpha < \infty$, is defined
as the set of holomorphic $f\colon B \rightarrow \bbd C$ for which
$$
\int_B \vert f(z) \vert^2 \left( \frac 1{\vert z\vert}
\log \frac 1{1  \vert z\vert } \right)^{\alpha}
\frac {d\nu (z)}{1\vert z\vert} < \infty
$$
if $0 < \alpha <\infty$ and ${\cal A}^{2, 1} (B) = H^2(B)$, the Hardy
space. Our objective of this note is to characterize, in terms of
the Bergman distance, those holomorphic $f\colon B \rightarrow U$ for
which the composition operator $C_f \colon {\cal B} \rightarrow
{\cal A}^{2, \alpha}(B)$ defined by $C_f (g) = g\circ f$,
$g \in {\cal B}$, is bounded. Our result has a corollary that
characterize the set of analytic functions of bounded mean
oscillation with respect to the Bergman metric.


104  Instabilité de vecteurs propres d'opérateurs linéaires Nikolskaia, Ludmila
We consider some geometric properties of eigenvectors of linear
operators on infinite dimensional Hilbert space. It is proved that
the property of a family of vectors $(x_n)$ to be eigenvectors
$Tx_n= \lambda_n x_n$ ($\lambda_n \noteq \lambda_k$ for $n\noteq k$)
of a bounded operator $T$ (admissibility property) is very instable
with respect to additive and linear perturbations. For instance,
(1)~for the sequence $(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$ to
be admissible for every admissible $(x_n)$ and for a suitable
choice of small numbers $\epsilon_n\noteq 0$ it is necessary and
sufficient that the perturbation sequence be eventually scalar:
there exist $\gamma_n\in \C$ such that $v_n= \gamma_n v_{k}$ for
$n\geq k$ (Theorem~2); (2)~for a bounded operator $A$ to transform
admissible families $(x_n)$ into admissible families $(Ax_n)$ it is
necessary and sufficient that $A$ be left invertible (Theorem~4).


118  Points of Weak$^\ast$Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$ Rao, T. S. S. R. K.
For a compact Hausdorff space with a dense set of isolated points, we
give a complete description of points of weak$^\ast$norm continuity
in the dual unit ball of the space of Banach space valued functions
that are continuous when the range has the weak topology. As an
application we give a complete description of points of weaknorm
continuity of the unit ball of the space of vector measures when
the underlying Banach space has the RadonNikodym property.


125  Modular Vector Invariants of Cyclic Permutation Representations Smith, Larry
Vector invariants of finite groups (see the introduction for an
explanation of the terminology) have often been used to illustrate the
difficulties of invariant theory in the modular case: see,
\eg., \cite{Ber}, \cite{norway}, \cite{fossum}, \cite{MmeB},
\cite{poly} and \cite{survey}. It is therefore all the more
surprising that the {\it unpleasant} properties of these invariants
may be derived from two unexpected, and remarkable, {\it nice}
properties: namely for vector permutation invariants of the cyclic
group $\mathbb{Z}/p$ of prime order in characteristic $p$ the
image of the transfer homomorphism $\Tr^{\mathbb{Z}/p} \colon
\mathbb{F}[V] \lra \mathbb{F}[V]^{\mathbb{Z}/p}$ is a prime ideal,
and the quotient algebra $\mathbb{F}[V]^{\mathbb{Z}/p}/ \Im
(\Tr^{\mathbb{Z}/p})$ is a polynomial algebra on the top Chern
classes of the action.


129  Hecke Operations and the Adams $E_2$Term Based on Elliptic Cohomology Baker, Andrew
Hecke operators are used to investigate part of the $\E_2$term of
the Adams spectral sequence based on elliptic homology. The main
result is a derivation of $\Ext^1$ which combines use of classical
Hecke operators and $p$adic Hecke operators due to Serre.


139  Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions Bonet, José; Domański, Paweł; Lindström, Mikael
Every weakly compact composition operator between weighted Banach
spaces $H_v^{\infty}$ of analytic functions with weighted supnorms is
compact. Lower and upper estimates of the essential norm of
continuous composition operators are obtained. The norms of the point
evaluation functionals on the Banach space $H_v^{\infty}$ are also
estimated, thus permitting to get new characterizations of compact
composition operators between these spaces.


149  A Note on Finite Dehn Fillings Boyer, S.; Zhang, X.
Let $M$ be a compact, connected, orientable 3manifold whose
boundary is a torus and whose interior admits a complete hyperbolic
metric of finite volume. In this paper we show that if the minimal
CullerShalen norm of a nonzero class in $H_1(\partial M)$ is
larger than $8$, then the finite surgery conjecture holds for $M$.
This means that there are at most $5$ Dehn fillings of $M$ which
can yield manifolds having cyclic or finite fundamental groups and
the distance between any slopes yielding such manifolds is at most
$3$.


155  NonCohenMacaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants Campbell, H. E. A.; Geramita, A. V.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L.
This paper contains two essentially independent results in the
invariant theory of finite groups. First we prove that, for any
faithful representation of a nontrivial $p$group over a field of
characteristic $p$, the ring of vector invariants of $m$ copies of
that representation is not \comac\ for $m\geq 3$. In the second
section of the paper we use Poincar\'e series methods to produce upper
bounds for the degrees of the generators for the ring of invariants as
long as that ring is Gorenstein. We prove that, for a finite
nontrivial group $G$ and a faithful representation of dimension $n$
with $n>1$, if the ring of invariants is Gorenstein then the ring is
generated in degrees less than or equal to $n(G1)$. If the ring of
invariants is a hypersurface, the upper bound can be improved to $G$.


162  LorentzSchatten Classes and Pointwise Domination of Matrices Cobos, Fernando; Kühn, Thomas
We investigate pointwise domination property in operator spaces
generated by Lorentz sequence spaces.


169  Heat Kernels of Lorentz Cones Ding, Hongming
We obtain an explicit formula for heat kernels of Lorentz cones, a
family of classical symmetric cones. By this formula, the heat
kernel of a Lorentz cone is expressed by a function of time $t$ and
two eigenvalues of an element in the cone. We obtain also upper and
lower bounds for the heat kernels of Lorentz cones.


174  Rings With Comparability Ferrero, Miguel; Sant'Ana, Alveri
The class of rings studied in this paper properly contains the
class of right distributive rings which have at least one
completely prime ideal in the Jacobson radical. Amongst other
results we study prime and semiprime ideals, right noetherian rings
with comparability and prove a structure theorem for rings with
comparability. Several examples are also given.


184  On Arithmetic Means of Sequences Generated by a Periodic Function Fiorito, Giovanni
In this paper we prove the convergence of arithmetic means of
sequences generated by a periodic function $\varphi (x) $, moreover
if $\varphi (x) $ satisfies a suitable symmetry condition, we prove
that their limit is $\varphi (0) $. Applications of previous
results are given to study other means of sequences and the
behaviour of a class of recursive series.


190  Topological Quantum Field Theory and Strong Shift Equivalence Gilmer, Patrick M.
Given a TQFT in dimension $d+1,$ and an infinite cyclic covering of
a closed $(d+1)$dimensional manifold $M$, we define an invariant
taking values in a strong shift equivalence class of matrices. The
notion of strong shift equivalence originated in R.~Williams' work
in symbolic dynamics. The TuraevViro module associated to a TQFT
and an infinite cyclic covering is then given by the Jordan form of
this matrix away from zero. This invariant is also defined if the
boundary of $M$ has an $S^1$ factor and the infinite cyclic cover
of the boundary is standard. We define a variant of a TQFT
associated to a finite group $G$ which has been studied by Quinn.
In this way, we recover a link invariant due to D.~Silver and
S.~Williams. We also obtain a variation on the SilverWilliams
invariant, by using the TQFT associated to $G$ in its unmodified form.


198  Commutators and Analytic Dependence of FourierBessel Series on $(0,\infty)$ Guadalupe, José J.; Pérez, Mario; Varona, Juan L.
In this paper we study the boundedness of the commutators $[b,
S_n]$ where $b$ is a $\BMO$ function and $S_n$ denotes the $n$th
partial sum of the FourierBessel series on $(0,\infty)$.
Perturbing the measure by $\exp(2b)$ we obtain that certain
operators related to $S_n$ depend analytically on the functional
parameter $b$.


209  Ample Vector Bundles of Curve Genus One Lanteri, Antonio; Maeda, Hidetoshi
We investigate the pairs $(X,\cE)$ consisting of a smooth complex
projective variety $X$ of dimension $n$ and an ample vector bundle
$\cE$ of rank $n1$ on $X$ such that $\cE$ has a section whose
zero locus is a smooth elliptic curve.


214  Conjugate Radius and Sphere Theorem Paeng, SeongHun; Yun, JongGug
Bessa [Be] proved that for given $n$ and $i_0$, there exists
an $\varepsilon(n,i_0)>0$ depending on $n,i_0$ such that if $M$
admits a metric $g$ satisfying $\Ric_{(M,g)} \ge n1$, $\inj_{(M,g)}
\ge i_0>0$ and $\diam_{(M,g)} \ge \pi\varepsilon$, then $M$ is
diffeomorphic to the standard sphere. In this note, we improve this
result by replacing a lower bound on the injectivity radius with a
lower bound of the conjugate radius.


221  Boundedness of the $q$MeanSquare Operator on VectorValued Analytic Martingales Liu, Peide; Saksman, Eero; Tylli, HansOlav
We study boundedness properties of the $q$meansquare operator
$S^{(q)}$ on $E$valued analytic martingales, where $E$ is a
complex quasiBanach space and $2 \leq q < \infty$. We establish
that a.s. finiteness of $S^{(q)}$ for every bounded $E$valued
analytic martingale implies strong $(p,p)$type estimates for
$S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new
characterizations (in terms of analytic and stochastic properties
of the function $S^{(q)}$) of the complex spaces $E$ that admit an
equivalent $q$uniformly PLconvex quasinorm. We also obtain a
vectorvalued extension (and a characterization) of part of an
observation due to Bourgain and Davis concerning the
$L^p$boundedness of the usual squarefunction on scalarvalued
analytic martingales.


231  Generating Ideals in Rings of IntegerValued Polynomials Rush, David E.
Let $R$ be a onedimensional locally analytically irreducible
Noetherian domain with finite residue fields. In this note it is
shown that if $I$ is a finitely generated ideal of the ring
$\Int(R)$ of integervalued polynomials such that for each $x \in
R$ the ideal $I(x) =\{f(x) \mid f \in I\}$ is strongly
$n$generated, $n \geq 2$, then $I$ is $n$generated, and some
variations of this result.


237  On Benson's Definition of Area in Minkowski Space Thompson, A. C.
Let $(X, \norm)$ be a Minkowski space (finite dimensional Banach
space) with unit ball $B$. Various definitions of surface area are
possible in $X$. Here we explore the one given by Benson
\cite{ben1}, \cite{ben2}. In particular, we show that this
definition is convex and give details about the nature of the
solution to the isoperimetric problem.


248  The Classification of $\Pin_4$Bundles over a $4$Complex Weber, Christian
In this paper we show that the Liegroup $\Pin_4$ is isomorphic to
the semidirect product $(\SU_2\times \SU_2)\timesv \Z/2$ where
$\Z/2$ operates by flipping the factors. Using this structure
theorem we prove a classification theorem for $\Pin_4$bundles over
a finite $4$complex $X$.


257  Homotopy of Knots and the Alexander Polynomial Austin, David; Rolfsen, Dale
Any knot in a 3dimensional homology sphere is homotopic to a knot
with trivial Alexander polynomial.


263  Mellin Transforms of Mixed Cusp Forms Choie, Youngju; Lee, Min Ho
We define generalized Mellin transforms of mixed cusp forms, show
their convergence, and prove that the function obtained by such a
Mellin transform of a mixed cusp form satisfies a certain
functional equation. We also prove that a mixed cusp form can be
identified with a holomorphic form of the highest degree on an
elliptic variety.


274  The Bockstein Map is Necessary Dădărlat, Marius; Eilers, Søren
We construct two nonisomorphic nuclear, stably finite,
real rank zero $C^\ast$algebras $E$ and $E'$ for which
there is an isomorphism of ordered groups
$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to
\bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible
with all the coefficient transformations. The $C^\ast$algebras
$E$ and $E'$ are not isomorphic since there is no $\Theta$
as above which is also compatible with the Bockstein operations.
By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair
of nonisomorphic, real rank zero, purely infinite $C^\ast$algebras
with similar properties.


285  On Kloosterman Sums with Oscillating Coefficients Deng, Peiming
In this paper we prove: for any positive integers $a$ and $q$ with
$(a,q) =1$, we have uniformly
$$
\sum_{\substack{n \leq N \\ (n,q) = 1, \,n\on \equiv 1 (\mod q)}}
\mu (n) e \left( \frac{a\on}{q} \right) \ll Nd (q) \left\{
\frac{\log^{\frac52} N}{q^{\frac12}} + \frac{q^{\frac15}
\log^{\frac{13}5} N}{N^{\frac15}} \right\}.
$$
This improves the previous bound obtained by D.~Hajela,
A.~Pollington and B.~Smith~\cite{5}.


291  Spaces of QuasiMeasures Grubb, D. J.; LaBerge, Tim
We give a direct proof that the space of Baire quasimeasures on a
completely regular space (or the space of Borel quasimeasures on a
normal space) is compact Hausdorff. We show that it is possible for
the space of Borel quasimeasures on a nonnormal space to be
noncompact. This result also provides an example of a Baire
quasimeasure that has no extension to a Borel quasimeasure. Finally,
we give a concise proof of the WheelerShakmatov theorem, which states
that if $X$ is normal and $\dim(X) \le 1$, then every
quasimeasure on $X$ extends to a measure.


298  Semigroup Algebras and Maximal Orders Jespers, Eric; Okniński, Jan
We describe contracted semigroup algebras of Malcev nilpotent
semigroups that are prime Noetherian maximal orders.


307  On the Moduli Space of a Spherical Polygonal Linkage Kapovich, Michael; Millson, John J.
We give a ``wallcrossing'' formula for computing the topology of
the moduli space of a closed $n$gon linkage on $\mathbb{S}^2$.
We do this by determining the Morse theory of the function
$\rho_n$ on the moduli space of $n$gon linkages which is given by
the length of the last sidethe length of the last side is
allowed to vary, the first $(n  1)$ sidelengths are fixed. We
obtain a Morse function on the $(n  2)$torus with level sets
moduli spaces of $n$gon linkages. The critical points of $\rho_n$
are the linkages which are contained in a great circle. We give a
formula for the signature of the Hessian of $\rho_n$ at such a
linkage in terms of the number of backtracks and the winding
number. We use our formula to determine the moduli spaces of all
regular pentagonal spherical linkages.


321  Averaging Operators and Martingale Inequalities in Rearrangement Invariant Function Spaces Kikuchi, Masato
We shall study some connection between averaging operators and
martingale inequalities in rearrangement invariant function spaces.
In Section~2 the equivalence between Shimogaki's theorem and some
martingale inequalities will be established, and in Section~3 the
equivalence between Boyd's theorem and martingale inequalities with
change of probability measure will be established.


335  Cyclic Subgroup Separability of HNNExtensions with Cyclic Associated Subgroups Kim, Goansu; Tang, C. Y.
We derive a necessary and sufficient condition for HNNextensions
of cyclic subgroup separable groups with cyclic associated
subgroups to be cyclic subgroup separable. Applying this, we
explicitly characterize the residual finiteness and the cyclic
subgroup separability of HNNextensions of abelian groups with
cyclic associated subgroups. We also consider these residual
properties of HNNextensions of nilpotent groups with cyclic
associated subgroups.


344  Positive Definite Distributions and Subspaces of $L_p$ With Applications to Stable Processes Koldobsky, Alexander 

354  A Real Holomorphy Ring without the Schmüdgen Property Marshall, Murray A.
A preordering $T$ is constructed in the polynomial ring $A = \R
[t_1,t_2, \dots]$ (countably many variables) with the following two
properties: (1)~~For each $f \in A$ there exists an integer $N$
such that $N \le f(P) \le N$ holds for all $P \in \Sper_T(A)$.
(2)~~For all $f \in A$, if $N+f, Nf \in T$ for some integer $N$,
then $f \in \R$. This is in sharp contrast with the
Schm\"udgenW\"ormann result that for any preordering $T$ in a
finitely generated $\R$algebra $A$, if property~(1) holds, then
for any $f \in A$, $f > 0$ on $\Sper_T(A) \Rightarrow f \in T$.
Also, adjoining to $A$ the square roots of the generators of $T$
yields a larger ring $C$ with these same two properties but with
$\Sigma C^2$ (the set of sums of squares) as the preordering.


359  A Generalized Rao Bound for Ordered Orthogonal Arrays and $(t,m,s)$Nets Martin, W. J.; Stinson, D. R.
In this paper, we provide a generalization of the classical Rao
bound for orthogonal arrays, which can be applied to ordered
orthogonal arrays and $(t,m,s)$nets. Application of our new bound
leads to improvements in many parameter situations to the strongest
bounds (\ie, necessary conditions) for existence of these objects.


371  Prime and Primary Ideals in a Prüfer Order in a Simple Artinian Ring with Finite Dimension over its Center Marubayashi, H.; Ueda, A.
Let $Q$ be a simple Artinian ring with finite dimension over its center.
An order $R$ in $Q$ is said to be {\it Pr\"ufer\/} if any onesided
$R$ideal is a progenerator. We study prime and primary ideals of a
Pr\"ufer order under the condition that the center is Pr\"ufer.
Also we characterize branched and unbranched prime ideals of a
Pr\"ufer order.


380  Asymptotic Behavior of Optimal Circle Packings in a Square Nurmela, Kari J.; Östergård, Patric R. J.; aus dem Spring, Rainer
A lower bound on the number of points that can be placed in a
square of side $\sigma$ such that no two points are within unit
distance from each other is proven. The result is constructive,
and the series of packings obtained contains many conjecturally
optimal packings.


386  Minimal Separators Polat, Norbert
A separator of a connected graph $G$ is a set of vertices whose
removal disconnects $G$. In this paper we give various conditions
for a separator to contain a minimal one. In particular we prove
that every separator of a connected graph that has no thick end, or
which is of bounded degree, contains a minimal separator.


393  A Class of Supercuspidal Representations of $G_2(k)$ Savin, Gordan
Let $H$ be an exceptional, adjoint group of type $E_6$ and split
rank 2, over a $p$adic field $k$. In this article we discuss the
restriction of the minimal representation of $H$ to a dual pair
$\PD^{\times}\times G_2(k)$, where $D$ is a division algebra of
dimension 9 over $k$. In particular, we discover an interesting
class of supercuspidal representations of $G_2(k)$.


401  Lie Derivations in Prime Rings With Involution Swain, Gordon A.; Blau, Philip S.
Let $R$ be a nonGPI prime ring with involution and characteristic
$\neq 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote
the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$
into itself. Then $\delta=\rho+\epsilon$ where $\epsilon$ is an
additive map into the skew elements of the extended centroid of $R$
which is zero on $[K,K]$, and $\rho$ can be extended to an ordinary
derivation of $\langle K\rangle$ into $RC$, the central closure.


412  Peirce Domains Tai, YungSheng
A theorem of Kor\'anyi and Wolf displays any Hermitian symmetric
domain as a Siegel domain of the third kind over any of its
boundary components. In this paper we give a simple proof that an
analogous realization holds for any selfadjoint homogeneous cone.


417  Some Properties of Rational Functions with Prescribed Poles AzizUlAuzeem, Abdul; Zarger, B. A.
Let $P(z)$ be a polynomial of degree not exceeding $n$ and let
$W(z) = \prod^n_{j=1}(za_j)$ where $a_j> 1$, $j =1,2,\dots, n$.
If the rational function $r(z) = P(z)/W(z)$ does not vanish in $z
< k$, then for $k=1$ it is known that
$$
r'(z) \leq \frac{1}{2} B'(z) \Sup_{z=1} r(z)
$$
where $B(Z) = W^\ast(z)/W(z)$ and $W^\ast (z) = z^n \overline
{W(1/\bar z)}$. In the paper we consider the case when $k>1$ and
obtain a sharp result. We also show that
$$
\Sup_{z=1} \biggl\{ \biggl \frac{r'(z)}{B'(z)} \biggr +\biggr
\frac{\bigl(r^\ast (z)\bigr)'}{B'(z)} \biggr \biggr\} =
\Sup_{z=1} r(z)
$$
where $r^\ast (z) = B(z) \overline{r(1/\bar z)}$, and as a consequence of
this result, we present a generalization of a theorem of O'Hara and
Rodriguez for selfinversive polynomials. Finally, we establish a similar
result when supremum is replaced by infimum for a rational function which
has all its zeros in the unit circle.


427  Ramanujan and the Modular $j$Invariant Berndt, Bruce C.; Chan, Heng Huat
A new infinite product $t_n$ was introduced by S.~Ramanujan on the
last page of his third notebook. In this paper, we prove
Ramanujan's assertions about $t_n$ by establishing new connections
between the modular $j$invariant and Ramanujan's cubic theory of
elliptic functions to alternative bases. We also show that for
certain integers $n$, $t_n$ generates the Hilbert class field of
$\mathbb{Q} (\sqrt{n})$. This shows that $t_n$ is a new class
invariant according to H.~Weber's definition of class invariants.


441  Product Bases for the Rationals Berrizbeitia, P.; Elliott, P. D. T. A.
A sequence of positive rationals generates a subgroup of finite
index in the multiplicative positive rationals, and group product
representations by the sequence need only a bounded number of
terms, if and only if certain related sequences have densities
uniformly bounded from below.


445  Smooth Maps and Real Algebraic Morphisms Bochnak, J.; Kucharz, W.
Let $X$ be a compact nonsingular real algebraic variety and let $Y$
be either the blowup of $\mathbb{P}^n(\mathbb{R})$ along a linear
subspace or a nonsingular hypersurface of $\mathbb{P}^m(\mathbb{R})
\times \mathbb{P}^n(\mathbb{R})$ of bidegree $(1,1)$. It is proved
that a $\mathcal{C}^\infty$ map $f \colon X \rightarrow Y$ can be
approximated by regular maps if and only if $f^* \bigl( H^1(Y,
\mathbb{Z}/2) \bigr) \subseteq H^1_{\alg} (X,\mathbb{Z}/2)$, where
$H^1_{\alg} (X,\mathbb{Z}/2)$ is the subgroup of $H^1 (X,
\mathbb{Z}/2)$ generated by the cohomology classes of algebraic
hypersurfaces in $X$. This follows from another result on maps
into generalized flag varieties.


452  Finite Rank Operators in Certain Algebras Bradley, Sean
Let $\Alg(\l)$ be the algebra of all bounded linear operators
on a normed linear space $\x$ leaving invariant each member
of the complete lattice of closed subspaces $\l$. We discuss
when the subalgebra of finite rank operators in $\Alg(\l)$ is
nonzero, and give an example which shows this subalgebra may
be zero even for finite lattices. We then give a necessary
and sufficient lattice condition for decomposing a finite rank
operator $F$ into a sum of a rank one operator and an operator
whose range is smaller than that of $F$, each of which lies in
$\Alg(\l)$. This unifies results of Erdos, Longstaff, Lambrou,
and Spanoudakis. Finally, we use the existence of finite rank
operators in certain algebras to characterize the spectra of
Riesz operators (generalizing results of Ringrose and Clauss)
and compute the Jacobson radical for closed algebras of Riesz
operators and $\Alg(\l)$ for various types of lattices.


463  A Generalized Characterization of Commutators of Parabolic Singular Integrals Hofmann, Steve; Li, Xinwei; Yang, Dachun
Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1,
\dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots
\le\az_n$. Denote $\az=\az_1+\cdots+\az_n$. We characterize those
functions $A(x)$ for which the parabolic Calder\'on commutator
$$
T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n}
K(xy)[A(x)A(y)]f(y)\,dy
$$
is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{\az1}K(x)$,
$K$ is smooth away from the origin and satisfies a certain cancellation
property.


478  A Remark On the MoserAubin Inequality For Axially Symmetric Functions On the Sphere Pruss, Alexander R.
Let $\scr S_r$ be the collection of all axially symmetric functions
$f$ in the Sobolev space $H^1(\Sph^2)$ such that $\int_{\Sph^2}
x_ie^{2f(\mathbf{x})} \, d\omega(\mathbf{x})$ vanishes for $i=1,2,3$.
We prove that
$$
\inf_{f\in \scr S_r} \frac12 \int_{\Sph^2} \nabla f^2 \, d\omega
+ 2\int_{\Sph^2} f \, d\omega \log \int_{\Sph^2} e^{2f} \, d\omega > \oo,
$$
and that this infimum is attained. This complements recent work of
Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang
concerning the MoserAubin inequality.


486  Spherical Functions on $\SO_0(p,q)/\SO(p)\times \SO(q)$ Sawyer, P.
An integral formula is derived for the spherical functions on the
symmetric space $G/K=\break
\SO_0(p,q)/\SO(p)\times \SO(q)$. This formula
allows us to state some results about the analytic continuation of
the spherical functions to a tubular neighbourhood of the
subalgebra $\a$ of the abelian part in the decomposition $G=KAK$.
The corresponding result is then obtained for the heat kernel of the
symmetric space $\SO_0(p,q)/\SO (p)\times\SO (q)$ using the Plancherel
formula.
In the Conclusion, we discuss how this analytic continuation can be
a helpful tool to study the growth of the heat kernel.


499  Characterizations of Simple Isolated Line Singularities Zaharia, Alexandru
A line singularity is a function germ $f\colon(\CC ^{n+1},0) \lra\CC$
with a smooth $1$dimensional critical set $\Sigma=\{(x,y)\in \CC\times
\CC^n \mid y=0\}$. An isolated line singularity is defined by the
condition that for every $x \neq 0$, the germ of $f$ at $(x,0)$ is
equivalent to $y_1^2 +\cdots+y_n ^2$. Simple isolated line
singularities were classified by Dirk Siersma and are analogous
of the famous $ADE$ singularities. We give two new
characterizations of simple isolated line singularities.


507  Author Index  Index des auteurs 1999, for 1999  pour
No abstract.

