In 1988 Rieger exhibited a differentiable function having a zero at
the golden ratio\break
$(-1+\sqrt5)/2$ for which when Newton's method for approximating
roots is applied with an initial value $x_0=0$, all approximates
are so-called ``best rational approximates''---in this case, of the
form $F_{2n}/F_{2n+1}$, where $F_n$ denotes the $n$-th Fibonacci
number. Recently this observation was extended by Komatsu to the
class of all quadratic irrationals whose continued fraction
expansions have period length $2$. Here we generalize these
observations by producing an analogous result for all quadratic
irrationals and thus provide an explanation for these phenomena.

We show that there exists a singular inner function $S$ which is
universal for noneuclidean translates; that is one for which the set
$\{S(\frac{z+z_n}{1+\bar z_nz}):n\in\mathbb{N}\}$ is locally uniformly dense
in the set of all zero-free holomorphic functions in $\mathbb{D}$ bounded by
one.

Using weighted Delsarte surfaces, we give examples of $K3$ surfaces
in positive characteristic whose formal Brauer groups have height
equal to $5$, $8$ or $9$. These are among the four values of the
height left open in the article of Yui \cite{Y}.

Leray's self-similar solution of the Navier-Stokes equations is
defined by
$$
u(x,t) = U(y)/\sqrt{2\sigma (t^*-t)},
$$
where $y = x/\sqrt{2\sigma (t^*-t)}$, $\sigma>0$. Consider the
equation for $U(y)$ in a smooth bounded domain $\mathcal{D}$ of
$\mathbb{R}^3$ with non-zero boundary condition:
\begin{gather*}
-\nu \bigtriangleup U + \sigma U +\sigma y \cdot \nabla U + U\cdot
\nabla U + \nabla P = 0,\quad y \in \mathcal{D}, \\
\nabla \cdot U = 0, \quad y \in \mathcal{D}, \\
U = \mathcal{G}(y), \quad y \in \partial \mathcal{D}.
\end{gather*}
We prove an existence theorem for the Dirichlet problem in Sobolev
space $W^{1,2} (\mathcal{D})$. This implies the local existence of
a self-similar solution of the Navier-Stokes equations which blows
up at $t=t^*$ with $t^* < +\infty$, provided the function
$\mathcal{G}(y)$ is permissible.

The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are
subspaces of $\BMOA$ for $0<p\le 1$. We obtain lower and upper
estimates for the essential norm of a composition operator from the
Bloch space into $Q_p$, in particular from the Bloch space into
$\BMOA$.

Let $g\colon M^{2n}\rightarrow M^{2n}$ be a smooth map of period $m>2$ which
preserves orientation. Suppose that the cyclic action defined by $g$ is regular
and that the normal bundle of the fixed point set $F$ has a $g$-equivariant
complex structure. Let $F\pitchfork F$ be the transverse self-intersection of
$F$ with itself. If the $g$-signature $\Sign (g,M)$ is a rational integer and
$n<\phi (m)$, then there exists a choice of orientations such that $\Sign(g,M)=
\Sign F=\Sign(F\pitchfork F)$.

In this paper, we give some conditions to judge when a system of
Hermitian quadratic forms has a real linear combination which is
positive definite or positive semi-definite. We also study some
related geometric and topological properties of the moduli space.

This paper is about the topologies arising from statistical
coincidence on locally finite point sets in locally compact Abelian
groups $G$. The first part defines a uniform topology
(autocorrelation topology) and proves that, in effect, the set of all
locally finite subsets of $G$ is complete in this topology. Notions
of statistical relative denseness, statistical uniform discreteness,
and statistical Delone sets are introduced.

The second part looks at the consequences of mixing the original and
autocorrelation topologies, which together produce a new Abelian
group, the autocorrelation group. In particular the relation between
its compactness (which leads then to a $G$-dynamical system) and pure
point diffractivity is considered. Finally for generic regular model
sets it is shown that the autocorrelation group can be identified with
the associated compact group of the cut and project scheme that
defines it. For such a set the autocorrelation group, as a
$G$-dynamical system, is a factor of the dynamical local hull.

Let $N$ be an integer which is larger than one. In this paper we
study invariant subspaces of $L^2 (\mathbb{T}^N)$ under the double
commuting condition. A main result is an $N$-dimensional version of
the theorem proved by Mandrekar and Nakazi. As an application of this
result, we have an $N$-dimensional version of Lax's theorem.

It is known that any non-archimedean Fr\'echet space of countable
type is isomorphic to a subspace of $c_0^{\mathbb{N}}$. In this
paper we prove that there exists a non-archimedean Fr\'echet space
$U$ with a basis $(u_n)$ such that any basis $(x_n)$ in a
non-archimedean Fr\'echet space $X$ is equivalent to a subbasis
$(u_{k_n})$ of $(u_n)$. Then any non-archimedean Fr\'echet space
with a basis is isomorphic to a complemented subspace of $U$. In
contrast to this, we show that a non-archimedean Fr\'echet space
$X$ with a basis $(x_n)$ is isomorphic to a complemented subspace
of $c_0^{\mathbb{N}}$ if and only if $X$ is isomorphic to one of
the following spaces: $c_0$, $c_0 \times \mathbb{K}^{\mathbb{N}}$,
$\mathbb{K}^{\mathbb{N}}$, $c_0^{\mathbb{N}}$. Finally, we prove
that there is no nuclear non-archimedean Fr\'echet space $H$ with
a basis $(h_n)$ such that any basis $(y_n)$ in a nuclear
non-archimedean Fr\'echet space $Y$ is equivalent to a subbasis
$(h_{k_n})$ of $(h_n)$.

Exponent information is proven about the Lie groups $SU(3)$,
$SU(4)$, $Sp(2)$, and $G_2$ by showing some power of the $H$-space
squaring map (on a suitably looped connected-cover) is null homotopic.
The upper bounds obtained are $8$, $32$, $64$, and $2^8$ respectively.
This null homotopy is best possible for $SU(3)$ given the number of
loops, off by at most one power of~$2$ for $SU(4)$ and $Sp(2)$, and
off by at most two powers of $2$ for $G_2$.

We prove that a class of perturbations of standard ${\rm CR}$
structure on the boundary of three-dimensional complex ellipsoid
$E_{p,q}$ can be realized as hypersurfaces on $\mathbb{C}^2$, which
generalizes the result of Burns and Epstein on the embeddability of
some perturbations of standard ${\rm CR}$ structure on $S^3$.

Voiculescu has previously established the uniqueness of the wave operator
for the problem of $\mathcal{C}^{(0)}$-perturbation of commuting tuples
of self-adjoint operators in the case where the norm ideal $\mathcal{C}$
has the property $\lim_{n\rightarrow\infty} n^{-1/2}\|P_n\|_{\mathcal{C}}=0$,
where $\{P_n\}$ is any sequence of orthogonal projections with $\rank(P_n)=n$.
We prove that the same uniqueness result holds true so long as $\mathcal{C}$
is not the trace class. (It is well known that there is no such uniqueness
in the case of trace-class perturbation.)

In this paper we investigate the uniqueness of transcendental
meromorphic function dealing with the shared values in some angular
domains instead of the whole complex plane.

Let $P$ be a transitive permutation group of order $p^m$, $p$ an odd prime,
containing a regular cyclic subgroup. The main result of this paper is a
determination of the suborbits of $P$. The main result is used to give a
simple proof of a recent result by J.~Morris on Cayley digraph isomorphisms.

Unlike the (classical) Kolakoski sequence on the alphabet $\{1,2\}$, its analogue
on $\{1,3\}$ can be related to a primitive substitution rule. Using this connection,
we prove that the corresponding bi-infinite fixed point is a regular generic model set
and thus has a pure point diffraction spectrum. The Kolakoski-$(3,1)$ sequence is
then obtained as a deformation, without losing the pure point diffraction property.

The congruences of a finite sectionally complemented lattice $L$ are
not necessarily uniform (any two congruence classes of a
congruence are of the same size). To measure how far a congruence
$\Theta$ of $L$ is from being uniform, we introduce $\Spec\Theta$, the
spectrum of $\Theta$, the family of cardinalities of the
congruence classes of $\Theta$. A typical result of this paper
characterizes the spectrum $S = (m_j \mid j < n)$ of a nontrivial
congruence $\Theta$ with the following two properties:
\begin{enumerate}[$(S_2)$]
\item[$(S_1)$] $2 \leq n$ and $n \neq 3$.

\item[$(S_2)$] $2 \leq m_j$ and $m_j \neq 3$, for all $j<n$.
\end{enumerate}

We show that Poincar\'e inequalities with reverse doubling weights hold in a
large class of irregular domains whenever the weights satisfy certain
conditions. Examples of these domains are John domains.

A discrete group $G$ is called identity excluding\/
if the only irreducible
unitary representation of $G$ which weakly contains the $1$-dimensional identity
representation is the $1$-dimensional identity representation itself. Given a
unitary representation $\pi$ of $G$ and a probability measure $\mu$ on $G$, let
$P_\mu$ denote the $\mu$-average $\int\pi(g) \mu(dg)$. The goal of this article
is twofold: (1)~to study the asymptotic behaviour of the powers $P_\mu^n$, and
(2)~to provide a characterization of countable amenable identity excluding groups.
We prove that for every adapted probability measure $\mu$ on an identity excluding
group and every unitary representation $\pi$ there exists and orthogonal projection
$E_\mu$ onto a $\pi$-invariant subspace such that $s$-$\lim_{n\to\infty}\bigl(P_\mu^n-
\pi(a)^nE_\mu\bigr)=0$ for every $a\in\supp\mu$. This also remains true for suitably
defined identity excluding locally compact groups. We show that the class of countable
amenable identity excluding groups coincides with the class of $\FC$-hypercentral
groups; in the finitely generated case this is precisely the class of groups of
polynomial growth. We also establish that every adapted random walk on a countable
amenable identity excluding group is ergodic.

We give a survey of old and new results concerning the expressibility
of the real roots of a solvable polynomial over a real number field by
real radicals. A characterization of Fermat primes is obtained in
terms of solvability by real radicals for certain ploynomials.

Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$.
The subset $X+X^2 k[[X]]$ of the ring $k[[X]]$ is a group under the substitution
law $\circ $ sometimes called the Nottingham group of $k$; it is denoted by
$\mathcal{R}_k$. The ramification of one series $\gamma\in\mathcal{R}_k$ is
caracterized by its lower ramification numbers: $i_m(\gamma)=\ord_X
\bigl(\gamma^{p^m} (X)/X - 1\bigr)$, as well as its upper ramification numbers:
$$
u_m (\gamma) = i_0 (\gamma) + \frac{i_1 (\gamma) - i_0(\gamma)}{p} +
\cdots + \frac{i_m (\gamma) - i_{m-1} (\gamma)}{p^m} , \quad (m \in
\mathbb{N}).
$$
By Sen's theorem, the $u_m(\gamma)$ are integers. In this paper, we determine
the sequences of integers $(u_m)$ for which there exists $\gamma\in\mathcal{R}_k$
such that $u_m(\gamma)=u_m$ for all integer $m \geq 0$.

Generalizing results from [MM1] referring
to the intersection body $IK$ and
the cross-section body $CK$ of a convex body
$K \subset \sR^d, \, d \ge 2$,
we prove theorems about maximal $k$-sections of convex bodies,
$k \in \{1, \dots, d-1\}$,
and, simultaneously, statements
about common maximal
$(d-1)$- and $1$-transversals of families
of convex bodies.

A band is a semigroup of idempotent operators. A nonnegative band
$\cls$ in $\clb(\cll^2 (\clx))$ having at least one element of finite
rank and with rank $(S) > 1 $ for all $S$ in $\cls$ is known to have a
special kind of common invariant subspace which is termed
a standard subspace (defined below).

Such bands are called decomposable. Decomposability has helped to
understand the structure of nonnegative bands with constant finite
rank. In this paper, a geometric characterization of maximal,
rank-one, indecomposable nonnegative bands is obtained which
facilitates the understanding of their geometric structure.

In this paper, we prove a general result computing the number of rational points
of bounded height on a projective variety $V$ which is covered by lines. The
main technical result used to achieve this is an upper bound on the number of
rational points of bounded height on a line. This upper bound is such that it
can be easily controlled as the line varies, and hence is used to sum the counting
functions of the lines which cover the original variety $V$.

We study the interplay between canonical heights and endomorphisms of an abelian
variety $A$ over a number field $k$. In particular we show that whenever the ring
of endomorphisms defined over $k$ is strictly larger than $\Z$ there will
be $\Q$-linear relations among the values of a canonical height pairing evaluated
at a basis modulo torsion of $A(k)$.

This article is a short introduction to super-Brownian motion. Some of
its properties are discussed but our main objective is to describe a
number of limit theorems which show super-Brownian motion is a
universal limit for rescaled spatial stochastic systems at criticality
above a critical dimenson. These systems include the voter model, the
contact process and critical oriented percolation.

In this paper we consider collections of
compact operators on a real or
complex Banach space including linear operators
on finite-dimensional vector spaces. We show
that such a collection is simultaneously
triangularizable if and only if it is arbitrarily
close to a simultaneously triangularizable
collection of compact operators. As an application
of these results we obtain an invariant subspace
theorem for certain bounded operators. We
further prove that in finite dimensions near
reducibility implies reducibility whenever
the ground field is $\BR$ or $\BC$.

The usual constructions of classifying spaces for monoidal categories
produce CW-complexes with
many cells that, moreover, do not have any proper geometric meaning.
However, geometric nerves of
monoidal categories are very handy simplicial sets whose simplices
have
a pleasing geometric
description: they are diagrams with the shape of the 2-skeleton of
oriented standard simplices. The
purpose of this paper is to prove that geometric realizations of
geometric nerves are classifying
spaces for monoidal categories.

Let $M$ be a complete flat Lorentz $3$-manifold $M$ with purely
hyperbolic holonomy $\Gamma$. Recurrent geodesic rays are completely
classified when $\Gamma$ is cyclic. This implies that for any pair of
periodic geodesics $\gamma_1$, $\gamma_2$, a unique geodesic forward
spirals towards $\gamma_1$ and backward spirals towards $\gamma_2$.

We construct new examples of non-nil algebras with any number of
generators, which are direct sums of two
locally nilpotent subalgebras. Like all previously known examples, our examples
are contracted semigroup algebras and the underlying semigroups are unions
of locally nilpotent subsemigroups.
In our constructions we make more
transparent
than in the past the close relationship between the considered problem
and combinatorics of words.

If $A$ is a set of $M$ positive integers, let $G(A)$ be the maximum
of $a_i/\gcd(a_i,a_j)$ over $a_i,a_j\in A$. We show that if
$G(A)$ is not too much larger than $M$, then $A$ must have a
special structure.

We show that the product of four or five consecutive positive
terms in arithmetic progression can never be a perfect power whenever the
initial term is coprime to the common difference of the arithmetic
progression. This is a generalization of the results of Euler and Obl\'ath
for the case of squares, and an extension of a theorem of Gy\H ory on three
terms in arithmetic progressions. Several other results concerning the
integral solutions of the equation of the title are also obtained. We extend
results of Sander on the rational solutions of the equation in $n,y$ when
$b=d=1$. We show that there are only finitely many solutions in $n,d,b,y$
when $k\geq 3$, $l\geq 2$ are fixed and $k+l>6$.

In this paper we give an inversion formula of the Radon transform on the
Heisenberg group by using the wavelets defined in [3]. In addition, we
characterize a space such that the inversion formula of the Radon transform
holds in the weak sense.

Let $V$ be a $K3$ surface defined over a number field $k$. The
Batyrev-Manin conjecture for $V$ states that for every nonempty open
subset $U$ of $V$, there exists a finite set $Z_U$ of accumulating
rational curves such that the density of rational points on $U-Z_U$ is
strictly less than the density of rational points on $Z_U$. Thus,
the set of rational points of $V$ conjecturally admits a stratification
corresponding to the sets $Z_U$ for successively smaller sets $U$.
In this paper, in the case that $V$ is a Kummer surface, we prove that
the Batyrev-Manin conjecture for $V$ can be reduced to the
Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$
induced by multiplication by $m$ on the associated abelian surface
$A$. As an application, we use this to show that given some restrictions
on $A$, the set of rational points of $V$ which lie on rational curves
whose preimages have geometric genus 2 admits a stratification of

We apply the method of complex scaling to give a natural
proof of a formula relating the multiplicity of a resonance to the
multiplicity of a pole of the scattering matrix.

For certain real quadratic number fields, we prove density results concerning
$4$-ranks of tame kernels. We also discuss a relationship between $4$-ranks of
tame kernels and %% $4$-class ranks of narrow ideal class groups. Additionally,
we give a product formula for a local Hilbert symbol.

The stable basin theorem was introduced by Basmajian and Miner as a
key step in their necessary condition for the discreteness of a
non-elementary group of complex hyperbolic isometries. In this
paper we improve several of Basmajian and Miner's key estimates and
so give a substantial improvement on the main inequality in the
stable basin theorem.

For a locally compact group $G$, the convolution product on
the space $\nN(L^p(G))$ of nuclear operators was defined by Neufang
\cite{Neuf_PhD}. We study homological properties of the convolution algebra
$\nN(L^p(G))$ and relate them to some properties of the group $G$,
such as compactness, finiteness, discreteness, and amenability.

D. J. Grubb [3] has shown that uniqueness holds, under a
mild growth condition, for Vilenkin series which converge almost
everywhere to zero. We show that, under even less restrictive
growth conditions, one can replace the limit function 0 by an
arbitrary $f\in L^q$, when $q>1$.

We give a new characterization of Hardy martingale cotype
property of complex quasi-Banach space by using the existence of a
kind of plurisubharmonic functions. We also characterize the best
constants of Hardy martingale inequalities with values
in the complex quasi-Banach space.

The main purpose of this paper is to determine
isotropic immersions of complex
space forms into real space forms with low
codimension. This is an improvement of a result of S. Maeda.

We prove an uniform H\"older continuity of the resolvent of
the Laplace-Beltrami operator on the real axis for a class
of asymptotically Euclidean Riemannian manifolds. As an application we
extend a result of Burq on the behaviour of the
local energy of solutions to the wave equation.

Dans cet article, \`a partir de la notion d'enlacement introduite
dans ~\cite{F} entre des paires d'ensembles $(B,A)$ et $(Q,P)$,
nous \'etablissons l'existence d'un point critique d'une
fonctionnelle continue sur un espace m\'etrique lorsqu'une de ces
paires enlace l'autre. Des renseignements sur la localisation du
point critique sont aussi obtenus. Ces r\'esultats conduisent \`a
une g\'en\'eralisation du th\'eor\`eme des trois points critiques.
Finalement, des applications \`a des probl\`emes aux limites pour
une \'equation quasi-lin\'eaire elliptique sont pr\'esent\'ees.

Order components of a finite simple group were introduced in [4].
It was proved that some non-abelian simple groups are uniquely determined
by their order components. As the main result of this paper, we
show that groups $PSU_{11}(q)$ are also uniquely determined by
their order components. As corollaries of this result, the
validity of a conjecture of J. G. Thompson and a conjecture of W.
Shi and J. Bi both on $PSU_{11}(q)$ are obtained.

We study a compactness property of the operators between weighted
Lebesgue spaces that average a function over certain domains involving
a star-shaped region. The cases covered are (i) when the average is
taken over a difference of two dilations of a star-shaped region in
$\RR^N$, and (ii) when the average is taken over all dilations of
star-shaped regions in $\RR^N$. These cases include, respectively,
the average over annuli and the average over balls centered at origin.

A basic problem in dynamics is to identify systems
with positive entropy, i.e., systems which are ``chaotic.'' While
there is a vast collection of results addressing this issue in
topological dynamics, the phenomenon of positive entropy remains by and
large a mystery within the broader noncommutative domain of $C^*$-algebraic
dynamics. To shed some light on the noncommutative situation we propose
a geometric perspective inspired by work of Glasner and Weiss on
topological entropy.
This is a written version of the author's talk
at the Winter 2002 Meeting of the Canadian Mathematical Society
in Ottawa, Ontario.

We show that the Prym map for 4-th cyclic \'etale covers of curves
of genus 4 is a dominant morphism to a Shimura variety for a family
of Abelian 6-folds of Weil type. According to the result of Schoen,
this implies algebraicity of Weil classes for this family.

We axiomatize the main properties of the classical Tur\'an Theorem
in order to apply it to a general context. We provide applications in the
cases of number fields, function fields, and geometrically irreducible
varieties over a finite field.

We axiomatize the main properties of the classical Erd\"os-Kac Theorem
in order to apply it to a general context. We provide applications in the
cases of number fields, function fields, and geometrically irreducible
varieties over a finite field.

Jeffrey and Kirwan gave expressions
for intersection pairings on the reduced space
$M_0=\mu^{-1}(0)/G$ of a Hamiltonian $G$-space $M$
in terms of multiple residues.
In this paper we prove a residue formula for
symplectic volumes of reduced spaces of a quasi-Hamiltonian
$\SU(2)$-space. The definition of quasi-Hamiltonian
$G$-spaces was recently introduced in .

Let $\cal A$ be a $C^*$-algebra and $E$ be a Banach space with
the Radon-Nikodym property. We prove that if $j$ is an embedding
of $E$ into an injective Banach space then for every absolutely
summing operator $T:\mathcal{A}\longrightarrow E$, the composition
$j \circ T$ factors through a diagonal operator from $l^{2}$ into
$l^{1}$. In particular, $T$ factors through a Banach space with
the Schur property. Similarly, we prove that for $2<p<\infty$, any
absolutely summing operator from $\cal{A}$ into $E$ factors
through a diagonal operator from $l^p$ into $l^2$.

In this paper, we consider Yang-Mills connections
on a vector bundle $E$ over a compact Riemannian manifold $M$ of
dimension $m> 4$, and we show that any set of Yang-Mills
connections with the uniformly bounded $L^{\frac{m}{2}}$-norm of
curvature is compact in $C^{\infty}$ topology.