The generating degree $\gdeg (A)$ of a topological commutative ring
$A$ with $\Char A = 0$ is the cardinality of the smallest subset $M$
of $A$ for which the subring $\Z[M]$ is dense in $A$. For a prime
number $p$, $\C_p$ denotes the topological completion of an algebraic
closure of the field $\Q_p$ of $p$-adic numbers. We prove that $\gdeg
(\C_p) = 1$, \ie, there exists $t$ in $\C_p$ such that $\Z[t]$ is
dense in $\C_p$. We also compute $\gdeg \bigl( A(U) \bigr)$ where
$A(U)$ is the ring of rigid analytic functions defined on a ball $U$
in $\C_p$. If $U$ is a closed ball then $\gdeg \bigl( A(U) \bigr) =
2$ while if $U$ is an open ball then $\gdeg \bigl( A(U) \bigr)$ is
infinite. We show more generally that $\gdeg \bigl( A(U) \bigr)$ is
finite for any {\it affinoid} $U$ in $\PP^1 (\C_p)$ and $\gdeg \bigl(
A(U) \bigr)$ is infinite for any {\it wide open} subset $U$ of $\PP^1
(\C_p)$.

This paper is concerned with the structure of the arithmetic sum of a
finite number of central Cantor sets. The technique used to study this
consists of a duality between central Cantor sets and sets of subsums
of certain infinite series. One consequence is that the sum of a finite
number of central Cantor sets is one of the following: a finite union
of closed intervals, homeomorphic to the Cantor ternary set or an
$M$-Cantorval.

We show that an $\RA$ loop has a torsion-free normal complement in the
loop of normalized units of its integral loop ring. We also
investigate whether an $\RA$ loop can be normal in its unit loop.
Over fields, this can never happen.

The Hamiltonian potentials of the bending deformations of $n$-gons
in $\E^3$ studied in \cite{KM} and \cite{Kl} give rise to a Hamiltonian
action of the Malcev Lie algebra $\p_n$ of the pure braid group
$P_n$ on the moduli space $M_r$ of $n$-gon linkages with the side-lengths
$r= (r_1,\dots, r_n)$ in $\E^3$. If $e\in M_r$ is a singular point we may
linearize the vector fields in $\p_n$ at $e$. This linearization yields a
flat connection $\nabla$ on the space $\C^n_*$ of $n$ distinct points on
$\C$. We show that the monodromy of $\nabla$ is the dual of a quotient
of a specialized reduced Gassner representation.

The paper is dealing with determination of the integral $\int_{\gamma}
f \,dz$ along the fractal arc $\gamma$ on the complex plane by terms
of polynomial approximations of the function~$f$. We obtain
inequalities for polynomials and conditions of integrability for
functions from the H\"older, Besov and Slobodetskii spaces.

Motivated by deformation theory of holomorphic maps between almost
complex manifolds we endow, in a natural way, the tangent bundle of
an almost complex manifold with an almost complex structure. We
describe various properties of this structure.

Let $p$ be prime and let $\vartheta\in\Z^*_p$ be of
multiplicative order $t$ modulo $p$. We consider exponential
sums of the form
$$
S(a) = \sum_{x =1}^{t} \exp(2\pi i a \vartheta^{x^2}/p)
$$
and prove that for any $\varepsilon > 0$
$$
\max_{\gcd(a,p) = 1} |S(a)| = O( t^{5/6 + \varepsilon}p^{1/8}) .
$$

A challenge by R.~Padmanabhan to prove by group theory the
commutativity of cancellative semigroups satisfying a particular
law has led to the proof of more general semigroup laws being
equivalent to quite simple ones.

The singular spectrum of $u$ in a convolution equation $\mu * u = f$,
where $\mu$ and $f$ are tempered ultradistributions of Beurling or
Roumieau type is estimated by
$$
SS u \subset (\mathbf{R}^n \times \Char \mu) \cup SS f.
$$
The same is done for $SS_{*}u$.

The purpose of this paper is to show the limitations of the
conjectures of algebraic approximation. For this, we construct
points of $\bC^m$ which do not admit good algebraic approximations
of bounded degree and height, when the bounds on the degree and the
height are taken from specific sequences. The coordinates of these
points are Liouville numbers.

Simple necessary conditions for weak type $(1,1)$ of
invariant operators on $L(\rr^d)$ and their applications to
rational Fourier multiplier are given.

Around 1995, Professors Lupacciolu, Chirka and Stout showed that a
closed subset of $\C^N$ ($N\geq 2$) is removable for holomorphic
functions, if its topological dimension is less than or equal to
$N-2$. Besides, they asked whether closed subsets of $\C^2$
homeomorphic to the real line (the simplest 1-dimensional sets) are
removable for holomorphic functions. In this paper we propose a
positive answer to that question.

In this paper the germ of neighborhood of a compact leaf in a
Lagrangian foliation is symplectically classified when the compact
leaf is $\bT^2$, the affine structure induced by the Lagrangian
foliation on the leaf is complete, and the holonomy of $\bT^2$ in
the foliation linearizes. The germ of neighborhood is classified by a
function, depending on one transverse coordinate, this function is
related to the affine structure of the nearly compact leaves.

We prove an algebraic ``no-go theorem'' to the effect that a
nontrivial \pa\ cannot be realized as an associative algebra with the
commu\-ta\-tor bracket. Using it, we show that there is an
obstruction to quantizing the \pa\ of polynomials generated by a
nilpotent \ba\ on a \sm. This result generalizes \gr 's famous
theorem on the impossibility of quantizing the Poisson algebra of
polynomials on $\r^{2n}$. Finally, we explicitly construct a
polynomial quantization of a \sm\ with a solvable \ba, thereby showing
that the obstruction in the nilpotent case does not extend to the
solvable case.

Let $B_N$ be the unit ball in $\mathbb{C}^N$ and let $f$ be a function
holomorphic and $L^2$-integrable in $B_N$. Denote by $E(B_N,f)$
the set of all slices of the form $\Pi =L\cap B_N$, where $L$ is a
complex one-dimensional subspace of $\mathbb{C}^N$, for which $f|_{\Pi}$
is not $L^2$-integrable (with respect to the Lebesgue measure on $L$).
Call this set the exceptional set for $f$. We give a characterization
of exceptional sets which are closed in the natural topology of slices.

James Arthur was awarded the Canada Gold Medal of the National
Science and Engineering Research Council in 1999. This
introduction to his work is an attempt to explain his methods and
his goals to the mathematical community at large.

We construct unbounded positive $C^2$-solutions of the equation
$\Delta u + K u^{(n + 2)/(n - 2)} = 0$ in $\R^n$ (equipped
with Euclidean metric $g_o$) such that $K$ is bounded between two
positive numbers in $\R^n$, the conformal metric $g=u^{4/(n-2)}g_o$
is complete, and the volume growth of $g$ can be arbitrarily fast
or reasonably slow according to the constructions. By imposing natural
conditions on $u$, we obtain growth estimate on the $L^{2n/(n-2)}$-norm
of the solution and show that it has slow decay.

A generalization of Schm\"udgen's Positivstellensatz is given which holds
for any basic closed semialgebraic set in $\mathbb{R}^n$ (compact or not).
The proof is an extension of W\"ormann's proof.

Let $G$ be an infinite discrete amenable group or a non-discrete
amenable group. It is shown how to construct a net $(f_\alpha)$ of
positive, normalized functions in $L_1(G)$ such that the net converges weak*
to invariance but does not converge strongly to invariance. The solution of
certain linear equations determined by colorings of the Cayley graphs of the
group are central to this construction.

The zeta function of a nonsingular pair of quadratic forms defined over a
finite field, $k$, of arbitrary characteristic is calculated. A.~Weil made
this computation when $\rmchar k \neq 2$. When the pair has even order, a
relationship between the number of zeros of the pair and the number of
places of degree one in an appropriate hyperelliptic function field is

Let $X$ be a reduced nonsingular quasiprojective scheme over ${\mathbb
R}$ such that the set of real rational points $X({\mathbb R})$ is dense
in $X$ and compact. Then $X({\mathbb R})$ is a real algebraic variety.
Denote by $H_k^{\alg}(X({\mathbb R}), {\mathbb Z}/2)$ the group of
homology classes represented by Zariski closed $k$-dimensional
subvarieties of $X({\mathbb R})$. In this note we show that $H_1^{\alg}
(X({\mathbb R}), {\mathbb Z}/2)$ is a proper subgroup of
$H_1(X({\mathbb R}), {\mathbb Z}/2)$ for a nonorientable hyperelliptic
surface $X$. We also determine all possible groups $H_1^{\alg}(X({\mathbb R}),
{\mathbb Z}/2)$ for a real ruled surface $X$ in connection with the previously
known description of all possible topological configurations of $X$.

We show that every compactum has cohomological dimension $1$ with respect
to a finitely generated nilpotent group $G$ whenever it has cohomological
dimension $1$ with respect to the abelianization of $G$. This is applied
to the extension theory to obtain a cohomological dimension theory condition
for a finite-dimensional compactum $X$ for extendability of every map from
a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely
generated homotopy groups over all of $X$.

Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$.
The $c$-numerical range of an $n \times n$ matrix $A$ is defined by
$$
W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\}
\text{ an orthonormal basis for } \IC^n \Bigr\},
$$
and the $c$-numerical radius of $A$ is defined by $r_c (A) = \max
\{|z| : z \in W_c (A)\}$. We determine the structure of those linear
operators $\phi$ on algebras of block triangular matrices, satisfying
$$
W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or}
\quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.
$$

Jacobi-like forms for a discrete subgroup $\G \subset \SL(2,\mbb R)$
are formal power series with coefficients in the space of functions on
the Poincar\'e upper half plane satisfying a certain functional
equation, and they correspond to sequences of certain modular forms.
We introduce Hecke operators acting on the space of Jacobi-like forms
and obtain an explicit formula for such an action in terms of modular
forms. We also prove that those Hecke operator actions on Jacobi-like
forms are compatible with the usual Hecke operator actions on modular
forms.

There is a theorem, usually attributed to Napoleon, which states that
if one takes any triangle in the Euclidean Plane, constructs
equilateral triangles on each of its sides, and connects the midpoints
of the three equilateral triangles, one will obtain an equilateral
triangle. We consider an analogue of this problem in the hyperbolic
plane.

In this paper the author prove that standard modules of classical
groups whose Langlands quotients are generic are irreducible. This
establishes a conjecture of Casselman and Shahidi for this important
class of groups.

We give an explicit recipe for the determination of the images
associated to the Galois action on $p$-torsion points of elliptic
curves. We present a table listing the image for all the elliptic
curves defined over $\QQ$ without complex multiplication with
conductor less than 200 and for each prime number~$p$.

Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan
$\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones
qui sont des perturbations polynomiales de $H_0$. On obtient alors
une condition n\'ecessaire d'isochronisme, et un crit\`ere de choix
pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant
une g\'en\'eralisation du caract\`ere isochrone des perturbations
hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S].

Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the
plane $\Rset^2$. We obtain a necessary condition for a system to be
isochronous. We can think of this result as a generalization of the
isochronous behaviour of the homogeneous polynomial perturbation of
the Hamiltonian $H_0$ considered in [L], [P], [S].

Irrational rotation $C^*$-algebras have an inductive limit
decomposition in terms of matrix algebras over the space of continuous
functions on the circle and this decomposition can be chosen to be
invariant under the flip automorphism. It is shown that the flip is
essentially the only toral automorphism with this property.

We continue to study the simplest closure conditions for chains of
spectral transformations of the Laurent biorthogonal polynomials
($\LBP$). It is shown that the 1-1-periodic $q$-closure condition
leads to the $\LBP$ introduced by Pastro. We introduce classes of
semi-classical and Laguerre-Hahn $\LBP$ associated to generic closure
conditions of the chain of spectral transformations.

In this paper, we establish the existence of positive solution of a
nonlinear subelliptic equation involving the critical Sobolev exponent on
the Heisenberg group, which generalizes a result of Brezis and
Nirenberg in the Euclidean case.

We examine Hilbert bimodules which possess a (generally unbounded)
involution. Topics considered include a linking algebra
representation, duality, locality, and the role of these bimodules
in noncommutative differential geometry

Motivated by a question of Per Enflo, we develop a hypercube criterion
for locating linear isometric copies of $\lone$ in an arbitrary real
normed space $X$.

The said criterion involves finding $2^{n}$ points in $X$ that satisfy
one metric equality. This contrasts nicely to the standard classical
criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric
equalities.

Let $F$ be a totally real number field and let $\GL_{n}$ be the
general linear group of rank $n$ over $F$. Let $\mathfrak{p}$
be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$
with respect to the valuation induced by $\mathfrak{p}$. We will
consider a finite quotient of the affine building of the group
$\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object
as a hypergraph and find a set of commuting operators whose sum will
be the usual adjacency operator of the graph underlying the hypergraph.

We find a lower bound on the number of imaginary quadratic extensions
of the function field $\F_q(T)$ whose class groups have an element of
a fixed order.

More precisely, let $q \geq 5$ be a power of an odd prime and let $g$
be a fixed positive integer $\geq 3$. There are $\gg q^{\ell
(\frac{1}{2}+\frac{1}{g})}$ polynomials $D \in \F_q[T]$ with $\deg(D)
\leq \ell$ such that the class groups of the quadratic extensions
$\F_q(T,\sqrt{D})$ have an element of order~$g$.

We classify finite subgroups of $\SO(4)$ generated by anti-unitary
involutions. They correspond to involutions fixing pointwise a
Lagrangian plane. Explicit descriptions of the finite groups and the
configurations of Lagrangian planes are obtained.

Functions defined on closed sets are simultaneously approximated and
interpolated by meromorphic functions with prescribed poles and zeros
outside the set of approximation.

In answer to a question posed in \cite{G}, we give sufficient
conditions on a Lie nilmanifold so that any ergodic rotation of the
nilmanifold is metrically conjugate to its inverse. The condition is
that the Lie algebra be what we call quasi-graded, and is weaker than
the property of being graded. Furthermore, the conjugating map can be
chosen to be an involution. It is shown that for a special class of
groups, the condition of quasi-graded is also necessary. In certain
examples there is a continuum of conjugacies.

In this paper we find a formula for the Alexander polynomial
$\Delta_{p_1,\dots,p_k} (x)$ of pretzel knots and links with
$(p_1,\dots,p_k, \nega 1)$ twists, where $k$ is odd and
$p_1,\dots,p_k$ are positive integers. The polynomial $\Delta_{2,3,7}
(x)$ is the well-known Lehmer polynomial, which is conjectured to have
the smallest Mahler measure among all monic integer polynomials. We
confirm that $\Delta_{2,3,7} (x)$ has the smallest Mahler measure among
the polynomials arising as $\Delta_{p_1,\dots,p_k} (x)$.

Let $\ce$ be an ample vector bundle of rank $r$ on a projective
variety $X$ with only log-terminal singularities. We consider the
nefness of adjoint divisors $K_X + (t-r) \det \ce$ when $t \ge \dim X$
and $t>r$. As an application, we classify pairs $(X,\ce)$ with
$c_r$-sectional genus zero.

Nous montrons que la A-cat\'egorie d'un espace simplement connexe de
type fini est inf\'erieure ou \'egale \`a $n$ si et seulement si son
mod\`ele d'Adams-Hilton est un r\'etracte homotopique d'une alg\`ebre
diff\'erentielle \`a $n$ \'etages. Nous en d\'eduisons que
l'invariant $\Acat$ augmente au plus de 1 lors de l'attachement
d'une cellule \`a un espace.

We show that the A-category of a simply connected space of finite type
is less than or equal to $n$ if and only if its Adams-Hilton model is
a homotopy retract of an $n$-stage differential algebra. We deduce
from this that the invariant $\Acat$ increases by at most 1 when a
cell is attached to a space.

In this article it is shown that every bounded linear operator
on a complex, infinite dimensional, separable Hilbert space is
a sum of at most eighteen unilateral (alternatively, bilateral)
weighted shifts. As well, we classify products of weighted shifts,
as well as sums and limits of the resulting operators.

We prove an identity between weighted orbital integrals of the unit
elements in the Hecke algebras of $\GL(r)$ and its $n$-fold
metaplectic covering, under the assumption that $n$ is relatively
prime to any proper divisor of every $1 \leq j \leq r$.

We give canonical resolutions of singularities of several cone
varieties arising from invariant theory. We establish a connection
between our resolutions and resolutions of singularities of closure of
conjugacy classes in classical Lie algebras.

We show that, if a Banach algebra $\A$ is a left ideal in its second
dual algebra and has a left bounded approximate identity, then the
weak amenability of $\A$ implies the ($2m+1$)-weak amenability of $\A$
for all $m\geq 1$.