In this paper, we study the $L^p$ mapping properties of a class of singular
integral operators with rough kernels belonging to certain block spaces. We
prove that our operators are bounded on $L^p$ provided that their kernels
satisfy a size condition much weaker than that for the classical
Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an
example showing that our size condition is optimal. As a consequence of our
results, we substantially improve a previously known result on certain maximal
functions.

We search for theorems that, given a $C_i$-field $K$ and a subfield $k$ of $K$, allow
us to conclude that $k$ is a $C_j$-field for some $j$. We give appropriate theorems in
the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$
is an algebraic extension. Here we are able to prove some results, and make conjectures. We
also point out the connection between these questions and Lang's conjecture on nonreal function
fields over a real closed field.

We extend the methods of Van der Poorten and Chapman
for
explicitly evaluating the Dede\-kind eta function at quadratic
irrationalities. Via evaluation of Hecke
$L$-series we obtain new evaluations at points in
imaginary quadratic number fields with
class numbers 3 and 4. Further, we overcome the limitations
of the earlier methods and via modular equations provide
explicit evaluations where the class number is 5 or 7.

Using a modification of Webster's proof of the Newlander--Nirenberg
theorem, it is shown that, for a weakly convergent sequence of
integrable unitary connections on a complex vector bundle over a
complex manifold, there is a subsequence of local holomorphic frames
that converges strongly in an appropriate Holder class.

The Lusternik--Schnirelmann category has been described in different ways.
Two major ones, the first by Ganea, the second by Whitehead, are presented here
with a number of variants. The equivalence of these variants relies on
the axioms of Quillen's model category, but also sometimes on an additional
axiom, the so-called ``cube axiom''.

In this article, we give an explicit formula to compute the
non abelian twisted sign-deter\-mined Reidemeister torsion of the
exterior of a fibered knot in terms of its monodromy. As an
application, we give explicit formulae for the non abelian
Reidemeister torsion of torus knots and of the figure eight knot.

In this note we consider $\overline\partial$-problem in
line bundles over complex projective space $\mathbb{CP}^1$
and prove that the
equation can be solved for $(0,1)$ forms with compact support. As a
consequence, any Cauchy-Riemann function on a compact real hypersurface in
such line bundles is a jump of two holomorphic functions defined on the
sides of the hypersurface. In particular, the results can be applied to
$\mathbb{CP}^2$ since by removing a point from it we get a line bundle over
$\mathbb{CP}^1$.

We characterize the weight functions
$u,v,w$ on $(0,\infty)$ such that
$$
\left(\int_0^\infty f^{*}(t)^
qw(t)\,dt\right)^{1/q}
\leq
C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t),
$$
where
$$
f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1}
\int_{0}^{t}f^*(s)u(s)\,ds.
$$
As an application we present a~new simple characterization of
the associate space to the space $\Gamma^ \infty(v)$, determined by the
norm
$$
\|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t),
$$
where
$$
f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds.
$$

We introduce roots of indecomposable modules over group algebras of finite groups,
and we investigate some of their properties. This allows us to correct an error
in Landrock's book which has to do with roots of simple modules.

We give a geometric proof of classical results that characterize
Pisot numbers as algebraic $\lambda>1$ for which
there is $x\neq0$ with $\lambda^nx \to 0 \mod$ and identify such
$x$ as members of $\Z[\lambda^{-1}] \cdot \Z[\lambda]^*$ where $\Z[\lambda]^*$ is the dual module of $\Z[\lambda]$.

We consider the w$^*$-closed operator algebra $\cA_+$ generated
by the image of the semigroup $SL_2(\R_+)$ under a unitary representation
$\rho$ of $SL_2(\R)$ on the Hilbert~space $L_2(\R)$.
We show that $\cA_+$ is a reflexive operator algebra and
$\cA_+=\Alg\cD$ where $\cD$ is a double triangle subspace
lattice. Surprisingly, $\cA_+$ is also generated as a
w$^*$-closed algebra by the image under $\rho$ of a strict
subsemigroup of $SL_2(\R_+)$.

Given a finite group $G$, we attach to the character degrees
of $G$ a graph whose vertex set is the set of primes dividing the degrees of
irreducible characters of $G$, and with an edge between $p$ and $q$ if
$pq$ divides the degree of some irreducible character of $G$.
In this paper, we describe which graphs occur when $G$ is
a solvable group of Fitting height $2$.

In this paper we give a characterization of real hypersurfaces of type $A$ in a complex
two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which are tubes over totally geodesic
$G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$ in terms of the {\it vanishing Lie
derivative\/} of the shape operator $A$ along the direction of the Reeb vector field $\xi$.

Given a non-negative, locally integrable function $V$
on $\RR^n$, we give a necessary and sufficient condition
that $-\Delta+V$ have purely discrete spectrum,
in terms of the scattering length
of $V$ restricted to boxes.

We investigate the geometry of manifolds with bounded Ricci
curvature in $L^1$-sense. In particular, we generalize the
classical volume comparison theorem to our situation and obtain a
generalized sphere theorem.

We give in this note a weighted version of Brianchon and Gram's
decomposition for a simple polytope. We can derive from this
decomposition the weighted polar formula of Agapito and a weighted
version of Brion's theorem, in a manner similar to Haase, where the
unweighted case is worked out. This weighted version of Brianchon
and Gram' decomposition
is a direct consequence of the ordinary Brianchon--Gram formula.

This paper investigates the relationship between a system of
differential equations and the underlying geometry associated with
it. The geometry of a surface determines shortest paths, or
geodesics connecting nearby points, which are defined as the
solutions to a pair of second-order differential equations: the
Euler--Lagrange equations of the metric. We ask when the converse
holds, that is, when solutions to a system of differential
equations reveals an underlying geometry. Specifically, when may
the solutions to a given pair of second order ordinary
differential equations $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and
$d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by
$t\rightarrow T(y,t)$ so as to give locally the geodesics of a
Euclidean space? Our approach is based upon Cartan's method of
equivalence. In the second part of the paper, the equivalence
problem is solved for a generic pair of second order ordinary
differential equations of the above form revealing the existence
of 24 invariant functions.

Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the Rogers--Shephard inequality we obtain that
$\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a cross-polytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified Banach--Mazur distance to the
regular hexagon.

We give a short proof of Totaro's theorem that every$E_8$-torsor over
a field $k$ becomes trivial over a finiteseparable extension of $k$of
degree dividing $d(E_8)=2^63^25$.

It is shown that the ergodic Hilbert transform
exists for a class of bounded symmetric admissible processes
relative to invertible measure preserving transformations. This
generalizes the well-known result on the existence of the ergodic
Hilbert transform.

An invariant is presented which classifies, up to
equivariant isomorphism, $C^*$-dynamical systems arising as limits from
inductive systems of elementary $C^*$-algebras on which the Euclidean
motion group acts by way of unitary representations that decompose into
finite direct sums of irreducibles.

A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian
for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence
of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a
corollary we prove: If the first four distinct eigenvalues have even multiplicities
then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This
leads to generalizations of some classical results in the theory of surfaces.

Continuous mappings defined from compact subsets $K$ of complex
Euclidean space $\cc^n$ into complex projective space $\pp^m$ are
approximated by rational mappings. The fundamental tool employed
is homotopy theory.

Let $X$ be a Polish space.
We will prove that
$$
\dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to
} X\},
$$
where $\dim_L X$ and $\dim_T X$ stand
for the concentration dimension and
the topological dimension of $X$, respectively.

A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) Bochnak--Siciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.

If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in
C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$-clean if every element is
the sum of a unit and a root of $g(x)$. If $V$ is a vector space of
countable dimension over a division ring $D,$ they showed that
$\end {}_{D}V$ is
$g(x)$-clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=x-x^{2}$
this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan.
In this paper we remove the countable condition, and in fact prove that
$\Mend {}_{R}M$ is $g(x)$-clean for any semisimple module $M$ over an arbitrary
ring $R$ provided that $g(x)\in (x-a)(x-b)C[x]$ where $a,b\in C$ and both $b$
and $b-a$ are units in $R$.

A well-known theorem states that if $f(z)$ generates a PF$_r$
sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two
counterexamples
which show that this is not true, and give a correct version of the theorem.
In the infinite limit the result is sound: if $f(z)$ generates a PF
sequence then $1/f(-z)$ generates a PF sequence.

We extend a result of Noritzsch,
which describes the orbit sizes in the action of a
Frobenius group $G$ on a finite vector space $V$ under
certain conditions, to a more general class of finite
solvable groups $G$.
This result has applications in computing
irreducible character degrees of finite groups.
Another application, proved here, is a result
concerning the structure of certain groups with
few complex irreducible character degrees.

This paper investigates the modularity of three
non-rigid Calabi--Yau threefolds with bad reduction at 11. They are
constructed as fibre products of rational elliptic surfaces,
involving the modular elliptic surface of level 5. Their middle
$\ell$-adic cohomology groups are shown to split into
two-dimensional pieces, all but one of which can be interpreted in
terms of elliptic curves. The remaining pieces are associated to
newforms of weight 4 and level 22 or 55, respectively. For this
purpose, we develop a method by Serre to compare the corresponding
two-dimensional 2-adic Galois representations with uneven trace.
Eventually this method is also applied to a self fibre product of
the Hesse-pencil, relating it to a newform of weight 4 and level
27.

We prove a self-adjoint analogue of the Marcus--Pisier inequality, comparing the
expected value of convex functionals on randomreflection matrices and on elements of
the Gaussian orthogonal (or unitary) ensemble.

We show that given a stable weighted configuration on the asymptotic
boundary of a
locally compact Hadamard space, there is a polygon with Gauss
map prescribed by the given weighted configuration.
Moreover, the same result holds for
semistable configurations on arbitrary Euclidean buildings.

It is shown that the group of compactly
supported, measure-preserving homeomorphisms of a
connected, second countable manifold is locally contractible in the direct limit topology.
Furthermore, this group is weakly homotopically equivalent to the more general group of
compactly supported homeomorphisms.

In this paper we study affine completeness of generalised dihedral
groups. We give a formula for the number of unary compatible
functions on these groups, and we characterise for every $k \in~\N$
the $k$-affine complete generalised dihedral groups. We find that
the direct product of a $1$-affine complete group with itself need not
be $1$-affine complete. Finally, we give an example of a nonabelian
solvable affine complete group. For nilpotent groups we find a
strong necessary condition for $2$-affine completeness.

We show that each point of the principal eigencurve of the
nonlinear problem
$$
-\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad
\text{in } \Omega,
$$
is stable (continuous) with respect to the exponent $p$ varying in
$(1,\infty)$; we also prove some convergence results
of the principal eigenfunctions corresponding.

This paper is concerned with the structure of
inner $E_0$-semigroups. We show that any inner
$E_0$-semigroup acting on an infinite factor
$M$ is completely determined by a continuous
tensor product system of Hilbert spaces in
$M$ and that the product system associated
with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.

It is known that the derivative of
a Blaschke product whose zero sequence lies in a Stolz angle
belongs to all the Bergman spaces $A^p$ with $0<p<3/2$.
The question
of whether this result is best possible remained open.
In this paper,
for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain
a number of conditions which are equivalent to the membership of $B'$ in the
space $A^p$ ($p>1$). As a consequence,
we prove that there exists a Blaschke product $B$
with zeros on a radius such that $B'\notin A^{3/2}$.

Free analogues of the logarithmic Sobolev inequality compare the relative
free Fisher information with the relative free entropy. In the present paper
such an inequality is obtained for measures on the circle. The method is
based on a random matrix approximation procedure, and a large deviation
result concerning the eigenvalue distribution of special unitary matrices is
applied and discussed.

This note shows that any set of cofibrations containing the standard
set of generating projective cofibrations determines a cofibrantly
generated proper closed model structure on the category of simplicial
presheaves on a small Grothendieck site, for which the weak
equivalences are the local weak equivalences in the usual sense.

We construct vector-valued modular forms of weight 2 associated to
Jacobi-like forms with respect to a symmetric tensor representation of
$\G$ by using the method of Kuga and Shimura as well as the
correspondence between Jacobi-like forms and sequences of modular forms.
As an application, we obtain vector-valued modular forms determined by
theta functions and by pseudodifferential operators.

We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal
if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed
that Littlewood polynomials with particularly high minimum modulus on
the unit
circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a
skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle,
as well as providing a new proof of the known result that a self-reciprocal
Littlewood polynomial must have a zero on the unit circle.

In this paper, we find a lower bound on the number of cyclic function
fields of prime degree~$l$ whose class numbers are divisible by a
given
integer $n$. This generalizes a previous result of D. Cardon and R.
Murty
which gives a lower bound on the number of quadratic function fields
with
class numbers divisible by $n$.

The purpose of this note is to show that the homologically trivial
cycles contructed by Clemens and their generalisations
due to Paranjape can be detected by the technique of
spreading out. More precisely, we associate to these cycles (and the
ambient varieties in which they live) certain families which arise
naturally and which are defined over $\bbC$ and show that these
cycles, along with their relations, can be detected in the singular
cohomology of the total space of these families.

The aim of this paper is to study sequences of integers
for which the second differences between their squares are
constant. We show that there are infinitely many nontrivial
monotone sextuples having this property and discuss some related
problems.

By applying the Cayley--Dickson process to the division algebra
of real octonions, one obtains a 16-dimensional real algebra
known as (real) sedenions. We denote this algebra by $\bA_4$.
It is a flexible quadratic algebra (with unit element 1) but
not a division algebra.

We classify the subalgebras of $\bA_4$ up to conjugacy (i.e.,
up to the action of the automorphism group $G$ of $\bA_4$)
with one exception: we leave aside the more complicated case
of classifying the quaternion subalgebras.
Any nonzero subalgebra contains 1 and we show that there are
no proper subalgebras of dimension 5, 7 or $>8$.
The proper non-division subalgebras have dimensions
3, 6 and 8. We show that in each of these dimensions
there is exactly one conjugacy class of such subalgebras.
There are infinitely many conjugacy classes of subalgebras in
dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.

We consider the growth norm of a measurable function $f$ defined by
$$\|f\|_{-\sigma}=\ess\{\delta_D(z)^\sigma|f(z)|:z\in D\},$$
where $\delta_D(z)$ denote the distance from $z$ to $\partial D$.
We prove some optimal growth norm estimates for $\bar\partial$
on convex domains of finite type.

Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero
and $\Gamma$ be a subgroup of $\Gamma_0$
of finite index of genus zero. We find universal recursive
relations giving the $q_{r}$-series coefficients of
$j_0$ by using those of the $q_{h_{s}}$-series of $j$, where $j$ is
the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul
for $\Gamma_0$ without zeros on the complex upper half plane
$\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for
$q$-series coefficients of any modular form on
$\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.

We prove that convex sets are measure convex and extremal sets are measure extremal
provided they are of low Borel complexity. We also present
examples showing that the positive results cannot be strengthened.

This paper studies Hausdorff--Young inequalities for certain group extensions,
by use of Mackey's theory. We consider the case in which the dual
action of the quotient group is free almost everywhere. This
result applies in particular to yield a Hausdorff--Young inequality for
nonunimodular groups.

In this article we will show that there are infinitely many
symmetric, integral $3 \times 3$ matrices, with zeros on the
diagonal, whose eigenvalues are all integral. We will do this by
proving that the rational points on a certain non-Kummer, singular
K3 surface
are dense. We will also compute the entire N\'eron--Severi group of
this surface and find all low degree curves on it.

In this paper we describe six pencils of $K3$-surfaces which have
large Picard number ($\rho=19,20$) and each contains precisely five
special fibers: four have A-D-E singularities and one is
non-reduced. In particular, we characterize these surfaces as cyclic
coverings of some $K3$-surfaces described in a recent paper by Barth
and the author.
In many cases, using
3-divisible sets, resp., 2-divisible sets, of rational curves and
lattice theory, we describe explicitly the Picard lattices.

Using the Polyak--Viro Gauss diagram formula for
the degree-4 Vassiliev invariant, we
extend some previous results on positive knots and
the non-triviality of the Jones polynomial
of untwisted Whitehead doubles.

For a non-trivial knot in the $3$-sphere,
only integral Dehn surgery can create a closed $3$-manifold containing a projective plane.
If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true.
In contrast to these, we show that non-integral surgery on a hyperbolic knot
can create a closed non-orientable surface of any genus greater than two.

Considering a mapping $g$ holomorphic on a neighbourhood of a rationally
convex set $K\subset\cc^n$, and range into the complex projective space
$\cc\pp^m$, the main objective of this paper is to show that we can
uniformly approximate $g$ on $K$ by rational mappings defined from
$\cc^n$ into $\cc\pp^m$. We only need to ask that the second \v{C}ech
cohomology group $\check{H}^2(K,\zz)$ vanishes.