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.
