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.