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's 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 ${{d}^{2}}{{y}^{1}}/d{{t}^{2}}=f\left( y,\dot{y},t \right)$ and ${{d}^{2}}{{y}^{2}}/d{{t}^{2}}=g\left( y,\dot{y},t \right)$ be reparameterized by $t\to T\left( y,t \right)$ 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 ${{\mathbb{E}}^{d}}$ , we denote by ${{\mathcal{M}}^{d}}\left( B \right)$ the Minkowski space (i.e., finite dimensional Banach space) with unit ball $B$ . Let $K$ be an arbitrary convex body in ${{\mathcal{M}}^{d}}\left( B \right)$ . The relationship between volume $V\left( K \right)$ and the Minkowskian thickness (= minimal width) ${{\Delta }_{B}}\left( K \right)$ of $K$ can naturally be given by the sharp geometric inequality $V\left( K \right)\ge \alpha \left( B \right)\cdot {{\Delta }_{B}}{{\left( K \right)}^{d}}$ , where $\alpha \left( B \right)>0$ . As a simple corollary of the Rogers-Shephard inequality we obtain that ${{\left( _{d}^{2d} \right)}^{-1}}\le \alpha \left( B \right)/V\left( B \right)\le {{2}^{-d}}$ 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 finite separable extension of $k$ of degree dividing $d\left( {{E}_{g}} \right)={{2}^{6}}{{3}^{2}}5$ .

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 ( ${{\mathbb{R}}^{3}}$ , can). As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in ( ${{\mathbb{R}}^{3}}$ , 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 ${{\mathbb{C}}^{n}}$ into complex projective space ${{\mathbb{P}}^{m}}$ are approximated by rational mappings. The fundamental tool employed is homotopy theory.

Let $X$ be a Polish space. We will prove that

$${{\dim}_{T}}X=\inf \left\{ {{\dim}_{L}}{X}'\,:\,{X}'\,\text{is homeomorphic to }X\, \right\},$$

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 $F\left( x,y \right)$ converges on a set of lines of positive capacity then $F\left( x,y \right)$ is convergent; (3) Abhyankar–Moh–Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

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

We give a characterization of products of projective spaces using unsplit covering families of rational curves.

A well-known theorem states that if $f\left( z \right)$ generates a $\text{P}{{\text{F}}_{r}}$ sequence then $1/f\left( -z \right)$ generates a $\text{P}{{\text{F}}_{r}}$ 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 $f\left( z \right)$ generates a $\text{PF}$ sequence then $1/f\left( -z \right)$ generates a $\text{PF}$ 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 random reflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.