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.

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.

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.

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 Calderón–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.

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.

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 ${{\text{A}}_{4}}$. It is a flexible quadratic algebra (with unit element 1) but not a division algebra.

We classify the subalgebras of ${{\text{A}}_{4}}$ up to conjugacy (i.e., up to the action of the automorphism group $G$ of ${{\text{A}}_{4}}$) 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.

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.

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 $\text{case }K=k\left( t \right)$ and $K=k\left( \left( t \right) \right)$. 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 Dedekind 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.

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 $k\in \mathbb{N}$ 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.

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 consider the growth norm of a measurable function $f$ defined by

$${{\left\| f \right\|}_{-\sigma }}=\text{ess}\,\,\text{sup}\left\{ {{\delta }_{D}}{{\left( z \right)}^{\sigma }}\left| f\left( z \right) \right|:z\in D \right\},$$

where ${{\delta }_{D}}\left( z \right)$ denote the distance from $z$ to $\partial D$. We prove some optimal growth norm estimates for $\bar{\partial }$ on convex domains of finite type.

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.

We show that each point of the principal eigencurve of the nonlinear problem

$$-{{\Delta }_{p}}u-\text{ }\lambda m(x){{\left| u \right|}^{p-2}}u=\mu {{\left| u \right|}^{p-2}}u\,\,\text{in}\Omega ,$$

is stable (continuous) with respect to the exponent $p$ varying in $\left( 1,\infty \right)$; we also prove some convergence results of the principal eigenfunctions corresponding.

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 $d\left( {{E}_{g}} \right)={{2}^{6}}{{3}^{2}}5$.

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 $\mathfrak{H}\left( \text{here}\,\,{{q}_{\ell }}\,:=\,{{e}^{2\pi iz/\ell }} \right)$. We find universal recursive formulas for $q$-series coefficients of any modular form on $\Gamma _{0}^{+}\left( p \right)$ in terms of those of the canonical Hauptmodul $j_{p}^{+}$.

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.

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.

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”.

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.