Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\left\lceil 2n/\left( n-d+2 \right) \right\rceil -1$ flips for all $n\ge d+2\ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension $d\ge 3$. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included.

Soient $F$ un corps commutatif localement compact non archimédien et $\psi$ un caractère additif non trivial de $F$. Soient $n$ et ${n}'$ deux entiers distincts, supérieurs à 1. Soient $\pi$ et ${\pi }'$ des représentations irréductibles supercuspidales de $\text{G}{{\text{L}}_{n}}\left( F \right)$, $\text{G}{{\text{L}}_{{{n}'}}}\left( F \right)$ respectivement. Nous prouvons qu’il existe un élément $c=c\left( \pi ,{\pi }',\psi \right)$ de ${{F}^{\times }}$ tel que pour tout quasicaractère modéré $\mathcal{X}$ de ${{F}^{\times }}$ on ait $\mathcal{E}\left( \chi \pi \times {\pi }',s,\psi \right)=\chi {{\left( c \right)}^{-1}}\mathcal{E}\left( \pi \times {\pi }',s,\psi \right)$. Nous examinons aussi certains cas où $n={n}',{\pi }'={{\pi }^{\text{v}}}$. Les résultats obtenus forment une étape vers une démonstration de la conjecture de Langlands pour $F$, qui ne fasse pas appel à la géométrie des variétés modulaires, de Shimura ou de Drinfeld.

We prove a strong variant of the Borwein-Preiss variational principle, and show that on Asplund spaces, Stegall's variational principle follows from it via a generalized Smulyan test. Applications are discussed.

We provide the reader with a uniform approach for obtaining various useful explicit upper bounds on residues of Dedekind zeta functions of numbers fields and on absolute values of values at $s=1$ of $L$-series associated with primitive characters on ray class groups of number fields. To make it quite clear to the reader how useful such bounds are when dealing with class number problems for CM-fields, we deduce an upper bound for the root discriminants of the normal CM-fields with (relative) class number one.

We show that the Elliott invariant is a classifying invariant for the class of ${{C}^{*}}$-algebras that are simple unital infinite dimensional inductive limits of finite direct sums of building blocks of the form

$$\left\{ f\in C\left( \mathbb{T} \right)\otimes {{M}_{n}}:f\left( {{x}_{i}} \right)\in {{M}_{{{d}_{i}}}},i=1,2,...,N \right\},$$

where ${{x}_{1}},\,{{x}_{2.}},...,{{x}_{N\,}}\in \mathbb{T},{{d}_{1}},{{d}_{2}},.\,.\,.,{{d}_{N}}$ are integers dividing $n$, and ${{M}_{{{d}_{i}}}}$ is embedded unitally into ${{M}_{n}}$. Furthermore we prove existence and uniqueness theorems for $*$-homomorphisms between such algebras and we identify the range of the invariant.

A theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact Kähler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.