On sait depuis les travaux de Bricard et de Connelly qu’il existe dans l’espace euclidien des polyèdres (non convexes) qui sont flexibles : on peut les déformer continûment sans changer la forme de leurs faces. La conjecture des soufflets affirme que le volume interieur de ces polyèdres est constant au cours de la déformation. Elle a été démontrée récemment par I. Sabitov, qui a pour cela utilisé des outils algébriques inattendus dans ce contexte.

This paper gives a shortest path algorithm for CFG (context free grammar) labeled and weighted digraphs where edge weights may be positive or negative, but negative-weight cycles are not allowed in the underlying unlabeled graph. These results build directly on an algorithm of Barrett et al. [SIAM J. Comput.30 (2000) 809–837]. In addition to many other results, they gave a shortest path algorithm for CFG labeled and weighted digraphs where all edges are nonnegative. Our algorithm is based closely...

A generalized s-star, s ≥ 1, is a tree with a root Z of degree s; all other vertices have degree ≤ 2. $S_i$ denotes a generalized 3-star, all three maximal paths starting in Z have exactly i+1 vertices (including Z). Let be a surface of Euler characteristic χ() ≤ 0, and m():= ⎣(5 + √49-24χ( ))/2⎦. We prove: (1) Let k ≥ 1, d ≥ m() be integers. Each polyhedral map G on with a k-path (on k vertices) contains a k-path of maximum degree ≤ d in G or a generalized s-star T, s ≤ m(), on d + 2- m() vertices...

Let $P_k$ be a path on $k$ vertices. In an earlier paper we have proved that each polyhedral map $G$ on any compact $2$-manifold $𝕄$ with Euler characteristic $\chi \left(𝕄\right)\le 0$ contains a path $P_k$ such that each vertex of this path has, in $G$, degree $\le k⌊\frac{5+\sqrt{49-24\chi \left(𝕄\right)}}{2}⌋$. Moreover, this bound is attained for $k=1$ or $k\ge 2$, $k$ even. In this paper we prove that for each odd $k\ge \frac{4}{3}⌊\frac{5+\sqrt{49-24\chi \left(𝕄\right)}}{2}⌋+1$, this bound is the best possible on infinitely many compact $2$-manifolds, but on infinitely many other compact $2$-manifolds the upper bound can be lowered to $⌊(k-\frac{1}{3})\frac{5+\sqrt{49-24\chi \left(𝕄\right)}}{2}⌋$.

Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation $f={\sum }_{C\in }{a}_C{\phi }_C$ with $a_C\in E$ and a partition of unity ${\phi }_C:C\in$ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction ${f|}_P$ attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space...