Le problème de l'enseignement des mesures des grandeurs géométriques à partir de l'exemple des aires

Nous tentons dans cet article à la fois d'élucider un certain nombre de points théoriques concernant la mesure de ces grandeurs et de proposer quelques réflexions didactiques aussi bien sur les difficultés que doivent surmonter les élèves que sur les problèmes qu'on veut les amener à pouvoir résoudre. Nous avons plus particulièrement étudié le problème de la mesure des aires, c'est pourquoi notre réflexion sera plus approfondie dans ce cas que dans celui des volumes

A weak equivalence between shifts of finite type

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A quadratic algorithm for road coloring

The Road Coloring Theorem states that every aperiodic directed graph with constant out-degree has a synchronized coloring. This theorem had been conjectured during many years as the Road Coloring Problem before being settled by A. Trahtman. Trahtman's proof leads to an algorithm that finds a synchronized labeling with a cubic worst-case time complexity. We show a variant of his construction with a worst-case complexity which is quadratic in time and linear in space. We also extend the Road Coloring Theorem to the periodic case

Application of non destructive testing to the detection of aeronautical defects in composite structures

A study of two Non-destructive Testing methods (NDT) was carried out in specimens with different kinds of simulated defects. Ultrasonics test (US) and Infrared Thermography (IRT) were applied with the aim to evaluate the detectability and the accuracy of each method.These techniques have acquired great importance in the aeronautics industry because they allow to control the aerostructures without intervening in their physical and mechanical integrity. In the second part of the study, a comparison of both techniques was achieved in order toanalyse their limits and advantages. It appeared that detectability of defects was much better in a sample with flat-bottomed holes defects in the case of Ultrasonic Test. However it was found that Infrared Thermography is much more limited to the thickness of the specimen than the ultrasonic waves. On the other hand, defects were all revealed with IRT in a sandwich composite including Teflon inserts, which was not the case for US

The syntactic graph of a sofic shift is invariant under shift equivalence

International audienceWe de ne a new invariant for shift equivalence of so fic shifts. This invariant, that we call the syntactic graph of a so fic shift, is the directed acyclic graph of characteristic groups of the non null regular D-classes of the syntactic semigroup of the shift

A hierarchy of irreducible sofic shifts

International audienceWe define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts