A model of professional competences in mathematics to update mathematical and didactic knowledge of teachers

This paper describes part of a research and development project carried out in public elementary schools. Its objective was to update the mathematical and didactic knowledge of teachers in two consecutive levels in urban and rural public schools of Region de Los Lagos and Region de Los Rios of southern Chile. To that effect, and by means of an advanced training project based on a professional competences model, didactic interventions based on types of problems and types of mathematical competences with analysis of contents and learning assessment were designed. The teachers’ competence regarding the didactic strategy used and its results, as well as the students’ learning achievements are specified. The project made possible to validate a strategy of lifelong improvement in mathematics, based on the professional competences of teachers and their didactic transposition in the classroom, as an alternative to consolidate learning in areas considered vulnerable in two regions of the country.     

Graphes et ordonnés démontables, propriété de la clique fixe

This work extends to dismantable graphs many properties of dismantable posets dealing with products, exponentiation, existence of paths in a graph of all homomorphisms, fixed clique property, etc. We show that a poset is dismantlable in the sense of Rival if and only if its comparability graph is dismantlable, thus proving that, for posets, dismantlability is a comparability invariant. We establish the analogues for graphs of results of Duffus, Poguntke and Rival about fixed point sets and cores in posets.     

L'intégration graphique des équations différentielles ordinaires

In the period which precedes the appearance of computers, needs in calculation of the scientists and engineers led to an important development of graphic methods of integration. To contribute to the study of this little known phenomenon, the article presents techniques and instruments used for the graphic integration of ordinary differential equations, and looks for their historic origins by going back to the beginning of calculus: processes of geometric calculation by the polygonal method or the method of radius of curvature, use of tractional motion for the conception of integraphs, reduction to graphic quadratures in finite or infinite number.     

Approximation dans un Problème de Viscoélasticité Périodique

We are concerned with equations which derive from a quasistatic periodic problem of viscoelasticity. We give a condition which yields to existence and uniqueness of a periodic solution. Then we prove a finite element method based on equilibrium elements for the space approximation and on the explicit Euler scheme for the time approximation.     

Sur la connexité en géométrie analytique complexe locale

In [3] A. Grothendieck shows that a subscheme Y of a local and complete n-dimensional irreductible scheme, defined by p equations is connected in dimension n − p. This means that Y cannot be disconnected by removing a subspace of dimension < n − p. The aim of this paper is essentially to give a local analytic version of this theorem and a direct proof using only analytic geometry methods.     

Étude asymptotique des phénomènes acoustiques dans un mélange fluide-solide thermoélastique

Asymptotic study of the macroscopie behaviour of a fluid-thermoelastic solid mixture. The homogenization theory is applied to the study of small perturbations in a mixture of a thermoelastic solid and a viscous compressible fluid. There appear memory phenomena in the bulk behaviour as well as a modification in the initial data regarding the original problem. The convergence of the formal asymptotic expansion to the solution of the homogenized equations is proved.     

Opérateurs elliptiques sur les variétés non compactes

Soient M une variété, V un ouvert de M, et P un opérateur différentiel elliptique du second ordre, à coefficients C^∞ et réels tel que P1 0. Soit A_V l'opérateur induit par P dans l'espace de Banach C₀(V) des fonctions continues sur V nulles au point à l'infini de V, muni de la norme du suprémum. On démontre que A_V engendre un semi-groupe fortement continu à contraction ssi il existe K compact de V, h fonction continue strictement positive dans VβK et nulle au point à l'infini de V telle que (1 − P) h soit la distribution associée à une fonction continue non négative dans VβK. On en déduit immédiatement un résultat bien connu: si M est une variété de Cartan-Hadamard, A_M engendre un semi-groupe fortement continu à contraction dans C₀(M).Let M be a manifold, V an open set of M, and P an elliptic differential operator of the second order, with real C^∞ coefficients and such that P1 0. Let A_V be the operator induced by P in the complex Banach space C₀(V) of all continuous functions vanishing at the point at infinity of V, endowed with the supremum norm. One proves that A_V generates a strongly continuous contraction semi-group iff there exists K compact of V, h continuous strictly positive in VβK and 0 at infinity of V such that (1 − P) h is the distribution associated to a nonnegative continuous function in VβK. One deduces immediately from that a well-known result: if M is a Cartan-Hadamard manifold, A_M generates a strongly continuous contraction semigroup in C₀(M).     

Opérateurs différentiels linéaires à coefficients constants et schémas de Hilbert ponctuels

We use algebraic methods to study systems of linear partial differential equations with constant coefficients. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sur les Propriétés Arithmétiques d'une Fonction Entière

Let q ∈ Q, ∣q∣> 1 and In this paper, we study the arithmetical nature of the values of f at algebraic points. We prove that if α belongs to a quadratic extension ?? of ?, then f(α) ? ??.     

Sémantique algébrique ďun système logique basé sur un ensemble ordonné fini

In order to modelize the reasoning of an intelligent agent represented by a poset T, H. Rasiowa introduced logic systems called “Approximation Logics”. In these systems a set of constants constitutes a fundamental tool. In this papers, we consider logic systems called L′_T without this kind of constants but limited to the case where T is a finite poset. We prove a weak deduction theorem. We introduce also an algebraic semantics using Hey ting algebra with operators. To prove the completeness theorem of the L′_T system with respect to the algebraic semantics, we use the method of H. Rasiowa and R. Sikorski for first order logic. In the propositional case, a corollary allows us to assert that it is decidable to know “if a propositional formula is valid”. We study also certain relations between the L′_T logic and the intuitionistic and classical logics.     

Fonctions de partitions à parité périodique

Let be the set of positive integers and a subset of . For , let denote the number of partitions of n with parts in . In the paper J. Number Theory 73 (1998) 292, Nicolas et al. proved that, given any and , there is a unique set , such that is even for n>N. Soon after, Ben Saïd and Nicolas (Acta Arith. 106 (2003) 183) considered , and proved that for all k≥0, the sequence is periodic on n. In this paper, we generalise the above works for any formal power series f in with f(0)=1, by constructing a set such that the generating function of is congruent to f modulo 2, and by showing that if f=P/Q, where P and Q are in with P(0)=Q(0)=1, then for all k≥0 the sequence is periodic on n.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.