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Philosophie, Méthodologie Et Applications De L'analyse Non Philosophie, méthodologie et applications de l’analyse non standard Augustin Fruchard, Veronique Gautheron, Tewfik Sari To cite this version: Augustin Fruchard, Veronique Gautheron, Tewfik Sari. Philosophie, méthodologie et applications de l’analyse non standard. Colloque à la mémoire de E. Isambert, Dec 2007, NA, France. Universite Paris 13, 2012, Publications de l’Université de Paris 13. hal-01190319v2 HAL Id: hal-01190319 https://hal.archives-ouvertes.fr/hal-01190319v2 Submitted on 8 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Actes du colloque à la mémoire d’Emmanuel Isambert 21-22 décembre 2007 , Université de Paris 7 Philoso phie, méthodolo g ie et applications de l’analyse non standard Éditeurs : Augustin Fruchard Véronique Gautheron Tewfik Sari 2010 Mathematics Subject Classification: 01, 03, 28, 34, 35, 39, 41, 49, 60, 92, 93. Publications de l'Université de Paris 13 Emmanuel Isambert était Professeur à l'université de Paris 13, lorsqu'il est décédé brutalement en septembre 2007. Logicien et mathématicien, spécialiste d'équations différentielles, il était un membre très actif du réseau Georges Reeb. La rencontre du réseau de décembre 2007 était dédiée à sa mémoire. La diversité des thèmes abordés dans ces Actes, philosophie, logique et théor ie des ensem bles, ma théma tiques financ ières, pro ba bilités e t équations différentielles, histoire des mathématiques, témoigne de l'ouverture d'esprit d'Emmanuel. http://www.math.univ-paris13.fr/laga/ Pr´eface Emmanuel Isambert allait sur ses soixante ans au moment de sa disparition brutale en septembre 2007. Apr`es une formation et une th`ese de logique, puis une assez br`eve incursion dans les espaces de Banach, un probl`eme de m´ecanique des solides l’a orient´evers l’utilisation de l’analyse non standard dans l’´etude des ´equations diff´erentielles ordinaires. Il a en particulier ´etudi´ede pr`es les fleuves, qui sont des trajectoires de champs de vecteurs o`use concentrent d’autres trajectoires, en collaboration avec Mich`ele Artigue et V´eronique Gautheron. Il a aussi ´etudi´eles canards d’´equations non diff´erentiables et explor´e les fondements de l’analyse non standard. Emmanuel ´etait un pilier du r´eseau Georges Reeb, participant activement aux rencontres, ayant toujours des questions pertinentes `aposer `ala fin des expos´es, avec une belle ´energie et sa bonne humeur inoxydable. La rencontre du r´eseau qui a eu lieu `aParis Chevaleret en d´ecembre 2007 ´etait d´edi´ee `a sa m´emoire. Les Actes contiennent d’une part la version ´ecrite – parfois largement compl´et´ee – des expos´es et d’autre part des contributions nouvelles. La diversit´edes th`emes abord´es dans ces Actes, allant de questions philosophiques, logiques ou de th´eorie des ensembles, aux math´ematiques financi`eres, probabilit´es et ´equations diff´erentielles, en passant par des th`emes relevant de l’histoire des math´ematiques, t´emoigne de l’ouverture d’esprit d’Emmanuel. Emmanuel est arriv´edans la toute jeune Universit´eParis 13 en 1971. Il y a toujours pris `a cœur son rˆole d’enseignant. El´ement stable et fiable de l’´equipe, il ´etait volontiers prˆet `afaire diff´eremment pour faire mieux. En particulier il a saisi toutes les opportunit´es d’introduire et d’exp´erimenter l’outil informatique dans les formations qui se sont ouvertes, comme en t´emoignent (page vii) deux de ses coll`egues du d´epartement de math´ematiques. Emmanuel venait de se remarier au printemps 2007, quelques mois avant son d´ec`es, et respirait le bonheur, empli de projets d’avenir, comme le montrent les photos de premi`ere page prises le jour de ce mariage. Deux autres passions l’habitaient : la musique comme hautbo¨ıste d’orchestre, puis choriste d’op´erette, et la montagne, qu’il a escalad´ee sur tous les continents. Sa fille, physicienne, n´ee d’une vie ant´erieure, venait de lui annoncer qu’il allait ˆetre grand- p`ere, mais il n’a malheureusement pas pu voir la petite Cl´emence. Les ´editeurs i Liste des participants et contributeurs Mich`ele Artigue (Paris) ................. [email protected] Rachid Bebbouchi (Alger) ......................... [email protected] Eric´ Benoˆıt (La Rochelle) ............................. [email protected] Imme van den Berg ( Evora)´ ................................ [email protected] Gilles Bernot (Sophia-Antipolis) . [email protected] Jacques Bosgiraud (Paris) . [email protected] Bernard Brighi (Mulhouse) ........................... [email protected] Agathe Chollet (La Rochelle) ............................ [email protected] Jean-Paul Comet (Sophia-Antipolis) . [email protected] Ren´eCori (Paris) .................................... [email protected] Aparna Das (Nice) ................................... [email protected] Antoine Delcroix (Pointe-`a-Pitre) ....................... [email protected] Francine Diener (Nice) ........................... [email protected] Marc Diener (Nice) .................................... ..... [email protected] Fr´ed´eric Eyssette (Sophia-Antipolis) .............. [email protected] Thomas Forget (La Rochelle) ............................. [email protected] Augustin Fruchard (Mulhouse) .................... [email protected] V´eronique Gautheron (Paris) ............................ [email protected] R´emi Goblot (Lille) ................................. [email protected] Luis Gonzaga Albuquerque (Lisbonne) . [email protected] Claude Lobry (Nice) .................................. [email protected] Robert Lutz (Mulhouse) ................................. [email protected] Gwenola Madec (Paris) ......................... [email protected] Jean-Andr´eMarti (Pointe-`aPitre) . [email protected] Fran¸cois Parreau (Paris) . [email protected] Pheakdei Mauk (Nice) .................................... [email protected] Yves P´eraire (Clermont-Ferrand) ..................... [email protected] Yvette Perrin (Clermont-Ferrand) .................. [email protected] Jean-Michel Salanskis (Paris) ............................ [email protected] Nadir Sari (La Rochelle) .................................. [email protected] Tewfik Sari (Mulhouse) .................................. tewfi[email protected] Dimitris Scarpalezos (Paris) ......................... [email protected] Reinhard Sch¨afke (Strasbourg) . [email protected] Jean-Marie Strelcyn (Paris-Villetaneuse) . [email protected] Guy Wallet (La Rochelle) ............................ [email protected] ii Table des mati`eres – Contents Gwenola Madec et Fran¸cois Parreau T´emoignages........................................... .............................. v V´eronique Gautheron Tests, rigueur et fantaisie.............................. .............................. 1 Mich`ele Artigue Souvenirs........................................... ................................. 3 Yvette Perrin Les vissicitudes d’un raisonnement par analogie . ..................... 5 Robert Lutz Le concept m´etaphysique de relations dynamique . .................... 13 Guy Wallet Les entiers naturels en th´eorie constructive des types . ......................... 17 Yves P´eraire Contextual approach of automatic deduction theory. Application to analysis. ........................... 29 Imme van den Berg Functions of limited accumulation. ......................... 39 Imme van den Berg Discretizations of higher order . .......................... 63 Jacques Bosgiraud Moderate deviations in Rk ................................................... 89 Jacques Bosgiraud Des lois log-normales presque normales. ...................... 97 Marc Diener, Pheakdei Mauk On the implicite interest in the Yunus equation............. ....................... 101 Bernard Brighi, Augustin Fruchard, Tewfik Sari The Blasius equation.................................... .......................... 105 Augustin Fruchard, Reinhard Sch¨afke De nouveaux d´eveloppements asymptotiques combin´es pour la perturbation singuli`ere.......................... ........................... 125 Claude Lobry, Tewfik Sari La mod´elisation de la persistance en ´ecologie. ...................... 163 Francine Diener, Aparna Das, Gilles Bernot, Jean-Paul Comet, Fr´ed´eric Eyssette Correspondence between discrete and piecewise linear models of gene regulatory networks . ....................... 185 iii iv T´emoignages Gwenola Madec et Fran¸cois Parreau Fran¸cois : D`es le d´ebut des ann´ees 80, Emmanuel avait compris que l’outil informatique ouvrait un champ consid´erable de possibilit´es pour l’enseignement des math´ematiques. Il existait alors tr`es peu, sinon pas du tout, de ressources disponibles et tout ´etait `ainventer dans ce domaine. Avec un petit groupe de coll`egues motiv´es, nous avons appris ensemble `aprogrammer et, sous l’impulsion de Jean-Fran¸cois M´ela, nous avons mis au point un diplˆome d’universit´e “Math´ematiques pour non-sp´ecialistes”. Le but ´etait d’aborder un certain nombre de notions `apartir de l’exp´erimentation sur machine et sans grands pr´erequis.
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