Rapports de stage
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Rapports de stage
The purpose of these notes is to present a motivating simple case of a joint project with Y Guivarc'h and J Ph Anker The project itself concerns the compactification of Bruhat Tits buildings that is the geometries naturally attached to semisimple groups over locally compact non Archimedean local fields We deal with the geometric the measure theoretic due to Furstenberg the group theoretic due to Guivarc'h and the polyhedral compactifications The point is to define them to identify them and then to use them to parametrize closed subgroups of dynamical interest Our guideline is the case of symmetric spaces
12 pages
English
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Rapports de stage
THEOREMS AS CONSTRUCTIVE VISIONS1 Giuseppe Longo CNRS Ecole Normale Supérieure et CREA Ecole Polytechnique Rue D'Ulm Paris France http: www di ens fr users longo Abstract This paper briefly reviews some epistemological perspectives on the foundation of mathematical concepts and proofs It provides examples of axioms and proofs from Euclid to recent “concrete incompleteness” theorems In reference to basic cognitive phenomena the paper focuses on order and symmetries as core “construction principles” for mathematical knowledge A distinction is then made between these principles and the “proof principles” of modern Mathemaical Logic The role of the blend of these different forms of founding principles will be stressed both for the purposes of proving and of understanding and communicating the proof THE CONSTRUCTIVE CONTENT OF EUCLID'S AXIOMS From the time of Euclid to the age of super computers Western mathematicians have continually tried to develop and refine the foundations of proof and proving Many of these attempts have been based on analyses logically and historically linked to the prevailing philosophical notions of the day However they have all exhibited more or less explcitly some basic cognitive principles for example the notions of symmetry and order Here I trace some of the major steps in the evolution of notion of proof linking them to these cognitive basics For this purpose let's take as a starting point Euclid's Aithemata Requests the minimal constructions required to do geometry: Invited lecture ICMI conference on Proof and Proving Taipei Taiwan May Hanna de Villiers eds Springer
19 pages
English
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Rapports de stage
The evolution of post communist countries: An interpretation from the perspective of cooperation Dimiter Ialnazov Kyoto University kyoto u ac jp Nikolay Nenovsky University of National and World Economy Sofia LEO STUDIUM and ICER1 com Summary: The basic hypothesis we present is the following: both the transition phases and the diversity of trajectories of post communist countries are the result of the difference in prevailing strategic behaviours This difference determines the models of cooperation namely the two main archetypes that of the prisoner's dilemma and the stag hunter applied at the socioeconomic context of post communist transition These two archetypal societal models are on their part conditional on the existence of external and internal anchors While with the prisoner's dilemma that became a model of total exchange under the conditions of high social heterogeneity and broken informational channels it is profitable not to cooperate under the stag hunter model a model involving a common goal a common project advantageous in general are cooperative strategies The various countries in different phases can be approximated to either one or the other game the prisoner's dilemma or the stag hunter A shift to the cooperative game becomes possible as a result of the operation of internal or external anchors For instance in the beginning of the transition with no clear vision in sight amid an outburst of large diversity of economic and social actors and old system's information channels falling to pieces the appropriate analytical model to apply would be the prisoner's dilemma And vice versa later on especially when a decision was adopted to join the EU and with pre accession chapters being opened and closed i e an external anchor emerging appropriate for analytical reasoning becomes the stag hunter game model Résume: Les trajectoires des pays ex communistes affichent une diversité grandissante Dans ce papier nous nous proposons de le reconsidérer par la théorie de la coopération sociale Nous interprétons ...
15 pages
English
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