Formalismes non classiques pour le traitement informatique de la topologie et de la géométrie discrète

Abstract : The aim of this work is to introduce new theoretical basis for the discretization of continuous objects using non classical formalisms. This is done using a discrete model of the continuum called the Harthong-Reeb line together with the related arithmetization method which is a discretisation process of continuous functions. This study stands on a nonstandard arithmetical framework. Firstly, we use an axiomatic version of nonstandard arithmetic. In order to improve the constructive content of our method, the next step is to use another approach of nonstandard arithmetic deriving from the theory of Ω-numbers by Laugwitzand Schmieden. This second approach leads to a discrete multi-resolution representation of continuous functions. Afterwards, we investigate to what extent the Harthong-Reeb line fits Bridges axioms of the constructive continuum.
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Agathe Chollet. Formalismes non classiques pour le traitement informatique de la topologie et de la géométrie discrète. Mathématiques générales [math.GM]. Université de La Rochelle, 2010. Français. ⟨NNT : 2010LAROS315⟩. ⟨tel-00579781⟩

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