Sur des aspects algébriques et combinatoires de l'Analyse Relationnelle. : Applications en classification automatique, en théorie du choix social et en théorie des tresses.

Abstract : Relational Analysis is concerned with the analysis of binary relations and their applications in different mathematical fields. The approach presented here and which has been developped initially by J.F. Marcotorchino and P. Michaud is particular as it represents the binary relations as pairwise comparisons matrices, and as it is basically related to different tools from graph theory, statistics and linear programming. The most usual application domains of Relational Analysis are clustering and multicriteria decision making which are respectively based upon the analysis of equivalence and order relations. We have been interested in extending Relational Analysis to binary relations algebra, multi- valued logics, combinatorics and combinatorial optimization. These researches have given new results in the usual application domains which have been mentionned beforehand and also to some new extensions namely, the axiomatic of social choice theory and braids and knots theo- ries. Finally, this work presents the following contributions : • A new clustering algorithm for which the number of clusters is not required • A unified framework for some similarity indices between numerical vectors (which are valid for binary or qualitative or quantitative variables), some categorical variables asso- ciation criteria and ordered variables association criteria • An aggregation process called the “consensus principle”, which is based on a generalized majority rule and which is derived from the combinatoric formulas of H. Poincaré and Ch. Jordan • A generalization of the D. Black, K. Inada and A.K. Sen restriction conditions in social choice theory, which aims at defining sufficient and necessary conditions for the transiti- vity of the social choice obtained from several individual preferences • A new modelling approach for braids that enables to take into account generic isotopic movements of strings and to present algorithmic problems in braids and knots theory as combinatorial optimization problems
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M Julien Ah-Pine. Sur des aspects algébriques et combinatoires de l'Analyse Relationnelle. : Applications en classification automatique, en théorie du choix social et en théorie des tresses.. Mathématiques [math]. Université de Paris VI, Pierre et Marie Curie, 2007. Français. 〈tel-01515257〉

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