A Survey on Galois Stratifications and Measures of Viability Risk

J.-P. Aubin ; O. Dordan

Journal of convex analysis, Tome 23 (2016) no. 1, pp. 181-225

Résumé

The hyperspace of an underlying space is the family of its subsets and hypermaps are maps from one hyperspace to another one. Set-valued maps generate some of them, either "pointwise", as image and focus hypermaps and their inverses, or indirectly, as "viability kernels" of subsets under a differential inclusion, and several other viability concepts (capture basins, invariant kernels and absorption basins, and many combinations of them). When the underlying space is itself the product of two spaces, the hyperspace is the space of graphs of set-valued maps, or binary relations. When the arrival space is the space of real numbers, the hyperspace of the product contains the graphs, epigraphs and hypographs of numerical functions, so that hypermaps actually operate on functions and provide adequate "solutions" of first-order partial differential equations, the graph of which are capture basins, for instance. When the underlying space is the product of more than two spaces, it contains the graphs of "relations": they pop-up whenever it is impossible to partition a group of variables in only two classes, the inputs and the outputs.

Détail

Classification : 34G25, 34A60, 49J27, 49J53, 93B03, 37B55, 28B20, 45N05
Mots-clés : hypermap, focus, mutational analysis, Galois stratification, Galois quotation, Galois transform, Galois measure, atypical logic, mathematical morphology, society, cultures, economic classes, dynamic economic behavior