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Journal of Convex Analysis 23 (2016), No. 1, 181--225
Copyright Heldermann Verlag 2016

A Survey on Galois Stratifications and Measures of Viability Risk

Jean-Pierre Aubin
VIMADES, 14 rue Domat, 75005 Paris, France

Olivier Dordan
Institut de Mathématiques, Université de Bordeaux, 351 Cours de la Libération, 33405 Talence, France

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.
The framework of hyperspaces and hypermaps is useful for studying many problems at this level. Some general results are surveyed in the first section. The main examples concern the powers of hypermaps from one hyperspace into itself. Under adequate assumptions, each hypermap leads to a stratification of the underlying space, associating with each set a denumerable partition telling "how deep" an element belongs to this set or "how far" it does not belong to it. This is in this sense that we speak of "Galois measure of viability risk". Indeed, the concept of "Galois measure" allows not only to measure the localization of a given element in a subset of this partition, but to locate any other subsets in the stratification. The simplest example is the hypermap mapping any subset to its interior: the partition of each states it generates is made of its interior, its topological boundary and its exterior. Other hypermaps induce more sophisticated stratifications, such as the ones generated by viability concepts, the elements of which are the iterations of "viability algorithms". The analysis of many examples can be done at the level of hyperspaces and hypermaps or hyperrelations: we briefly mention the atypical logics and mathematical morphology. Sociology (studying societies of individual agents sharing a same subset of cultural codes) and, in particular, economics, when the economic codes are made of commodities on one hand, prices and units of numéraire measuring them, and the societies are the agents who consumer or producer, buyer or seller, debtor or creditor. This opens avenues for further research in the sciences of life.

Keywords: hypermap, focus, mutational analysis, Galois stratification, Galois quotation, Galois transform, Galois measure, atypical logic, mathematical morphology, society, cultures, economic classes, dynamic economic behavior.

MSC: 34G25, 34A60, 49J27, 49J53, 93B03, 37B55, 28B20, 45N05

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