In this paper we develop a differential analogue of o-minimal cell decomposition for the theory CODF of closed ordered differential fields. Thanks to this differential cell decomposition we define a well-behaving dimension function on the class of definable sets in CODF. We conclude this paper by proving that this dimension is closely related to both the usual differential transcendence degree and the topological dimension associated, in this case, with a natural differential topology on ordered differential fields.

In [T. Brihaye, C. Michaux, C. Rivière, Cell decomposition and dimension function in the theory of closed ordered differential fields, Ann. Pure Appl. Logic .] the authors proved a cell decomposition theorem for the theory of closed ordered differential fields which generalizes the usual Cell Decomposition Theorem for o-minimal structures. As a consequence of this result, a well-behaving dimension function on definable sets in CODF was introduced. Here we continue the study of this cell decomposition in CODF by proving three (...) additional results. We first discuss the relation between the δ-cells introduced in the above-mentioned reference and the notion of Kolchin polynomial in differential algebra. We then prove two generalizations of classical decomposition theorems in o-minimal structures. More exactly we give a theorem of decomposition into definably d-connected components and a differential cell decomposition theorem for a particular class of definable functions in CODF. (shrink)

In this paper we give a differential lifting principle which provides a general method to geometrically axiomatize the model companion (if it exists) of some theories of differential topological fields. The topological fields we consider here are in fact topological systems in the sense of van den Dries, and the lifting principle we develop is a generalization of the geometric axiomatization of the theory DCF₀ given by Pierce and Pillay. Moreover, it provides a geometric alternative to the axiomatizations obtained by (...) Tressl and Guzy/Point in separate papers where the authors also build general schemes of axioms for some model complete theories of differential fields. We first characterize the existentially closed models of a given theory of differential topological fields and then, under an additional hypothesis of largeness, we show how to modify this characterization to get a general scheme of first-order axioms for the model companion of any large theory of differential topological fields. We conclude with an application of this lifting principle proving that, in existentially closed models of a large theory of differential topological fields, the jet-spaces are dense in their ambient topological space. (shrink)

In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable .In this paper we study the case where the fields are expanded with finitely (...) many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language Lequation image = {+, –, ·, D, <1, …, shrink)

Tout pouvoir politique se manifeste et par sa maîtrise de la contrainte et par la possibilité qu'il a de se dramatiser dans des cérémonies qui visent à intégrer et à mobiliser les administrés pour des actions communes. Il affirme simultanément sa légitimité, ses hiérarchies et ses priorités. Les exemples des rites du protocole, des élections municipales, des inaugurations et intronisations énoncent comment le politique se donne en spectacle et quels effets sociaux produisent de telles démonstrations. Outre les conditions les plus (...) favorables à l'apparition des rites, sont examinées la dynamique et l'obsolescence de ceux-ci selon les régimes politiques, ainsi que les raisons expliquant l'intérêt actuel des anthropologues pour les liturgies politiques.All political power is manifested and his mastery of stress and the ability he has to dramatize in ceremonies designed to integrate and mobilize administered for joint actions. He says its legitimacy simultaneously, hierarchies and priorities. Examples of rites protocol, municipal elections, inaugurations and state inductions how politics makes a spectacle and what social effects produce such demonstrations. In addition to the conditions most favorable to the emergence of rites are examined the dynamics and obsolescence as these political regimes, as well as the reasons for the current interest anthropologists liturgies for politicians. (shrink)