Théorème de préparation pour les fonctions logarithmico-exponentielles

Lion, Jean-Marie; Rolin, Jean-Philippe

doi:10.5802/aif.1583

Cited by 66 publications

(74 citation statements)

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“…A similar extension of subanalytic sets ((Ran^P)" 0^1118 ' 0^ sets ) was ^en treated by L. van den Dries, A. Macintyre, D. Marker in [DMM] (geometric proofs of these facts were found recently by J-M. Lion and J.-P. Rolin [LR1], [LR2]). Another type of o-minimal structure ((R^)-definable sets) was obtained by C. Miller [Mi], by adding to subanalytic sets all functions x ->-x r ^ r € K, where K is a subfield of M. We give a list and examples of o-minimal structures in section 1.…”

Section: Introductionmentioning

confidence: 93%

On gradients of functions definable in o-minimal structures

Kurdyka¹

1998

Annales De L’institut Fourier

434

414

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Section: Introductionmentioning

confidence: 93%

On gradients of functions definable in o-minimal structures

Kurdyka¹

1998

Annales De L’institut Fourier

434

414

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“…The phase portrait is similar to the one in the flat case but the section defined by (28) and corresponding to y = 0 is here : dθ ds = εαcos θ. In particular if α = 0 (strict case) there exist both oscillating and rotating trajectories corresponding to projections of geodesics starting from 0, see It is important to observe that our description of the behavior of t −→ y(t) is true for the gradated form of order 0, but also of any order when λ −→ ∞.…”

Section: Analysis Of the Foliation Fmentioning

confidence: 66%

“…Using [28], the elimination of the parameter k ′ is allowed in this category and will lead to a log-exp graph. The precise algorithm to evaluate C 1 has been established in [2] and we proceed as follows.…”

Section: Estimation Ofcmentioning

confidence: 99%

On the role of abnormal minimizers in sub-riemannian geometry

Bonnard¹,

Trélat²

2001

Annales De La Faculté Des Sciences De Toulouse Mathématiques

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Consider a sub-Riemannian geometry (U, D, g) where U is a neighborhood at 0 in IR n , D is a rank-2 smooth (C ∞ or C ω ) distribution and g is a smooth metric on D. The objective of this article is to explain the role of abnormal minimizers in SR-geometry. It is based on the analysis of the Martinet SR-geometry.Key words : optimal control, singular trajectories, sub-Riemannian geometry, abnormal minimizers, sphere and wave-front with small radii.Résumé On considère un problème sous-Riemannien (U, D, g) où U est un voisinage de 0 dans IR n , D une distribution lisse de rang 2 et g une métrique lisse sur D. L'objectif de cet article est d'expliquer le rôle des géodésiques anormales minimisantes en géométrie SR. Cette analyse est fondée sur le modèle SR de Martinet.Titre Le rôle des géodésiques anormales minimisantes en géométrie sousRiemannienne.

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“…[12,26]), namely it is equivalent to the preparation theorem in the sense of Parusiński-Lion-Rolin [32,19,33], which says that every definable function of several variables depends piecewise on (or can be prepared with respect to) any fixed variable in a certain simple fashion. The preparation theorem, in turn, yields many geometric, differential and integral applications, like the Lipschitz structure of subanalytic sets (cf.…”

Section: Valuation Property For L-terms We Begin Withmentioning

confidence: 99%

Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

Nowak

2009

Ann. Polon. Math.

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Abstract. This paper investigates the geometry of the expansion RQ of the real field R by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusiński-Lion-Rolin). To this end, we study nonstandard models R of the universal diagram T of RQ in the language L augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our earlier article [Ann. Polon. Math. 96 (2009), 65-74], combines modifications by blowing up with a suitable partitioning. Via an analysis of L-terms and infinitesimals, we prove the valuation property for functions given by L-terms, and next the exchange property for substructures of a given model R. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for T is established through model-theoretic compactness. The universal theory T is thus complete and o-minimal, and RQ is its prime model. Under the circumstances, every definable function is piecewise given by L-terms, and therefore the previous results concerning L-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux's theorem with parameter is demonstrated.

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Théorème de préparation pour les fonctions logarithmico-exponentielles

Cited by 66 publications

References 0 publications

On gradients of functions definable in o-minimal structures

On gradients of functions definable in o-minimal structures

On the role of abnormal minimizers in sub-riemannian geometry

Quantifier elimination, valuation property and preparation theorem in quasianalytic geometry via transformation to normal crossings

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