A generic condition implying o-minimality for restricted C$^{\infty }$-functions

Olivier Le Gal

Displaying similar documents to “A generic condition implying o-minimality for restricted C $^{\infty}$ -functions”

Algebraic approximation of analytic sets definable in an o-minimal structure

Marcin Bilski, Kamil Rusek (2010)

Annales Polonici Mathematici

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Let K,R be an algebraically closed field (of characteristic zero) and a real closed field respectively with K=R(√(-1)). We show that every K-analytic set definable in an o-minimal expansion of R can be locally approximated by a sequence of K-Nash sets.

Extending Tamm's theorem

Lou van den Dries, Chris Miller (1994)

Annales de l'institut Fourier

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We extend a result of M. Tamm as follows: Let $f : A \to ℝ, A \subseteq ℝ^{m + n}$ , be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions $x \mapsto x^{r} : (0, \infty) \to ℝ, r \in ℝ$ . Then there exists $N \in ℕ$ such that for all $(a, b) \in A$ , if $y \mapsto f (a, y)$ is $C^{N}$ in a neighborhood of $b$ , then $y \mapsto f (a, y)$ is real analytic in a neighborhood of $b$ .

On gradients of functions definable in o-minimal structures

Krzysztof Kurdyka (1998)

Annales de l'institut Fourier

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We prove the o-minimal generalization of the Łojasiewicz inequality $∥ grad f ∥ \geq | f |^{α}$ , with $α < 1$ , in a neighborhood of $a$ , where $f$ is real analytic at $a$ and $f (a) = 0$ . We deduce, as in the analytic case, that trajectories of the gradient of a function definable in an o-minimal structure are of uniformly bounded length. We obtain also that the gradient flow gives a retraction onto levels of such functions.

A note on Bierstone-Milman-Pawłucki's paper "Composite differentiable functions"

Krzysztof Jan Nowak (2011)

Annales Polonici Mathematici

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We demonstrate that the composite function theorems of Bierstone-Milman-Pawłucki and of Glaeser carry over to any polynomially bounded, o-minimal structure which admits smooth cell decomposition. Moreover, the assumptions of the o-minimal versions can be considerably relaxed compared with the classical analytic ones.

$C^{1}$ -minimal subsets of the circle

Dusa McDuff (1981)

Annales de l'institut Fourier

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Necessary conditions are found for a Cantor subset of the circle to be minimal for some $C^{1}$ -diffeomorphism. These conditions are not satisfied by the usual ternary Cantor set.

Extending analyticK-subanalytic functions

Artur Piękosz (2004)

Open Mathematics

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Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.

A multidimensional Taylor's theorem with minimal hypothesis

J. Marshall Ash, A. Eduardo Gatto, Stephen Vági (1990)

Colloquium Mathematicae

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Topological types of fewnomials

Michel Coste (1998)

Banach Center Publications

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Ellis groups of quasi-factors of minimal flows

Joseph Auslander (2000)

Colloquium Mathematicae

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A quasi-factor of a minimal flow is a minimal subset of the induced flow on the space of closed subsets. We study a particular kind of quasi-factor (a 'joining' quasi-factor) using the Galois theory of minimal flows. We also investigate the relation between factors and quasi-factors.

Displaying similar documents to “A generic condition implying o-minimality for restricted C ∞ -functions”

Displaying similar documents to “A generic condition implying o-minimality for restricted C $^{\infty}$ -functions”