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21 results on '"Gal, Olivier"'

1. Trajectories of vector fields asymptotic to formal invariant curves

Author

Gal, Olivier Le and Sánchez, Fernando Sanz

Subjects
Mathematics - Dynamical Systems, Mathematics - Classical Analysis and ODEs, 34E05, 34C20, 34C08, 34A25, 37D10, 37C25
Abstract

We prove that a formal curve $\Gamma$ that is invariant by a $C^\infty$ vector field $\xi$ of $\mathbb{R}^m$ has a geometrical realization, as soon as the Taylor expansion of $\xi$ is not identically zero along $\Gamma$. This means that there is a trajectory $\gamma$ of $\xi$ which is asymptotic to $\Gamma$. This result solves a natural question proposed by Bonckaert nearly forty years ago. We also construct an invariant $C^0$ manifold $S$ in some open horn around $\Gamma$ which is composed entirely of trajectories asymptotic to $\Gamma$, and contains the germ of any such trajectory. If $\xi$ is analytic, we prove that there exists a trajectory asymptotic to $\Gamma$ which is, moreover, non-oscillating with respect to subanalytic sets.

Published
2023

2. Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy

Author

Gal, Olivier Le, Matusinski, Mickaël, and Sánchez, Fernando Sanz

Subjects
Mathematics - Dynamical Systems, Mathematics - Classical Analysis and ODEs, Mathematics - Differential Geometry, Mathematics - Logic, 34C08, 34D05, 03C64 (Primary), 14P15, 32B20 (Secondary)
Abstract

We introduce a notion of regular separation for solutions of systems of ODEs $y'=F(x,y)$, where F is definable in a polynomially bounded o-minimal structure and $y = (y_1,y_2)$. Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property introduced by J.-P. Rolin, R. Shaefke and the third author., Comment: 25 pages

Published
2020
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3. Powers are easy to avoid

Author

Jones, Gareth and Gal, Olivier Le

Subjects
Mathematics - Logic
Abstract

Suppose that $\widetilde{\mathbb R}$ is an o-minimal expansion of the real field in which restricted power functions are definable. We show that if $\widehat{\mathbb R}$ is both a reduct (in the sense of definability) of the expansion $\widetilde{\mathbb R}^{\mathbb R}$ of $\widetilde{\mathbb R}$ by all real power functions and an expansion (again in the sense of definability) of $\widetilde{\mathbb R}$, then, provided that $\widetilde{\mathbb R}$ and $\widehat{\mathbb R}$ have the same field of exponents, they define the same sets. This can be viewed as a polynomially bounded version of an old conjecture of van den Dries and Miller.

Published
2020

4. Limits of tangent spaces to definable sets

Author

Dinh, Si Tiep, Gal, Olivier Le, and Pham, Tien Son

Subjects
Mathematics - Algebraic Geometry
Abstract

We study the set of tangent limits at a given point to a set definable in any o-minimal structure by characterizing the set of exceptional rays in the tangent cone to the set at that point and investigating the set of tangent limits along these rays. Several criteria for determining exceptional rays will be given. The main results of the paper generalize, to the o-minimal setting and to arbitrary dimension, the main results of O'Shea--Wilson which deals with algebraic surfaces in $\mathbb R^3$.

Published
2020

5. On local definability of holomorphic functions

Author

Jones, Gareth, Kirby, Jonathan, Gal, Olivier Le, and Servi, Tamara

Subjects
Mathematics - Logic, 03C64, 14P10
Abstract

Given a collection A of holomorphic functions, we consider how to describe all the holomorphic functions locally definable from A. The notion of local definability of holomorphic functions was introduced by Wilkie, who gave a complete description of all functions locally definable from A in the neighbourhood of a generic point. We prove that this description is no longer complete in the neighbourhood of non-generic points. More precisely, we produce three examples of holomorphic functions which suggest that at least three new operations need to be added to Wilkie's description in order to capture local definability in its entirety. The constructions illustrate the interaction between resolution of singularities and definability in the o-minimal setting.

Published
2017
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6. Tangent cones and $C^1$ regularity of definable sets

Author

Kurdyka, Krzysztof, Gal, Olivier Le, and Nguyen, Nhan

Subjects
Mathematics - Geometric Topology, Mathematics - Differential Geometry
Abstract

Let $X\subset \mathbb R^n$ be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) $X$ is a $C^1$ manifold, (ii) the tangent cone and the paratangent cone of $X$ coincide at every point in $X$, (iii) for every $x \in X$, the tangent cone of $X$ at the point $x$ is a $k$-dimensional linear subspace of $\mathbb R^n$ ($k$ does not depend on $x$) varies continuously in $x$, and the density $\theta(X, x) < 3/2$., Comment: 11 pages

Published
2017

7. Optical elastic scattering for early label-free identification of clinical pathogens

Author

Genuer, Valentin, Gal, Olivier, Méteau, Jérémy, Marcoux, Pierre, Schultz, Emmanuelle, Lacot, Éric, Maurin, Max, and Dinten, Jean-Marc

Subjects
Physics - Biological Physics, Physics - Optics
Abstract

We report here on the ability of elastic light scattering in discriminating Gram+, Gram-and yeasts at an early stage of growth (6h). Our technique is non-invasive, low cost and does require neither skilled operators nor reagents. Therefore it is compatible with automation. It is based on the analysis of the scattering pattern (scatterogram) generated by a bacterial microcolony growing on agar, when placed in the path of a laser beam. Measurements are directly performed on closed Petri dishes. The characteristic features of a given scatterogram are first computed by projecting the pattern onto the Zernike orthogonal basis. Then the obtained data are compared to a database so that machine learning can yield identification result. A 10-fold cross-validation was performed on a database over 8 species (15 strains, 1906 scatterograms), at 6h of incubation. It yielded a 94% correct classification rate between Gram+, Gram-and yeasts. Results can be improved by using a more relevant function basis for projections, such as Fourier-Bessel functions. A fully integrated instrument has been installed at the Grenoble hospital's laboratory of bacteriology and a validation campaign has been started for the early screening of SA and MRSA (Staphylococcus aureus, methicillin-resistant S. aureus) carriers. Up to now, all the published studies about elastic scattering were performed in a forward mode, which is restricted to transparent media. However, in clinical diagnostics, most of media are opaque, such as blood-supplemented agar. That is why we propose a novel scheme capable of collecting back-scattered light which provides comparable results.

Published
2016
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8. Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal

Author

Gal, Olivier Le, Sanz, Fernando, and Speissegger, Patrick

Subjects
Mathematics - Classical Analysis and ODEs, Mathematics - Logic, 34C08, 03C64, 34M30
Abstract

Let X be an analytic vector field defined in a neighborhood of the origin of R^3, and let I be an analytically non-oscillatory integral pencil of X; that is, I is a maximal family of analytically non-oscillatory trajectories of X at the origin all sharing the same iterated tangents. We prove that if I is interlaced, then for any trajectory T in I, the expansion of the structure generated over the real field by T and all globally subanalytic sets is model-complete, o-minimal and polynomially bounded., Comment: 25 pages; refereed version

Published
2013

12. Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy.

Author

Le Gal, Olivier, Matusinski, Mickaël, and Sánchez, Fernando Sanz

Subjects
VECTOR fields, EXPONENTIAL dichotomy
Abstract

We introduce a notion of regular separation for solutions of systems of ODEs y' = F(x, y), where F is definable in a polynomially bounded o-minimal structure and y = (y1, y2). Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author. [ABSTRACT FROM AUTHOR]

Published
2022
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13. Solutions of definable ODEs with regular separation and dichotomy interlacement versus Hardy

Author

Le Gal, Olivier and Le Gal, Olivier

Abstract

Producción Científica, We introduce a notion of regular separation for solutions of systems of ODEs y'=F(x; y), where F is definable in a polynomially bounded o-minimal structure and y=(y1,y2). Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author., F. Sanz Sánchez was partially supported by Ministerio de Ciencia, Spain, process MTM2016-77642-C2-1-P and PID2019-105621GB-I00

Published
2021

15. Mass Spectrometry Imaging visualization tools developed during the Computis European project

Author

Robbe, Marie-France, Thévenot, Etienne A, Stoeckli, Markus, Hester, Alfons, Roempp, Andreas, Kharchenko, Andriy, Both, Jean-Pierre, Gal, Olivier, Haan, Serge, Laboratoire Outils d'Analyse des Données (LOAD), Département Métrologie Instrumentation & Information (DM2I), Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA)), Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction de Recherche Technologique (CEA) (DRT (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Laboratoire d'Intégration des Systèmes et des Technologies (LIST (CEA)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Novartis Pharma AG, Justus-Liebig-Universität Gießen = Justus Liebig University (JLU), FOM Institute for Atomic and Molecular Physics (AMOLF), Financial support from the European Commission (FP6 STREP project LSHG-CT-2005-518194), European Project: 32887,COMPUTIS, Robbe, Marie-France, Molecular Imaging in Tissue and Cells by Computer-assisted Innovative Multimode Mass Spectrometry - COMPUTIS - 32887 - OLD, Laboratoire d'Intégration des Systèmes et des Technologies (LIST), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Laboratoire d'Intégration des Systèmes et des Technologies (LIST), and Justus-Liebig-Universität Gießen (JLU)

Subjects
Moran index, [INFO.INFO-TS] Computer Science [cs]/Signal and Image Processing, spatial filtering, Regions Of Interest, zooming, ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION, standardardization, data interpretation, mass spectrometry imaging, spectrum analysis, peak picking, molecule identification, spectrum, temporal filtering, statistical analysis, [INFO.INFO-TS]Computer Science [cs]/Signal and Image Processing, peak list, [PHYS.PHYS.PHYS-INS-DET]Physics [physics]/Physics [physics]/Instrumentation and Detectors [physics.ins-det], signal processing, visualization, data format, m/z correlation, [CHIM.ORGA]Chemical Sciences/Organic chemistry, [CHIM.ORGA] Chemical Sciences/Organic chemistry, Magnetic Resonance Imaging, baseline correction, image processing, relative variance, pixel picking, classification, [PHYS.PHYS.PHYS-INS-DET] Physics [physics]/Physics [physics]/Instrumentation and Detectors [physics.ins-det], realignment of images. image coregistration, spectrum treatment, data processing, detrending, clustering
Abstract

International audience; The Computis European project (2006-2009) was aimed at developing innovative experimental mass spectrometry imaging (MSI) techniques and software tools for data treatment and visualization, and at validating them in key biological applications (neurobiology, pharmaceutical drug development). The project succeeded in defining a specific standard format for MSI, imzML, in collaboration with the HUPO consortium. The four main MSI software tools developed by the project all handle the imzML format.Data Cube Explorer provides an easy spectral and spatial exploration of MS images: spectrum zooming, scrolling through the dataset masses with a manual contrast tuning for images, selection of Regions Of Interest with the display of the associated spectra. The self-organizing map functionality classifies images according to the intensity of all pixel places and automatically selects images as different as possible. Mirion is a simple visualization module displaying spectra by pixel and for the total image, with zooming and scrolling functions. Histogram of the total ion count of each pixel can be calculated, using different input parameters. Images are displayed for each peak of the total spectrum, with a manual intensity tuning and a comparison of the intensity distributions by pixel between several images.EasyMSI enables spatial and spectral visualization of mass spectrometry imaging datasets (spectrum and image display, peak and pixel picking, zooming on spectra and images, ROI selection), as well as an assistance for the interpretation of data: This assistance includes indicators (relative variance, Moran index, m/z correlation) to highlight peaks that bring interesting information, peak list for molecule identification, spectrum denoising or structure analysis by clustering methods (K-means, fuzzy, hierarchical clustering, diffusion map). EasyMSI offers the advantage of processing and displaying the original data (i.e. without binning).BioMap is an image analysis platform for MSI and Magnetic Resonance Imaging. It includes viewing functions (spectrum and image display, intensity adjustment, zoom, treatment of multiple ROIs, geometrical transformations and operations), and spectrum treatment (spatial or temporal filtering, baseline correction, detrending). More elaborated functions enable a simultaneous view of all dataset images, creation of a movie, statistical and histogram analysis, co-registration of images of one or two dataset(s), and realignment of images. The use and capacities of these tools are presented through a comparative analysis of a rodent urinary bladder dataset in imzML format.

Published
2011

16. An o-minimal structure which does not admit C-infinity cellular decomposition

Author

Le Gal, Olivier, Rolin, Jean-Philippe, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics::Popular Physics, Mathematics::History and Overview, Nonlinear Sciences::Cellular Automata and Lattice Gases, 26E40, Computer Science::Computers and Society
Abstract

International audience; We build an o-minimal expansion of the real field which does not admit C-infinity cellular decomposition

Published
2008
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17. Model completeness in o-minimal polynomially bounded structures

Author

Le Gal , Olivier, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG], 03C64, [ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG]
Abstract

International audience; We show an explicit theorem of the complement "Gabrielov's '96 like" for o-minimal polynomially bounded structures. In model theoretic terms, this is equivalent to the model completeness of (R, >, +, *, F) where F is a global differential algebra of maps definable in an o-minimal polynomially bounded structure.

Published
2008
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18. Direct identification of clinically relevant bacterial and yeast microcolonies and macrocolonies on solid culture media by Raman spectroscopy

Author

Espagnon, Isabelle, primary, Ostrovskii, Denis, additional, Mathey, Raphaël, additional, Dupoy, Mathieu, additional, Joly, Pierre L., additional, Novelli-Rousseau, Armelle, additional, Pinston, Frédéric, additional, Gal, Olivier, additional, Mallard, Frédéric, additional, and Leroux, Denis F., additional

Published
2014
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20. Model completness of o-minimal polynomally bounded structures

Author

Géométrie analytique ; Institut de Recherche Mathématique de Rennes (IRMAR) ; CNRS - École normale supérieure (ENS) - Cachan - Institut National des Sciences Appliquées [INSA] - Université de Rennes 1 - Université de Rennes II - Haute Bretagne - CNRS - École normale supérieure (ENS) - Cachan - Institut National des Sciences Appliquées [INSA] - Université de Rennes 1 - Université de Rennes II - Haute Bretagne, Université Rennes 1, Jean-Marie Lion, Le Gal, Olivier, Géométrie analytique ; Institut de Recherche Mathématique de Rennes (IRMAR) ; CNRS - École normale supérieure (ENS) - Cachan - Institut National des Sciences Appliquées [INSA] - Université de Rennes 1 - Université de Rennes II - Haute Bretagne - CNRS - École normale supérieure (ENS) - Cachan - Institut National des Sciences Appliquées [INSA] - Université de Rennes 1 - Université de Rennes II - Haute Bretagne, Université Rennes 1, Jean-Marie Lion, and Le Gal, Olivier

Abstract

O-minimal structures were introduced in the 80' by Van den Dries answering the Grotendick's request for a tame geomety framework, and were largely studied by Wilkie and Macintyre. This thesis shows an explicit theorem of the complement for o-minimal polynomially bounded structures, result equivalent to the model-completness in model theorie. In 1968, Gabrielov shows a theorem of the complement for sub-analytic sets, which implice the tameness of global sub-analytics sets. He gives in 1996 an explicit version of this result. A generalisation of this theorem is introduced here. By valuation's arguments (due to Lojasiewicz in the analytic case and to Miller for the o-minimal case), some quasi-analytic's properties are exhibits, which permit to adapt the classical proof of model-completness sheme. This result is a step for better understand how o-minimal structures are generated and gives a reduced language on which an o-minimal polynomially bounded structure is model-complete., Les structures o-minimales, introduites dans les années '80 par Van den Dries et largement étudiées par Wilkie et Macintyre répondent à Grothendick en donnant le cadre d'une géométrie modérée. Cette thèse montre un théorème du complémentaire explicite pour les structures o-minimales polynomialement bornées, ce qui équivault à la modèle-complétude en théorie des modèles. En 1968, Gabrielov montre un théorème du complémentaire pour les sous-analytiques globaux, qui en implique la o-minimalité. Il améliore ce résultat en 96, avec un théorème explicite. Une généralisation de celui-ci est présentée ici. Par des arguments de valuation dus à Lojaciewicz et à Miller, des propriétés de quasi-analycité sont exhibées, qui permettent d'adapter le schéma classique des preuves de modèle-complétude. Ce résultat permet de mieux comprendre la façon dont sont générées les structures o-minimales et donne un langage réduit sur lequel une structure polynomialement bornée est modèle-complète.

21. Modèle complétude des structures o-minimales polynomialement bornées

Author

Olivier Le Gal, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université Rennes 1, Jean-Marie Lion, Le Gal, Olivier, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
[ MATH ] Mathematics [math], geometry, théorie des modèles, quasi-analycité, o-minimality, [MATH] Mathematics [math], théorème du complémentaire, 01 natural sciences, elimination of quantifiers, géométrie, stratification, 0103 physical sciences, o-minimalité, 0101 mathematics, [MATH]Mathematics [math], Mathematics, noetherianity, élimination des quantificateurs, 010102 general mathematics, model complete, General Medicine, modèle complétude, noetherianité, model theory, theorem of the complement, quasi-analycity, 010307 mathematical physics, Humanities
Abstract

O-minimal structures were introduced in the 80' by Van den Dries answering the Grotendick's request for a tame geomety framework, and were largely studied by Wilkie and Macintyre. This thesis shows an explicit theorem of the complement for o-minimal polynomially bounded structures, result equivalent to the model-completness in model theorie.In 1968, Gabrielov shows a theorem of the complement for sub-analytic sets, which implice the tameness of global sub-analytics sets. He gives in 1996 an explicit version of this result. A generalisation of this theorem is introduced here.By valuation's arguments (due to Lojasiewicz in the analytic case and to Miller for the o-minimal case), some quasi-analytic's properties are exhibits, which permit to adapt the classical proof of model-completness sheme. This result is a step for better understand how o-minimal structures are generated and gives a reduced language on which an o-minimal polynomially bounded structure is model-complete., Les structures o-minimales, introduites dans les années '80 par Van den Dries et largement étudiées par Wilkie et Macintyre répondent à Grothendick en donnant le cadre d'une géométrie modérée. Cette thèse montre un théorème du complémentaire explicite pour lesstructures o-minimales polynomialement bornées, ce qui équivault à la modèle-complétude en théorie des modèles.En 1968, Gabrielov montre un théorème du complémentaire pourles sous-analytiques globaux, qui en implique la o-minimalité. Il améliore ce résultat en 96, avec un théorème explicite. Une généralisation de celui-ci est présentée ici.Par des arguments de valuation dus à Lojaciewicz et à Miller, des propriétés de quasi-analycité sont exhibées, qui permettent d'adapter le schéma classique des preuves de modèle-complétude. Ce résultat permet de mieux comprendre la façon dont sont générées les structures o-minimales et donne un langage réduit sur lequel une structure polynomialement bornée est modèle-complète.

Published
2006

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