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An introduction to o-minimal geometry
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A parallel course on the construction of o-minimal structures was given by A. Macintyre in Pisa in Spring 1999 as mentioned in this paper, where the content of these notes owes a great deal to the excellent book by L. van den Dries [vD], some interesting topics contained in this book are not included here, such as the Vapnik-Chervonenkis property.Abstract:
Preface These notes have served as a basis for a course in Pisa in Spring 1999. A parallel course on the construction of o-minimal structures was given by A. Macintyre. The content of these notes owes a great deal to the excellent book by L. van den Dries [vD]. Some interesting topics contained in this book are not included here, such as the Vapnik-Chervonenkis property. Part of the material which does not come from [vD] is taken from the paper [Co1]. This includes the sections on the choice of good coordinates and the triangulation of functions in Chapter 4 and Chapter 5. The latter chapter contains the results on triviality in families of sets or functions which were the main aim of this course. The last chapter on smoothness was intended to establish property " DC k-all k " which played a crucial role in the course of Macintyre (it can be easily deduced from the results in [vDMi]). It is also the occasion to give a few results on tubular neighborhoods. I am pleased to thank Francesca Acquistapace, Fabrizio Broglia and all colleagues of the Dipartimento di Matematica for the invitation to give this course in Pisa and their friendly hospitality. Also many thanks to Antonio Ponchio for reading these notes and correcting mistakes.read more
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References
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Real Algebraic Geometry
TL;DR: The Tarski-Seidenberg Principle as a Transfer Tool for Real Algebraic Geometry as mentioned in this paper is a transfer tool for real algebraic geometry, and it can be used to solve the Hilbert's 17th Problem.
MonographDOI
Tame Topology and O-minimal Structures
TL;DR: In this article, the authors give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis, and cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the ominimal setting and show how these notions are easier to handle than in ordinary topology.
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Geometric categories and o-minimal structures
Lou van den Dries,Chris Miller +1 more
TL;DR: In this article, the authors make available an extension of the category of subanalytic sets that has these sets among its objects, and that behaves much like the categories of sub analytic sets, and they apply directly only to the cartesian spaces R and not to arbitrary real analytic manifolds.
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Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
TL;DR: In this paper, it was shown that the class of sub-analytic sets is closed under first-order logical definability (where, as well as boolean operations, the quantifiers ∃x ∈ R... ” and ∀x ∆ ∆, ∆, q(~ α) > 0), where p(x), q(x) are n-variable polynomials with real coefficients.
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Definable sets in ordered structures. i
Anand Pillay,Charles Steinhorn +1 more
TL;DR: In this article, a model theory for a class of linearly ordered structures, called min-minimal structures, has been proposed, which is based on the stability theory of minimal structures and strongly minimal theories.