The purpose of this paper is to extend a theorem of Speissegger [J. Reine Angew. Math. 508 (1999)], which states that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Specifically, we display a collection of properties possessed by the real numbers that suffices for a version of the proof of this theorem to go through. The degree of flexibility revealed in this study permits the use of certain model-theoretic arguments for the first time, e.g. the compactness theorem....

A continuous linear extension operator, different from Whitney’s, for ${𝒞}^p$-Whitney fields (p finite) on a closed o-minimal subset of ${ℝ}^n$ is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.

In this note we bind together Wilkie's complement theorem with Lion's theorem on geometric, regular and 0-regular families of functions.

We express the Euler-Poincaré characteristic of a semi-algebraic set, which is the intersection of a non-singular complete intersection with two polynomial inequalities, in terms of the signatures of appropriate bilinear symmetric forms.

We show that in the class of compact, piecewise $C^1$ curves K in ${ℝ}^n$, the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.

This article is focused on calculating the trajectory of an industrial robot in the production of composites for the automotive industry. The production technology is based on the winding of carbon fibres on a polyurethane frame. The frame is fastened to the end-effector of the robot arm (i.e. robot-end-effector, REE). The passage of the frame through the fibre processing head is determined by the REE trajectory. The position of the fibre processing head is fixed and is composed of three fibre guide...

We show that in the class of compact sets K in ${ℝ}^n$ with an analytic parametrization of order m, the sets with Zariski dimension m are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for tangential derivatives of (the traces of) polynomials on K.

Applying general results on separation of semialgebraic sets and spaces of orderings, we produce a catalogue of all possible geometric obstructions for separation of 3-dimensional semialgebraic sets and give some hints on how separation can be made decidable.

Soit $\Gamma$ un groupe de type fini non élémentaire. On note $D^n\left(\Gamma \right)$ l’ensemble des structures hyperboliques de dimension $n$ sur $\Gamma$. $D^n\left(\Gamma \right)$ peut se réaliser comme fermé dans un espace semi-algébrique qui admet une compactification naturelle par le spectre réel. On note ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$ le compactifié via le $\underset{\_}{spectre}$ réel de $D^n\left(\Gamma \right)$. L’objet de cet article est de décrire les points ajoutés dans ${\overline{D^n\left(\Gamma \right)}}^{\mathrm{sp}}$. La compactification obtenue de cette manière permet d’interpréter “les points frontières” comme des représentations de $\Gamma$ dans ${\mathrm{SO}}_F^{+}(n,1)$ où $F(\supset ℝ)$ est un corps réel...

This paper is devoted to the computation of distance to set, called S, defined by polynomial equations. First we consider the case of quadratic systems. Then, application of results stated for quadratic systems to the quadratic equivalent of polynomial systems (see [5]), allows us to compute distance to semi-algebraic sets. Problem of computing distance can be viewed as non convex minimization problem: $d(u,S)={inf}_{x\in S}{\parallel x-u\parallel }^2$, where u is in ${ℛ}^n$. To have, at least, lower approximation of distance, we consider the dual...

Deux types de courbures sont associés à un sous-ensemble compact et définissable d'une variété riemannienne analytique réelle. Si la variété est de courbure constante, il y a des relations linéaires entre ces mesures. Comme application, nous démontrons une formule cinématique, définissons des densités locales, et nous étudions les volumes des simplexes réguliers.

We present a tameness property of sets definable in o-minimal structures by showing that Morse functions on a definable closed set form a dense and open subset in the space of definable $C^p$ functions endowed with the Whitney topology.

In a previous paper by Koike and Paunescu, it was introduced the notion of direction set for a subset of a Euclidean space, and it was shown that the dimension of the common direction set of two subanalytic subsets, called the directional dimension, is preserved by a bi-Lipschitz homeomorphism, provided that their images are also subanalytic. In this paper we give a generalisation of the above result to sets definable in an o-minimal structure on an arbitrary real closed field. More precisely, we...