Grothendieck ring of semialgebraic formulas and motivic real Milnor fibres

We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas. We give a realization of our ring that allows to express a class as a Z[1/2]- linear combination of classes of real algebraic formulas, so this realization gives rise to a notion of virtual Poincar\'e polynomial for basic semialgebraic formulas. We then define zeta functions with coefficients in our ring, built on semialgebraic formulas in arc spaces. We show that they are rational and relate them to the topology of real Milnor fibres.Comment: 30 pages, 1 figur

Equisingularité réelle II : invariants locaux et conditions de régularité

International audienceFor germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the vanishing Euler characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is a linear combination of the invariants of the other sequence. We then connect our invariants to the geometry of the discriminants of all dimension. Finally we prove that these invariants are continuous along Verdier strata of a closed subanalytic set

Linearly Supporting Feature Extraction For Automated Estimation Of Stellar Atmospheric Parameters

We describe a scheme to extract linearly supporting (LSU) features from stellar spectra to automatically estimate the atmospheric parameters $T_{eff}$, log$~g$, and [Fe/H]. "Linearly supporting" means that the atmospheric parameters can be accurately estimated from the extracted features through a linear model. The successive steps of the process are as follow: first, decompose the spectrum using a wavelet packet (WP) and represent it by the derived decomposition coefficients; second, detect representative spectral features from the decomposition coefficients using the proposed method Least Absolute Shrinkage and Selection Operator (LARS)$_{bs}$; third, estimate the atmospheric parameters $T_{eff}$, log$~g$, and [Fe/H] from the detected features using a linear regression method. One prominent characteristic of this scheme is its ability to evaluate quantitatively the contribution of each detected feature to the atmospheric parameter estimate and also to trace back the physical significance of that feature. This work also shows that the usefulness of a component depends on both wavelength and frequency. The proposed scheme has been evaluated on both real spectra from the Sloan Digital Sky Survey (SDSS)/SEGUE and synthetic spectra calculated from Kurucz's NEWODF models. On real spectra, we extracted 23 features to estimate $T_{eff}$, 62 features for log$~g$, and 68 features for [Fe/H]. Test consistencies between our estimates and those provided by the Spectroscopic Sarameter Pipeline of SDSS show that the mean absolute errors (MAEs) are 0.0062 dex for log$~T_{eff}$ (83 K for $T_{eff}$), 0.2345 dex for log$~g$, and 0.1564 dex for [Fe/H]. For the synthetic spectra, the MAE test accuracies are 0.0022 dex for log$~T_{eff}$ (32 K for $T_{eff}$), 0.0337 dex for log$~g$, and 0.0268 dex for [Fe/H].Comment: 21 pages, 7 figures, 8 tables, The Astrophysical Journal Supplement Series (accepted for publication

Integration of Oscillatory and Subanalytic Functions

We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This paper extends the investigation started in [J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages analytiques et th\'eor\`eme de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under integration of sums of products of real globally subanalytic functions and their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.Comment: Final version. Accepted for publication in Duke Math. Journal. Changes in proofs: from Section 6 to the end, we now use the theory of continuously uniformly distributed modulo 1 functions that provides a uniform technical point of view in the proofs of limit statement

Mellin transforms of power-constructible functions

We consider several systems of algebras of real- and complex-valued functions, which appear in o-minimal geometry and related geometrically tame contexts. For each such system, we prove its stability under parametric integration and we study the asymptotics of the functions as well as the nature of their parametric Mellin transforms

Obras completas de Buffon Tomo I, Discursos preliminares

247 p. ; 16 cmDonación J.L. Estrada (Biblioteca General)Digitalizada en formato pd