arXiv subjects

Georges Comte

Publications and source records attributed to Georges Comte.

17 recordsLinked to original sources

arXiv2025

Decay of Fourier transforms and analytic continuation of power-constructible functions

For a subfield K of C, we denote by C^K the category of algebras of functions defined on the globally subanalytic sets that are generated by all K-powers and logarithms of positively-valued globally subanalytic functions. For any function f in C^\K(R), we study links between holomorphic extensions of f and the decay of its Fourier transform F[f] by using tameness properties of the globally subanalytic functions from which f is constructed. We first prove a number of theorems about analytic continuation of functions in C^K, including the fact that f in C^K(R) extends meromorphically to C if and only if f is rational. We then characterize the exponential rate of decay of F[f] by the maximal width of a horizontal strip in the plane about the real axis to which f extends holomorphically. Finally, we show that F[f] is integrable if f is integrable and continuous.

math.AG↗

arXiv2025

An equisingular heritage of Bernard Teissier

We give a brief and partial overview of Bernard Teissier's work in complex equisingularity theory, and a perspective on its legacy; in particular, we focus on the development of the theory in the real and the non-Archimedean contexts. Our aim is not to go into technical details, but, hopefully, rather to give a flavour of the forms taken by these developments, while providing enough definitions and references to give the reader access to old and new reference articles in the field.

math.AG↗

arXiv2024

Rational and lacunary algebraic curves

We give a bound on the number $\mathcal{Z}$ of intersection points in a ball of the complex plane, between a rational curve and a lacunary algebraic curve $Q=0$. This bound depends only on the lacunarity diagram of $Q$, and in particular is uniform in the coefficients of $Q$. Our bound shows that $\mathcal{Z}=O(dm)$, where $d$ is the degree of $Q$ and $m$ is the number of its monomials.

math.AG↗

arXiv2023

Parametric Fourier and Mellin transforms of power-constructible functions

We enrich the class of power-constructible functions, introduced in [CCRS23], to a class of algebras of functions which contains all complex powers of subanalytic functions, their parametric Mellin and Fourier transforms, and which is stable under parametric integration. By describing a set of generators of a special prepared form we deduce information on the asymptotics and on the loci of integrability of the functions of the class. We furthermore identify a subclass which is the smallest class containing all power-constructible functions and stable under parametric Fourier transforms and right-composition with subanalytic maps. This subclass is also stable under parametric integration, under taking pointwise and $L^p$limits, and under parametric Fourier-Plancherel transforms. Finally, we give a full asymptotic expansion in the power-logarithmic scale, uniformly in the parameters, for functions in this subclass.

math.CA↗

arXiv2023

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.

math.AG↗

arXiv2022

M Subdwarf Research III. Spectroscopic Diagnostics for Breaking Parameter Degeneracy

To understand the parameter degeneracy of M subdwarf spectra at low resolution, we assemble a large number of spectral features in the wavelength range of 0.6-2.5 {\mu}m with band strength quantified by narrowband indices. Based on the index trends of BT-Settl model sequences, we illustrate how the main atmospheric parameters (Teff, log g, [M/H], and [alpha/Fe]) affect each spectral feature differently. Furthermore, we propose a four-step process to determine the four parameters sequentially, which extends the basic idea proposed by Jao et al. Each step contains several spectral features that break the degeneracy effect when determining a specific stellar parameter. Finally, the feasibility of each spectroscopic diagnostic with different spectral qualities is investigated. The result is resolution-independent down to R~200.

astro-ph.SR↗

arXiv2022

Motivic Vitushkin invariants

We prove the nonarchimedean counterpart of a real inequality involving the metric entropy and measure geometric invariants $V_i$, called Vitushkin's variations. Our inequality is based on a new convenient partial preorder on the set of constructible motivic functions, extending the one considered by R. Cluckers and F. Loeser in Constructible motivic functions and motivic integration, Invent. Math., 173 (2008). We introduce, using motivic integration theory and the notion of riso-triviality, nonarchimedean substitutes of the Vitushkin variations $V_i$, and in particular of the number $V_0$ of connected components. We also prove the nonarchimedean global Cauchy-Crofton formula for definable sets of dimension $d$, relating $V_d$ and the motivic measure in dimension $d$.

math.AG↗

arXiv2020

M Subdwarf Research. II. Atmospheric Parameters and Kinematics

Applying the revised M subdwarf classification criteria discussed in Paper I to LAMOST DR7, combining the M subdwarf sample from Savcheva et al, a new M subdwarf sample was constructed for further study. The atmospheric parameters for each object were derived fitting with the PHOENIX grid, combining with Gaia DR2, the relationship between the gravity and metallicity were explored according to the locus both in the color-absolute magnitude diagram and the reduced proper motion diagram. Objects that have both the largest gravity and the lowest metallicity are located away from the main-sequence cloud and may be considered as the intrinsic M subdwarfs, which can be classified as luminosity class VI. Another group of objects whose spectra show typical M subdwarf characters have lower gravity and relatively moderate metal deficiency and occupy part of the ordinary M dwarf region in both diagrams. The Galactic U , V , W space velocity components and their dispersion show that the local Galactic halo population sampled in the solar neighborhood is represented by objects of high gravity and an inconspicuous bimodal metallicity distribution, with a fraction of prograde orbits. The other M subdwarfs seem to partly belong to the thick disk component with a significant fraction of thin disk moderately metal-poor objects intricately mixed with them. However, the selection effects, especially the favored anti-center direction of investigation in the LAMOST sub-sample, but also contamination by multiplicity and parameter coupling could play important roles and need to be further investigated.

astro-ph.SR↗

arXiv2018

M subdwarf research. I. Identification, modified classification system, and sample construction

We propose a revision of the system developed by L'epine et al. (2007) for spectroscopic M subdwarf classification. Based on an analysis of subdwarf spectra and templates from Savcheva et al. (2014), we show thatthe CaH1 feature originally proposed by Gizis (1997) is important in selecting reliable cool subdwarf spectra. This index should be used in combination with the [TiO5, CaH2+CaH3] relation provided by L\'epine et al. (2007) to avoid misclassification results. In the new system, the dwarf-subdwarf separators are first derived from a sample of more than 80,000 M dwarfs and a "labeled" subdwarf subsample, these objects being all visually identified from their optical spectra. Based on these two samples, we re-fit the initial [TiO5, CaH1] relation, and propose a new [CaOH, CaH1] relation supplementing the [TiO5, CaH1] relation to reduce the impact of uncertainty in flux calibration on classification accuracy. In addition, we recalibrate the $\zeta_{TiO/CaH}$ parameter defined in L'epine et al. (2007) to enable its successful application to LAMOST spectra. Using this new system, we select candidates from LAMOST Data Release 4 and finally identify a set of 2791 new M subdwarf stars, covering the spectral sequence from type M0 to M7. This sample contains a large number of objects located at low Galactic latitudes, especially in the Galactic anti-center direction, expanding beyond previously published halo- and thick disk-dominated samples. Besides, we detect magnetic activity in 141 objects. We present a catalog for this M subdwarf sample, including radial velocities, spectral indices and errors, activity flags, with a compilation of external data (photometric and GAIA DR2 astrometric parameters). The catalog is provided on-line, and the spectra can be retrieved from the LAMOST Data Release web portal.

astro-ph.SR↗

arXiv2016

Zeroes and rational points of analytic functions

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the graph $\Gamma_f$ of $f_{\vert D}$ and algebraic curves of degree $d$ is polynomially bounded in $d$. In particular, we show these assumptions are satisfied for random power series, for some explicit classes of lacunary series, and for solutions of linear differential equations with coefficients in ${\bf Q}[z]$. As a consequence, for any function $f$ in these families, $\Gamma_f$ has less than $\beta \log^\alpha T$ rational points of height at most $T$, for some $\alpha, \beta >0$.

math.AG↗

arXiv2016

Points of bounded height on oscillatory sets

We show that transcendental curves in $\mathbb R^n$ (not necessarily compact) have few rational points of bounded height provided that the curves are well behaved with respect to algebraic sets in a certain sense and can be parametrized by functions belonging to a specified algebra of infinitely differentiable functions. Examples of such curves include logarithmic spirals and solutions to Euler equations $x^2y''+xy'+cy=0$ with $c>0$.

math.AG↗

arXiv2016

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.

math.AG↗

arXiv2015

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].

astro-ph.SR↗

arXiv2014

A Search for Double-peaked narrow emission line Galaxies and AGNs in the LAMOST DR1

LAMOST has released more than two million spectra, which provide the opportunity to search for double-peaked narrow emission line (NEL) galaxies and AGNs. The double-peaked narrow-line profiles can be well modeled by two velocity components, respectively blueshifted and redshifted with respect to the systemic recession velocity. This paper presents 20 double-peaked NEL galaxies and AGNs found from LAMOST DR1 using a search method based on multi-gaussian fit of the narrow emission lines. Among them, 10 have already been published by other authors, either listed as genuine double-peaked NEL objects or as asymmetric NEL objects, the remaining 10 being first discoveries. We discuss some possible origins for double-peaked narrow-line features, as interaction between jet and narrow line regions, interaction with companion galaxies and black hole binaries. Spatially resolved optical imaging and/or follow-up observations in other spectral bands are needed to further discuss the physical mechanisms at work.

astro-ph.GA↗

arXiv2013

Deformation of singularities and additive invariants

In this survey on local additive invariants of real and complex definable singular germs we systematically present classical or more recent invariants of different nature as emerging from a tame degeneracy principle. For this goal, we associate to a given singular germ a specific deformation family whose geometry degenerates in such a way that it eventually gives rise to a list of invariants attached to this germ. Complex analytic invariants, real curvature invariants and motivic type invariants are encompassed under this point of view. We then explain how all these invariants are related to each other as well as we propose a general conjectural principle explaining why such invariants have to be related. This last principle may appear as the incarnation in definable geometry of deep finiteness results of convex geometry, according to which additive invariants in convex geometry are very few.

math.AG↗

arXiv2007

Equisingularite reelle : invariants locaux et conditions de regularite

For 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 evanescent 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.

math.DG↗

arXiv2006

Rotation of Trajectories of Lipschitz Vector Fields

We prove that in finite time a trajectory of a Lipschitz vector field in $\hbox{\bbbb R}^{\hbox{\tmm n}}$ can not have infinite rotation around a given point. This result extends to the mutual rotation of two trajectories of a field in $\hbox{\bbbb R}^{\hbox{\tmm 3}}$: this rotation is bounded from above on any finite time interval. The bounds we give are only in terms of the Lipschitz constant of the field and the length of the time interval.

math.CA↗