Oscillation of linear ordinary differential equations: on a theorem by A. Grigoriev

We give a simplified proof and an improvement of a recent theorem by A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions to systems of linear equations with polynomial or rational coefficients.Comment: 16 page

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

International Asteroid Warning Network Timing Campaign : 2019 XS

Peer reviewe

On Functions And Curves Defined By Ordinary Differential Equations

. These notes constitute a substantially extended version of a talk given in the Fields Institute (Toronto) during the semester "Singularities and Geometry", that culminated by Arnoldfest in celebration of V. I. Arnold's 60th anniversary. We give a survey of different results showing how an upper bound for the number of isolated zeros for functions satisfying ordinary differential equations, may be obtained without solving these equations. The main source of applications is the problem on zeros of complete Abelian integrals, one of the favorite subjects discussed on Arnold's seminar in Moscow for over quarter a century. Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. Isaac Newton It is useful to solve differential equations. Translation by Vladimir Arnold x1. Introduction 1.1. Equations and solutions. One of the illusions that are pleasant to nourish is the claim that simple equations cannot have complicated solutions. Though completely re..

International Asteroid Warning Network Timing Campaign : 2019 XS

Peer reviewe

The Second International Asteroid Warning Network Timing Campaign: 2005 LW3

The Earth close approach of near-Earth asteroid 2005 LW3 on 2022 November 23 represented a good opportunity for a second observing campaign to test the timing accuracy of astrometric observation. With 82 participating stations, the International Asteroid Warning Network collected 1046 observations of 2005 LW3 around the time of the close approach. Compared to the previous timing campaign targeting 2019 XS, some individual observers were able to significantly improve the accuracy of their reported observation times. In particular, U.S. surveys achieved good timing performance. However, no broad, systematic improvement was achieved compared to the previous campaign, with an overall negative bias persisting among the different observers. The calibration of observing times and the mitigation of timing errors should be important future considerations for observers and orbit computers, respectively.Funder: Institute of Cosmos SciencesUniversity of Barcelona (CEX2019-000918-M); European Union (PID2021-122842OB-C21);Full text license: CC BY</p