Computation of maximal local (un)stable manifold patches by the parameterization method

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.Comment: Revised version, typos corrected, references adde

Formal inverses of the generalized Thue-Morse sequences and variations of the Rudin-Shapiro sequence

A formal inverse of a given automatic sequence (the sequence of coefficients of the composition inverse of its associated formal power series) is also automatic. The comparison of properties of the original sequence and its formal inverse is an interesting problem. Such an analysis has been done before for the Thue{Morse sequence. In this paper, we describe arithmetic properties of formal inverses of the generalized Thue-Morse sequences and formal inverses of two modifications of the Rudin{Shapiro sequence. In each case, we give the recurrence relations and the automaton, then we analyze the lengths of strings of consecutive identical letters as well as the frequencies of letters. We also compare the obtained results with the original sequences.Comment: 20 page

Automatic Classification of Restricted Lattice Walks

We propose an experimental mathematics approach leading to the computer-driven discovery of various structural properties of general counting functions coming from enumeration of walks

Computational Particle Physics for Event Generators and Data Analysis

High-energy physics data analysis relies heavily on the comparison between experimental and simulated data as stressed lately by the Higgs search at LHC and the recent identification of a Higgs-like new boson. The first link in the full simulation chain is the event generation both for background and for expected signals. Nowadays event generators are based on the automatic computation of matrix element or amplitude for each process of interest. Moreover, recent analysis techniques based on the matrix element likelihood method assign probabilities for every event to belong to any of a given set of possible processes. This method originally used for the top mass measurement, although computing intensive, has shown its power at LHC to extract the new boson signal from the background. Serving both needs, the automatic calculation of matrix element is therefore more than ever of prime importance for particle physics. Initiated in the eighties, the techniques have matured for the lowest order calculations (tree-level), but become complex and CPU time consuming when higher order calculations involving loop diagrams are necessary like for QCD processes at LHC. New calculation techniques for next-to-leading order (NLO) have surfaced making possible the generation of processes with many final state particles (up to 6). If NLO calculations are in many cases under control, although not yet fully automatic, even higher precision calculations involving processes at 2-loops or more remain a big challenge. After a short introduction to particle physics and to the related theoretical framework, we will review some of the computing techniques that have been developed to make these calculations automatic. The main available packages and some of the most important applications for simulation and data analysis, in particular at LHC will also be summarized.Comment: 19 pages, 11 figures, Proceedings of CCP (Conference on Computational Physics) Oct. 2012, Osaka (Japan) in IOP Journal of Physics: Conference Serie