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DOI : 10.24033/bsmf.2757; International audience; We explain Ecalle's ''arbomould formalism'' in its simplest instance, showing how it allows one to give explicit formulas for the operators naturally attached to a germ of holomorphic map in one dimension. When applied to the classical linearization problem of non-resonant germs, which contains the well-known difficulties due to the so-called small divisor phenomenon, this elegant and concise tree formalism yields compact formulas, from which one easily recovers the classical analytical results of convergence of the solution under suitable arithmetical conditions on the multiplier. We rediscover this way Yoccoz's lower bound for the radius of convergence of the linearization and can even reach a global regularity result with respect to the multiplier (C^1-holomorphy) which improves on Carminati-Marmi's result. An appendix is devoted to the relationship between Ecalle's formalism and other algebraic constructions involving trees (the Connes-Kreimer Hopf algebra, a pre-Lie algebra, and representations thereof).

We describe the resurgence properties of some partition functions corresponding to field theories in dimension 0. We show that these functions satisfy linear differential equations with polynomial coefficients and then use elementary stability results or holonomic functions to prove resurgence properties, enhancing previously known results on growth estimates for the formal series involved, which had been obtained through a delicate combinatorics., Comment: 38 pages, 7 figures

We exhibit an internal coproduct on the Hopf algebra of finite topologies recently defined by the second author, C. Malvenuto and F. Patras, dual to the composition of "quasi-ormoulds", which are the natural version of J. Ecalle's moulds in this setting. All these results are displayed in the linear species formalism., Comment: 23 pages, one axodraw figure, Acknowledgements added

In this paper, we give new characterizations for the eigenvalues of the prolate wave equation as limits of the zeros of some families of polynomials: the coefficients of the formal power series appearing in the solutions near 0, 1 or ∞ (in the variables x, x − 1 or 1/x respectively). The result, which seems to be true for all values of the parameter τ , according to our numerical experiments, is here proved for small values of the parameter τ .

An internal coproduct is described, which is compatible with Hoffman's quasi-shuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with J. Ecalle's mould calculus, i.e. mould composition and contracting arborification, is precised., revised version, accepted for publication in Journal of Algebra

International audience; We give a natural and complete description of Ecalle's mould-comould formalism within a Hopf-algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes-Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogeneous coarborification, using the explicit duality between decorated Connes-Kreimer and Grossman-Larson algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.

These are the proceedings of a one-week international conference centered on asymptotic analysis and its applications. They contain major contributions dealing with: mathematical physics: PT symmetry, perturbative quantum field theory, WKB analysis, local dynamics: parabolic systems, small denominator questions, new aspects in mould calculus, with related combinatorial Hopf algebras and application to multizeta values, and a new family of resurgent functions related to knot theory.

Special Issue: Approximation and extrapolation of convergent and divergent sequences and series (CIRM, Luminy - France, 2009); International audience; The prolate spheroidal wave functions can be characterized as the eigenfunctions of a differential operator of order 2: (t2−τ2)φ″+2tφ′+σ2t2φ=μφ. In this article we study the formal solutions of this equation in the neighborhood of the singularities (the regular ones ±τ, and the irregular one, at infinity) and perform some numerical experiments on the computation of Stokes matrices and monodromy, using formal/numerical algorithms we developed recently in the Maple package Desir. This leads to the following conjecture: the series appearing in the formal solutions at infinity, depending on the parameter μ, are in general divergent; they become convergent for some particular values of the parameter, corresponding exactly to the eigenvalues of the prolate operator. We give the proof of this result and its interpretation in terms of differential Galois groups.

This book contains nine refereed research papers in various areas, from combinatorics to dynamical systems, with computer algebra as an underlying and unifying theme. Topics covered are irregular connections, summability of solutions and rank reduction of differential systems, asymptotic behaviour of divergent series, integrability of Hamiltonian systems, multiple zeta values, quasi polynomial formalism, Pade approximants related to analytic integrability, hybrid systems. The volume as a whole gives a presentation of productive interactions between ideas stemming from computer algebra and questions arising in dynamical systems or combinatorics. As such, it should be useful for both mathematicians and theoretical physicists who are interested in effective computation.

This article is an introduction to some aspects of \'Ecalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler's vector field $x^2\frac{\pa}{\pa x}+(x+y)\frac{\pa}{\pa y}$, and for which the formal normalisation is shown to be resurgent in $1/x$. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus., Comment: 78 pages