Foreword / José M. Gracia-Bondí­a -- Pre-Lie algebras and systems of Dyson–Schwinger equations / Loïc Foissy -- Five interpretations of Faà di Bruno’s formula / Alessandra Frabetti, Dominique Manchon -- A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems / W. Steven Gray, Luis A. Duffaut Espinosa -- On algebraic structures of numerical integration on vector spaces and manifolds / Alexander Lundervold, Hans Z. Munthe-Kaas -- Simple and contracting arborification / Emmanuel Vieillard-Baron -- Strong QCD and Dyson–Schwinger equations / Craig D. Roberts.

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Since the early works of G.-C. Rota and his school, Hopf algebras have been instrumental in algebraic combinatorics. In a seminal 1998 paper, A. Connes and D. Kreimer presented a Hopf algebraic approach to renormalization in perturbative Quantum Field Theory (QFT). This work triggered an abundance of new research on applications of Hopf algebraic techniques in QFT as well as other areas of theoretical physics. Furthermore, these new developments were complemented by progress made in other domains of applications, such as control theory, dynamical systems, and numerical integration methods. Especially in the latter context, it became clear that J. Butcher’s work from the early 1970s was well ahead of its time. The present volume emanated from a conference hosted in June 2011 by IRMA at Strasbourg University in France. Researchers from different scientific communities who share similar techniques and objectives gathered at this meeting to discuss new ideas and results on Faà di Bruno algebras, Dyson–Schwinger equations, and Butcher series. The purpose of this book is to present a coherent set of lectures reflecting the state of the art of research on combinatorial Hopf algebras relevant to high energy physics, control theory, dynamical systems, and numerical integration methods. More specifically, connections between Dyson–Schwinger equations, Faà di Bruno algebras, and Butcher series are examined in great detail. This volume is aimed at researchers and graduate students interested in combinatorial and algebraic aspects of QFT, control theory, dynamical systems and numerical analysis of integration methods. It contains introductory lectures on the various constructions that are emerging and developing in these domains.