Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. I

Editors: O. Costin, F. Fauvet, F. Menous, D. Sauzin

Publisher: Edizioni della Normale

Book Series : CRM Series

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

About this book

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, - a new family of resurgent functions related to knot theory.

Frontmatter

Complex elliptic pendulum

Abstract

This paper briefly summarizes previous work on complex classical mechanics and its relation to quantum mechanics. It then introduces a previously unstudied area of research involving the complex particle trajectories associated with elliptic potentials.

Carl M. Bender, Daniel W. Hook, Karta Singh Kooner

Parabolic attitude

Abstract

Being parabolic in complex dynamics is not a state of fact, but it is more an attitude. In these notes we explain the philosophy under this assertion.

Filippo Bracci

Power series with sum-product Taylor coefficients and their resurgence algebra

Abstract

The present paper is devoted to power series of SP type, i.e. with coefficients that are syntactically sum-product combinations. Apart from their applications to analytic knot theory and the so-called “Volume Conjecture”, SP-series are interesting in their own right, on at least four counts: (i) they generate quite distinctive resurgence algebras (ii) they are one of those relatively rare instances when the resurgence properties have to be derived directly from the Taylor coefficients (iii) some of them produce singularities that unexpectedly verify finite-order differential equations (iv) all of them are best handled with the help of two remarkable, infinite-order integral-differential transforms, mir and nir.

Jean Ecalle, Shweta Sharma

Brjuno conditions for linearization in presence of resonances

Abstract

We present a new proof, under a slightly different (and more natural) arithmetic hypothesis, and using direct computations via power series expansions, of a holomorphic linearization result in presence of resonances originally proved by Rüssmann.

Jasmin Raissy

Noncommutative symmetric functions and combinatorial Hopf algebras

Abstract

We present on a few examples a class of algebras which are increasingly popular in Combinatorics, and tend to permeate other fields as well. In particular, some of these algebras have connections with mould calculus and resurgence theory. They can be approached in many different ways. Here, they will be regarded as generalizations of the algebra of symmetric functions.

Jean-Yves Thibon

Backmatter

Title: Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. I
Editors: O. Costin
F. Fauvet
F. Menous
D. Sauzin
Publisher: Edizioni della Normale
Electronic ISBN: 978-88-7642-379-6
Print ISBN: 978-88-7642-374-1
DOI: https://doi.org/10.1007/978-88-7642-379-6

Springer Professional

Asymptotics in Dynamics, Geometry and PDEs; Generalized Borel Summation vol. I

About this book

Table of Contents

Frontmatter

Complex elliptic pendulum

Parabolic attitude

Power series with sum-product Taylor coefficients and their resurgence algebra

Brjuno conditions for linearization in presence of resonances

Noncommutative symmetric functions and combinatorial Hopf algebras

Backmatter

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