Analyzable Functions and Applications.

Intro -- Contents -- Preface -- A singularly perturbed Riccati equation -- On global aspects of exact WKB analysis of operators admitting infinitely many phases -- Asymptotic differential algebra -- Introduction -- 1. Hardy Fields -- 2. The Field of Logarithmic-Exponential Series -- 3. H-Fields and Asymptotic Couples -- 4. Algebraic Differential Equations over H-Fields -- References -- Formally well-posed Cauchy problems for linear partial differential equations with constant coefficients -- Non-oscillating integral curves and O-minimal structures -- 1. Introduction -- 2. Definitions and examples -- 2.1 Hardy fields and o-minimal structures -- 2.2 Boshernitzan's example -- 2.3 Quasianalytic Denjoy-Carleman classes -- 3. Euler's equation -- 3.1 Euler's equation in the real plane -- 3.2 Euler's equation in the complex plane -- 3.3 Formal conjugation -- Asymptotics and singularities for a class of difference equations -- Topological construction of transseries and introduction to generalized Borel summability -- 1. Introduction -- 1.1. Abstract multiseries -- 1.2. Topology on multiseries -- 1.3. Contractive operators -- 1.4. Inductive construction of logarithm-free transseries -- 1.5. The space T of general transseries -- 2. Equations in T: examples -- 2.1. Multidimensional systems: transseries solutions at irregular singularities of rank one -- 3. Borel summation techniques -- 3.1. Borel summation of transseries: a first order example -- 3.2. Generalized Borel summation for rank one ODEs -- 3.3. Difference equations and PDEs -- 3.4. More general irregular singularities and multisummability -- References -- Addendum to the hyperasymptotics for multidimensional Laplace integrals -- Higher-order terms for the de Moivre-Laplace theorem -- Twisted resurgence monomials and canonical-spherical synthesis of local objects.

1. Introduction: Object Analysis and Object Synthesis -- 1.1 The notion of Local Analytic Object -- 1.2 Object Analysis: the Bridge Equation -- 1.3 Object Synthesis: semi-formal candidates -- 1.4 Object Synthesis: from semi-formal to effective -- 2. Reminders about moulds, resurgent functions, alien derivations -- 2.1 Moulds/comoulds -- 2.2 Resurgent functions -- 2.3 Alien derivations or automorphisms. Their weights -- 2.4 Resurgence monomials -- 3. Object Analysis: six basic examples -- 3.1 Example 1: shift-like diffeomorphism -- 3.2 Example 2: Euler-like differential equation -- 3.3 Example 3: monocritical linear differential system -- 3.4 Example 4: monocritical non-linear differential system -- 3.5 Example 5: polycritical linear differential system -- 3.6 Example 6: polycritical non-linear differential system -- 4. The reverse problem: Object Synthesis -- 4.1 Standard or hyperlogarithmic resurgence monomials and monics -- 4.2 Semi-formal synthesis in Example 1 -- 4.3 Semi-formal synthesis in Example 2 -- 4.4 Semi-formal synthesis in the remaining examples -- 4.5 Inadequacy of the standard resurgence monomials for the purpose of synthesis -- 4.6 First intimations of "antipodality -- 4.7 The need for one free parameter at least -- 5. Methods for non-canonical Object Synthesis -- 5.1 Main and earliest method -- 5.2 Malgrange's method -- 5.3 The quasiconformal method -- 5.4 Comparison -- 6. Four closely related challenges -- 6.1 The main obstacle: faster-than-lateral growth -- 6.2 Challenge 1 : Searching for well-behaved averages -- 6.3 Challenge 2 : Searching for well-behaved alien derivations -- 6.4 Challenge 3 : Searching for well-behaved resurgence monomials -- 6.5 Challenge 4 : Searching for the proper notion of ramified exponential growth -- 6.6 Proofs -- 7. Well-behaved uniformising averages -- 7.1 Reminder about the standard or median average.

7.2 Diffusion-induced averages -- 7.3 Scale-invariance -- 7.4 The standard and organic averages as limit-cases -- 7.5 Proofs and comments -- 7.6 When exactly must we take recourse to well-behaved averages? -- 8. Well-behaved alien derivations -- 8.1 Reminder about the standard alien derivations -- 8.2 Diffusion-induced alien derivations -- 8.3 Scale-invariance -- 8.4 Standard and organic alien derivations as limit-cases -- 8.5 Proofs and comments -- 8.6 Tables of averages and derivations -- 8. 7 Pinpointing the difference between "good" and "bad -- 9. Proper notion of ramified-exponential growth -- 9.1 Bounded singularities -- 9.2 Manageable singularities -- 9.3 Arbitrary singularities. An open question -- 10. Well-behaved resurgence monomials -- 10.1 Reminder about the standard resurgence monomials -- 10.2 Well-behaved resurgence monomials induced by prodiffusions -- 10.3 Canonical well-behaved resurgence monomials -- 10.4 Proofs and comments -- 10.5 Extension to the multicritical case -- 11. Applications to canonical Object Synthesis -- 11.1 Outline of the construction -- 11.2 Proof of the convergence for a positive or large enough twist c -- 11.3 What is so special about the twistless case ? -- 11.4 The antipodal pairing -- 11.5 Iso-invariant deformations -- 11.6 Remarks and complements -- 12. Hyper-, peri-, para-logarithmic monomials and monies: total closure -- 12.1 Main objects and notations -- 12.2 Hyperlogarithmic monomials and monics -- 12.3 Perilogarithmic monomials and monics -- 12.4 Paralogarithmic monomials and monics -- Matching and singularities of canard values -- On the renormalization method of Chen, Goldenfeld, and Oono -- Generalizing surreal numbers -- Two examples of resurgence.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.