An Integro-Differential Structure for Dirac Distributions

We develop a new algebraic setting for treating piecewise functions and distributions together with suitable differential and Rota-Baxter structures. Our treatment aims to provide the algebraic underpinning for symbolic computation systems handling such objects. In particular, we show that the Green's function of regular boundary problems (for linear ordinary differential equations) can be expressed naturally in the new setting and that it is characterized by the corresponding distributional differential equation known from analysis.Comment: 38 page

Algebraic Birkhoff decomposition and its applications

Central in the Hopf algebra approach to the renormalization of perturbative quantum field theory of Connes and Kreimer is their Algebraic Birkhoff Decomposition. In this tutorial article, we introduce their decomposition and prove it by the Atkinson Factorization in Rota-Baxter algebra. We then give some applications of this decomposition in the study of divergent integrals and multiple zeta values.Comment: 39 pages. To appear in "Automorphic Forms and Langlands Program

Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology

In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.Comment: 29 pages, eps figures; minor changes; final versio