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

On non-multiaffine consistent-around-the-cube lattice equations

We show that integrable involutive maps, due to the fact they admit three integrals in separated form, can give rise to equations, which are consistent around the cube and which are not in the multiaffine form assumed in papers [1, 2]. Lattice models, which are discussed here, are related to the lattice potential KdV equation by nonlocal transformations (discrete quadratures).Comment: Isaac Newton Institute for Mathematical Sciences Preprint No NI11010-DIS 201

Spartan Daily, October 4, 1945

Volume 34, Issue 2https://scholarworks.sjsu.edu/spartandaily/3641/thumbnail.jp

Analogs of the M-Function in the Theory of Orthogonal Polynomials on the Unit Circle

We show that the multitude of applications of the Weyl-Titchmarsh m-function leads to a multitude of different functions in the theory of orthogonal polynomials on the unit circle that serve as analogs of the m-function

Models of advance directives in mental health care: stakeholder views

<i>Objective</i>: The aim of this study was to examine perceptions of the place of advance directives in mental health care. <i>Methods</i>: Postal survey of stakeholders was carried out to assess their views on different models of advance directives in mental health care. A total of 473 responded. <i>Results</i>: In all, 28% of psychiatrists thought advance directives were needed compared to 89% of voluntary organisations and above two–thirds of the other stakeholder groups. There were clear tensions between patient autonomy and right to treatment which underpin many of the concerns raised. Autonomy provided by advance directive can be contrasted with a co–operative partnership approach to advance planning. The legal status of advance directives is important for some people in relation to treatment refusal. There was general concern about the practical issues surrounding their implementation. <i>Conclusion</i>: There is a wide range of views in all stakeholder groups about the possible form advance directives should take. Although there is a widespread desire to increase patient involvement in treatment decisions, which advance directives could possibly help to realise, they may also have unwanted consequences for mental health services and individuals

Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT

In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras.Comment: 19 pages, 2 figure

Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion

We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.Comment: accepted for publication in Comm. in Math. Phy