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In [[mathematics]], the '''Butcher group''', named after the New Zealand mathematician [[John C. Butcher]], is an algebraic formalism involving [[rooted tree]]s that provides [[formal power series]] solutions of the non-linear [[ordinary differential equation]]s modeling the flow of a [[vector field]]. It was [[Arthur Cayley]], in response to a question of [[James Joseph Sylvester]], who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. In [[numerical analysis]], Butcher's formalism provides a method for analysing solutions of ordinary differential equations by the [[Runge-Kutta method]]. It was later realised that his group and the associated [[Hopf algebra]] of rooted trees underlie the Hopf algebra introduced by [[Dirk Kreimer]] and [[Alain Connes]] in their work on [[renormalization]] in [[quantum field theory]]. |
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[[Arthur Cayley]], in response to a question by [[James Joseph Sylvester]], first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. In [[numerical analysis]], Butcher's formalism provides a method for analysing solutions of ordinary differential equations by the [[Runge-Kutta method]]. It was later realised that his group and the associated [[Hopf algebra]] of rooted trees underlie the Hopf algebra introduced by [[Dirk Kreimer]] and [[Alain Connes]] in their work on [[renormalization]] in [[quantum field theory]]. |
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==References== |
==References== |
Revision as of 08:42, 24 June 2009
In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher, is an algebraic formalism involving rooted trees that provides formal power series solutions of the non-linear ordinary differential equations modeling the flow of a vector field. It was Arthur Cayley, in response to a question of James Joseph Sylvester, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. In numerical analysis, Butcher's formalism provides a method for analysing solutions of ordinary differential equations by the Runge-Kutta method. It was later realised that his group and the associated Hopf algebra of rooted trees underlie the Hopf algebra introduced by Dirk Kreimer and Alain Connes in their work on renormalization in quantum field theory.
References
- Cayley, Arthur (1857), "On the theory of analytic forms called trees", Philosophical Magazine, XIII: 172–176 (also in Volume 3 of the Collected Works of Cayley, pages 242-246)
- Butcher, J.C (2009), "Trees and numerical methods for ordinary differential equations", Numerical Algorithms, Springer online
- Hairer, E.; Wanner, G. (1974), "On the Butcher group and general multi-value methods", Computing, 13: 1–15
- Connes, Alain; Kreimer, Dirk (1998), "Hopf Algebras, Renormalization and Noncommutative Geometry", Communications in Mathematical Physics, 199: 203–242
- Brouder, Christian (2000), "Runge-Kutta methods and renormalization", Eur.Phys.J., C12: 521–534