| Revision as of 05:14, 24 June 2009 editChildofMidnight (talk | contribs)43,041 edits tweaks← Previous edit | Revision as of 08:42, 24 June 2009 edit undoMathsci (talk | contribs)Autopatrolled, Extended confirmed users, Pending changes reviewers66,107 edits Undid revision 298284484 by ChildofMidnight (talk) rv senseless edit to stubNext edit → | ||
| Line 1: | Line 1: | ||
| In ], the '''Butcher group''', named after the New Zealand mathematician ], is an algebraic formalism involving ]s that provides ] solutions of the non-linear ]s modeling the flow of a ]. It was ], in response to a question of ], who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. In ], Butcher's formalism provides a method for analysing solutions of ordinary differential equations by the ]. It was later realised that his group and the associated ] of rooted trees underlie the Hopf algebra introduced by ] and ] in their work on ] in ]. | |||
| ], in response to a question by ], first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. In ], Butcher's formalism provides a method for analysing solutions of ordinary differential equations by the ]. It was later realised that his group and the associated ] of rooted trees underlie the Hopf algebra introduced by ] and ] in their work on ] in ]. | |||
| ==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
 | This mathematics-related article is a stub. You can help Misplaced Pages by expanding it. |