Butcher group

http://dbpedia.org/resource/Butcher_group an entity of type: Abstraction100002137

In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by , is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was , prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. rdf:langString
rdf:langString Butcher group
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rdf:langString In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by , is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was , prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. pointed out that the Butcher group is the group of characters of the Hopf algebra of rooted trees that had arisen independently in their own work on renormalization in quantum field theory and Connes' work with Moscovici on local index theorems. This Hopf algebra, often called the Connes–Kreimer algebra, is essentially equivalent to the Butcher group, since its dual can be identified with the universal enveloping algebra of the Lie algebra of the Butcher group. As they commented: We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results.
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