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== elementary differentials definition ==

Pardon my confusion, but the definition of the elementary differentials is not completely clear (to me). Is the n here the same as the dimension of the domain of f? I assume so, from the partial derivatives on the right-hand side. Then we are assuming that every node of the tree in question has either 0 or n children?

Are the j_i on the right-hand side each being summed from 1 to n? Or is something else going on here? ] (]) 10:50, 3 July 2009 (UTC)

:''n'' is the number of children, which is arbitrary. The j_i are summed from 1 to ''N'', where ''N'' is the dimension of the domain. This should probably be explained in more detail. -- ] (]) 15:05, 3 July 2009 (UTC)
::Thanks, that makes perfect sense; I was confusing n and N. I've tentatively added bounds to the sum in the formula; I'm not sure what the style convention is when there are multiple indices so please revert if incorrect. ] (]) 15:47, 3 July 2009 (UTC)

== The connection to the Connes-Kreimer Hopf algebra ==

The second paragraph of the introduction is a bit misleading: I believe the connection between the Butcher group and Connes and Kreimer's work on renormalization was first pointed out in "Brouder (2000)", not in "Connes & Kreimer (1999)". Cf. "Connes & Kreimer (1999)" p. 90:
: ''(...) allowed C. Brouder to recognize a far-reaching relation with the combinatorics of numerical analysis as worked out by Butcher several decades ago.''
] (]) 12:58, 1 February 2011 (UTC)

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elementary differentials definition

Pardon my confusion, but the definition of the elementary differentials is not completely clear (to me). Is the n here the same as the dimension of the domain of f? I assume so, from the partial derivatives on the right-hand side. Then we are assuming that every node of the tree in question has either 0 or n children?

Are the j_i on the right-hand side each being summed from 1 to n? Or is something else going on here? TotientDragooned (talk) 10:50, 3 July 2009 (UTC)

n is the number of children, which is arbitrary. The j_i are summed from 1 to N, where N is the dimension of the domain. This should probably be explained in more detail. -- Jitse Niesen (talk) 15:05, 3 July 2009 (UTC)
Thanks, that makes perfect sense; I was confusing n and N. I've tentatively added bounds to the sum in the formula; I'm not sure what the style convention is when there are multiple indices so please revert if incorrect. TotientDragooned (talk) 15:47, 3 July 2009 (UTC)

The connection to the Connes-Kreimer Hopf algebra

The second paragraph of the introduction is a bit misleading: I believe the connection between the Butcher group and Connes and Kreimer's work on renormalization was first pointed out in "Brouder (2000)", not in "Connes & Kreimer (1999)". Cf. "Connes & Kreimer (1999)" p. 90:

(...) allowed C. Brouder to recognize a far-reaching relation with the combinatorics of numerical analysis as worked out by Butcher several decades ago.

Alu042 (talk) 12:58, 1 February 2011 (UTC)

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