Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas

Mathematics – Differential Geometry

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

38 pages, 0 figure

Scientific paper

In 1979, building on S. Lie's theory of symmetries of (partial) differrential equations, P.J. Olver formulated inductive formulas which are appropriate for the computation of the prolongations of an infinitesimal Lie symmetry to jet spaces, for an arbitrary number n\geq 1 of independent variables (x^1, ..., x^n) and for an arbitrary number m \geq 1 of dependent variables (y^1, ..., y^m). This paper is devoted to elaborate a formalism based on multiple Kronecker symbols which enables one to handle these ``unmanageable'' prolongations and to discover the underlying complicated combinatorics. Proceeding progressively, we write down closed explicit formulas in four cases: n=m=1; n\geq 1, m=1; n=1, m\geq 1; general case n\geq 1, m\geq 1. As a subpart of the obtained formulas, we recover four possible versions of the (multivariate) Fa\`a di Bruno formula. We do not employ the classical formalism based on the symmetric algebra ({\it cf.} e.g. H. Federer's book, p. 222), because it hides several explicit sums in symbolic compactifications, and because the presence of supplementary complexities ({\it e.g.} splitting of indices, combinatorics of partial derivatives) impedes us to apply such compactifications coherently. Our method of exposition is inductive: we conduct our reasonings by analyzing several thoroughly organized formulas, by comparing them together and by ``drifting'' towards generality, in homology with the classical style of L. Euler.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Four explicit formulas for the prolongations of an infinitesimal Lie symmetry and multivariate Faa di Bruno formulas will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-14159

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.