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Computation of Infinitesimals for a Group Action on a Multispace of One Independent Variable

Peter Rock

Abstract

This paper expands upon the work of Peter Olver's paper [Appl. Algebra Engrg. Comm. Comput. 11 (2001), 417-436], wherein Olver uses a moving frames approach to examine the action of a group on a curve within a generalization of jet space known as multispace. Here we seek to further study group actions on the multispace of curves by computing the infinitesimals for a given action. For the most part, we proceed formally, and produce in the multispace a recursion relation that closely mimics the previously known prolongation recursion relations for infinitesimals of a group action on jet space.

Computation of Infinitesimals for a Group Action on a Multispace of One Independent Variable

Abstract

This paper expands upon the work of Peter Olver's paper [Appl. Algebra Engrg. Comm. Comput. 11 (2001), 417-436], wherein Olver uses a moving frames approach to examine the action of a group on a curve within a generalization of jet space known as multispace. Here we seek to further study group actions on the multispace of curves by computing the infinitesimals for a given action. For the most part, we proceed formally, and produce in the multispace a recursion relation that closely mimics the previously known prolongation recursion relations for infinitesimals of a group action on jet space.
Paper Structure (14 sections, 12 theorems, 59 equations)

This paper contains 14 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Jet space
  4. Polynomial interpolation
  5. The Olver multispace
  6. Infinitesimals of group actions on jet space
  7. Differential structure
  8. Notation and divided differences
  9. Derivatives of discrete equations
  10. Computation of the infinitesimals
  11. Infinitesimal actions
  12. The coalescent limit
  13. Examples
  14. Conclusion and future work

Key Result

Proposition 2.9

If $z_i = (x_i, u_i)$ where all of the $(x_{i})$ are distinct, then the unique interpolating polynomial at the points $z_0,\dots,z_n$ of degree $\leq n$ is given by We will call this unique polynomial the $n$-th order Newton approximation of the function $u(x)$ based at the points $(x_i)$.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9: olver2001geometric
  • Theorem 2.10: olver2001geometric
  • ...and 14 more