By S. Zaidman
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Extra info for Almost-periodic functions in abstract spaces
Moreover, as functions of the parameters describing the surface, the product map µ and the inverse map g → g −1 are smooth. The rigorous definition requires a Lie group to be a manifold, that is, locally Euclidean. We will discuss the rigorous definition in detail in Chapter 2. 13 A set T of invertible maps taking some space X to itself is a transformation group, with the group product being composition of mappings, if, (i) for all f , g ∈ T , f ◦ g ∈ T , (ii) the identity map id : X → X, id(x) = x for all x ∈ X, is in T , and (iii) if f ∈ T then its inverse f −1 ∈ T .
19 follows from iterative use of the chain rule. These formulae and their derivations can be found in every textbook on symmetries of differential equations, for example Olver (1993) or Bluman and Kumei (1989), in a seemingly endless variety of notational styles. It is well worth taking the time to calculate a selection of prolongations of infinitesimals, not only to be sure which index is which in the preferred notation, but then also to implement it in the preferred computer algebra system. The software will be needed to do the calculations in Chapter 4.
Back-substituting = t − t into the first expression and rearranging terms, we obtain that exp(−3µ2 t) sin(µx + κ) = exp(−3µ2 t) sin(µx + κ), in other words, I is an invariant. To verify the group action property for the variable x, set x1 = x( ). Note that sin(µx1 + κ) = exp(3µ2 ) sin(µx + κ) and sin(µx1 (δ) + κ) = exp(3µ2 δ) sin(µx1 + κ) and thus sin(µx1 (δ) + κ) = exp(3µ2 δ) sin(µx1 + κ) = exp(3µ2 δ) exp(3µ2 ) sin(µx + κ) = exp(3µ2 ( + δ)) sin(µx + κ) = sin(µx( + δ) + κ) so that x1 (δ) = x( + δ) as required (for small enough δ and ).