Miscellaneous

Abstract

The first section of this chapter fixes some notations for multi indices that are widely used in the main text. Except for the distinction into multi and “repeated” indices these are standard notations in multivariate analysis. We also show an important finiteness result for multi indices, namely Dickson’s Lemma, and discuss term orders. The next section recalls those properties of (real-)analytic functions that are needed in the proof of the Cauchy–Kovalevskaya Theorem. Then some elementary operations with differential equations like reduction to first order or quasilinearisation are discussed within the context of the formal theory. Finally, a few simple facts about the modified Stirling numbers appearing in manipulations of Cartan characters and Hilbert functions are presented.