Lie systems: theory, generalisations, and applications

Lie systems form a class of systems of first-order ordinary differential equations whose general solutions can be described in terms of certain finite families of particular solutions and a set of constants, by means of a particular type of mapping: the so-called superposition rule. Apart from this fundamental property, Lie systems enjoy many other geometrical features and they appear in multiple branches of Mathematics and Physics, which strongly motivates their study. These facts, together with the authors' recent findings in the theory of Lie systems, led to the redaction of this essay, which aims to describe such new achievements within a self-contained guide to the whole theory of Lie systems, their generalisations, and applications.Comment: 161 pages, 2 figure

Remarks on superdifferential equations

We show that the term `superdifferential equation' has been employed in the literature to refer to different types of differential equations with even and odd variables. It is justified on physical and mathematical grounds that a subclass of them, the hereafter called Grassmann-valued differential equations, cannot be effectively described through supergeometric techniques. Instead, we analyse them in terms of standard differential equations on Grassmann algebra bundles. Our considerations are illustrated through examples of physical and mathematical relevance.Comment: This paper includes substantial errors and is therefore withdrawed by the author