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

This work concerns the definition and analysis of a new class of Lie systems on Poisson manifolds enjoying rich geometric features: the Lie-Hamilton systems. We devise methods to study their superposition rules, time independent constants of motion and Lie symmetries, linearisability conditions, etc. Our results are illustrated by examples of physical and mathematical interest.

Abstract. We review some recent results of the theory of Lie systems in order to apply such results to study Ermakov systems. The fundamental properties of Ermakov systems, i.e. their superposition rules, the Lewis-Ermakov invariants, etc., are found from this new perspective. We also obtain new results, such as a new superposition rule for the Pinney equation in terms of three solutions of a related Riccati equation.

No abstract

We use the geometric approach to the theory of Lie systems of differential equations in order to study dissipative Ermakov systems. We prove that there is a superposition rule for solutions of such equations. This fact enables us to express the general solution of a dissipative Milne–Pinney equation in terms of particular solutions of a system of second-order linear differential equations and a set of constants.

A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the socalled Lie systems, out of generic families of particular solutions and a set of constants. The first aim of this work is to propose several generalisations of this notion to secondorder differential equations. Next, several results on the existence of such generalisations are given and relations with the theories of Lie systems and quasi-Lie schemes are found. Finally, our methods are used to study second-order Riccati equations and other secondorder differential equations of mathematical and physical interest.

No abstract

ResumenEl artículo sostiene la tesis de que, aunque la culturalización del debate migratorio tiene aspectos positivos, la prioridad se halla en la política, puesto que, si no hay participación polí-tica de los inmigrantes en la sociedad de acogida, no hay sentimiento de pertenencia, ni conciencia de identidad, y difícilmente cabe hablar de integración social de los inmigrantes. Palabras clave: filosofía del derecho, filosofía política, migraciones, cultura. Abstract. Immigration, cultural diversity and political reconnaissanceThis article makes the claim that although a culturalist orientation in the discussion of migration has positive aspects, the real priority is in the political level, since without political participation of inmigrants in their new society, there is no feeling of belonging or identity, and therefore, it is difficult to talk about inmigrants' social integration. Key words: philosophy of law, politics philosophy, migrations, culture.