No abstract

No abstract

The basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated, leading to the conclusion that if they are differential equations on its phase space M , they are at most of the 2nd order. For open systems interacting with a bath at canonical equilibrium they have a particular form of an equation of a generalized Fokker-Planck type. We show that it is possible to obtain them as Liouville equations of Hamiltonian dynamics on M with a particular non-commutative differential structure, provided certain geometric in character, conditions are fulfilled. To this end, symplectic geometry on M is developped in this context, and an outline of the required tensor analysis and differential geometry is given. Certain questions for the possible mathematical interpretation of this structure are also discussed.

This paper associates the findings of a historical study with those of an empirical one with 16 years-old students (1st year of the Greek Lyceum). It aims at examining critically the much-discussed and controversial relation between the historical evolution of mathematical concepts and the process of their teaching and learning. The paper deals with the order relation on the number line and the algebra of inequalities, trying to elucidate the development and functioning of this knowledge both in the world of scholarly mathematical activity and the world of teaching and learning mathematics in secondary education. This twofold analysis reveals that the old idea of a "parallelism" between history and pedagogy of mathematics has a subtle nature with at least two different aspects (metaphorically named "positive" and "negative"), which are worth further exploration.