History of Mathematics in Mathematics Education: Recent devlopments

International audience<p>This is a concise survey on the recent developments (since 2000) concerning research on the relations between History and Pedagogy of Mathematics (the <i>HPM domain</i>). Section 1 explains the rationale of the study and formulates the key issues. Section 2 gives a brief historical account of the development of the <i>HPM domain</i> with focus on the main activities in its context and their outcomes. Section 3 provides a sufficiently comprehensive bibliographical survey of the work done in this area since 2000. Finally, section 4 summarizes the main points of this study.</p

Generalized Moyal structures in phase-space, master equations and their classical limit I. General formalism

Generalized Wigner and Weyl transformations of quantum operators are defined and their properties, as well as those of the algebraic structure induced on phase-space, are reviewed. Using such transformations, quantum linear evolution equations are given a phase-space representation; particularly, the general master equation of the Lindblad type generating quantum dynamical semigroups. The resulting expressions are better suited for deriving quantum corrections, taking the classical limit and for a general comparison of classical and quantum systems. We show that under quite general conditions, the classical limit of this master equation exists, is independent of the particular generalized Wigner transformation used and is an equation of the Fokker-Planck type (i.e. with nonnegative-definite leading coefficient) generating a classical Markov semigroup. © 1998 Elsevier Science B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Classical markovian kinetic equations: Explicit form and H-theorem

The probabilistic description of finite classical systems often leads to linear kinetic equations. A set of physically motivated mathematical requirements is accordingly formulated. We show that it necessarily implies that solutions of such a kinetic equation in the Heisenberg representation, define Markov semigroups on the space of observables. Moreover, a general H-theorem for the adjoint of such semigroups is formulated and proved provided that at least locally, an invariant measure exists. Under a certain continuity assumption, the Markov semigroup property is sufficient for a linear kinetic equation to be a second order differential equation with nonegative-definite leading coefficient. Conversely it is shown that such equations define Markov semigroups satisfying an H-theorem, provided there exists a nonnegative equilibrium solution for their formal adjoint.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Generalized Moyal structures in phase-space, master equations and their classical limit II. Applications to harmonic oscillator models

The formalism of the phase-space representation of quantum master equations via generalized Wigner transformations developed in a previous paper, is applied to the Lindblad-type kinetic equation, for a quantum harmonic oscillator coupled to an equilibrium bath of oscillators. The resulting equation is derived without introducing the rotating-wave approximation. In the classical limit, the equation reduces to a Fokker-Planck equation, which coincides with the one derived from the corresponding classical Hamiltonian. The formalism is also applied to other oscillator model equations often used in quantum optics. © 1998 Elsevier Science B.V. All rights reserved.SCOPUS: ar.jinfo:eu-repo/semantics/publishe