Constantinos Tzanakis

Integrating history of mathematics in the classroom: an analytic survey

An analytical survey of how history of mathematics has been and can be integrated into the mathematics classroom provides a range of models for teachers and mathematics educators to use or adapt.

The political context

People have studied, learned and used mathematics for over four thousand years. Decisions on what is to be taught in schools, and how, are ultimately political, influenced by a number of factors including the experience of teachers, expectations of parents and employers, and the social context of debates about the curriculum. The ICMI study is posited on the experience of many mathematics teachers across the world that its history makes a difference: that having history of mathematics as a resource for the teacher is beneficial.

Kinetic theory in the weak-coupling approximation

Abstract A general formalism, where irreversible processes are related to singularities of the resolvent of the Liouville operator, is applied to classical open systems. For a system weakly coupled to a thermal reservoir, a kinetic equation is derived. It is shown that the method leads to equations defining a positivity-preserving semigroup with the Maxwell-Boltzmann distribution as a stationary solution and obeying an H-theory. It is pointed out that these properties are not always shared by irreversible equations obtained as asymptotic approximations of the so-called generalized master equation.

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.

Discovering by analogy: the case of Schrödinger's equation

Schr?dingers equation is derived by using Hamiltons mathematically unified treatment of geometrical optics and analytical mechanics. This is done by a didactically appropriate reconstruction of Schr?dingers approach, in which the formal similarity of the two theories is considered in the strict sense of the proportionality of the corresponding quantities in the two theories. This is taken as an example emphasizing the crucial role analogy arguments can play in the formulation of new physical ideas or theories and illustrating the role the knowledge of the historical development of a subject can play in its presentation.

Kinetic equation for a classical test-particle model in the weak-coupling approximation

A kinetic equation for a harmonically bound classical particle, weakly coupled to a thermal reservoir, is presented. The derivation, based on the formalism of master equations developed in Brussels, is outlined and compared to other approaches. Some comments are also made as to the consistency of the so-called Kramers equation in the presence of an external field.

Unfolding interrelations between mathematics and physics, in a presentation motivated by history: two examples

A genetic approach revealing interrelations between mathematics and physics, in which history plays a prominent role by motivating their presentation, is outlined. This approach is illustrated by a detailed analysis of two examples, indicative of the intimate connection between these disciplines: (a) Newtons gravitational law derived from Keplers laws, as an application of differential calculus; (b) an account of Special Relativitys foundations, as an example of the use of simple matrix algebra in formulating and deriving physically important results.

Historical support for particular subjects

This chapter provides further specific examples of using historical mathematics in the classroom, both to support and illustrate the arguments in chapter 7, and to indicate the ways in which the teaching of particular subjects may be supported by the integration of historical resources.

Rotations, complex numbers and quaternions

The relation between complex numbers and plane rotations is given. In an effort to extend this relation to space rotations and some generalizations of complex numbers, one is led to Hamiltons quaternions and the group of unitary transformations SU(2). The presentation is inspired by the historical development of the subject and aims at giving a natural formulation of important algebraic concepts, based on the study of mathematically relevant concrete examples.

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

Abstract 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.