Generalised Scalar-tensor Theory in the Bianchi Type I Model

We use a conformal transformation to find solutions to the generalised scalar-tensor theory, with a coupling constant dependent on a scalar field, in an empty Bianchi type I model. We describe the dynamical behaviour of the metric functions for three different couplings: two exact solutions to the field equations and a qualitative one are found. They exhibit non-singular behaviours and kinetic inflation. Two of them admit both General Relativity and string theory in the low-energy limit as asymptotic cases.     

Dynamical Study of the Empty Bianchi Type I Model in Generalised Scalar-tensor Theory

A dynamical study of the generalised scalar-tensor theory in the empty Bianchi type I model is made. We use a method from which we derive the sign of the first and second derivatives of the metric functions and examine three different theories that can all tend towards relativistic behaviours at late time. We determine conditions so that the dynamic is in expansion and decelerated at late time.     

The Gibbs' phenomenon from a signal processing point of view

Recently, Fay and Kloppers gave two proofs to show that the well-known Gibbs' phenomenon for Fourier series at a jump discontinuity depends only on the size of the jump and is a multiple of the integral 1/π ∫₀ ^π (sin x / x) dx. We give another proof, based upon low-pass filtering of the Fourier transform, that uses the observation that a truncated Fourier series for a function ? (x) is ‘very nearly’ equal to the convolution integral 1/π ∫ _-∞ ^+∞ ? (x - t)(sin nt / t) dt.     

Convergence for Fourier series solutions of the forced harmonic oscillator II

This paper compliments two recent articles by the author in this journal concerning solving the forced harmonic oscillator equation when the forcing is periodic. The idea is to replace the forcing function by its Fourier series and solve the differential equation term-by-term. Herein the convergence of such series solutions is investigated when the forcing function is bounded, piecewise continuous, and piecewise smooth. The series solution and its term-by-term derivative converge uniformly over the entire real line. The term-by-term differentiation produces a series for the second derivative that converges pointwise and uniformly over any interval not containing a jump discontinuity of the forcing function.     

Using the Knowledge Quartet to develop mathematics content knowledge: the role of reflection on professional development

In this paper I report findings from a four year study of beginning elementary school teachers which investigated development in their mathematical knowledge for teaching (MKT). The study took a developmental research approach, in that the teachers and the researcher collaborated to develop the mathematics teaching of the teachers, while also trying to understand how such development occurred and might be facilitated. The Knowledge Quartet (KQ) framework was used as a tool to support focused reflection on the mathematical content of teaching, with the aim of promoting development in mathematical content knowledge. Although I focused primarily on whether and how focused reflection using the KQ would promote development, it was impossible to separate this from other influences, and in this paper I discuss the ways in which reflection was found to interrelate with other areas of influence. I suggest that by helping the teachers to focus on the content of their mathematics teaching, within the context of their experience in classrooms and of working with others, the KQ framework supported development in the MKT of teachers in the study.