Bases for rings of coinvariants

We study the multiplicative structure of rings of coinvariants for finite groups. We develop methods that give rise to natural monomial bases for such rings over their ground fields and explicitly determine precisely which monomials are zero in the ring of coinvariants. We apply our methods to the Dickson, upper triangular and symmetric coinvariants. Along the way, we recover theorems of Steinberg [17] and E. Artin [1]. Using these monomial bases we prove that the image of the transfer for a general linear group over a finite field is a principal ideal in the ring of invariants. This research is supported in part by the Natural Sciences and Engineering Research Council of Canada     

Fundamental symmetries and precision measurements on trapped particles

The tests of fundamental symmetries (CPT symmetry and the equivalence principle for antiparticles) that can be deduced from high-precision frequency measurements on trapped particles are reviewed.     

Techniques which accelerate the convergence of first order iterations automatically

This paper describes a way of approximating the optimal extrapolation of iterative techniques for solving equation systems. The approximations avoid auxiliary eigenvalue calculations and can be revised automatically during the iterative process. These extrapolations are cheap to compute and are particularly suited to solving nonlinear systems where the iteration matrix is path dependent.     

Experimental conditions required to verify the existence of a photon medium

If a scalar field incorporating a photon/environment interaction exists in nature, it should be possible to create a photon medium by focusing intense laser beams either in a vacuum or in a plasma. Such a medium should sustain many of the non-linear optical effects observable in material media. Techniques are proposed for verifying the existence of a photon medium via its effect on both the velocity of light and the scattering of light by light in intense laser beams.     

Analogs of renormalization group transformations in random processes

We review the properties of a real-space renormalization group transformation of the free energy, including the existence of oscillatory terms multiplying the non-analytic part of the free energy. We then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation. (The essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary.) These random processes are shown to be generated directly from Cantor sets. In a spatial representation, the ensuing random process exhibits a transition between Gaussian and fractal behavior. In the fractal regime, the trajectories will, in an average sense, form self-similar clusters. In a temporal representation, the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior. The fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses.     

Positronium: Precision determination of the ground-state fine-structure interval Δv,and measurement of density shifts in the noble gases

The positronium n = 1 fine-structure interval, Δv, has been measured in nitrogen yielding Δv = 203.3849(12) GHz. Comparison with theory shows the need to calculate higher-order theoretical terms. The positronium density shifts in the noble gases were also measured.     

Teachers' funds of knowledge and the teaching and learning of mathematics in multi-ethnic primary schools: Two teachers' views of linking home and school

In this paper we explore two teachers' views on the role(s) of parents, the local community and children's home lives in the learning of mathematics in primary school. We use Moll and Greenberg's concept of ‘funds of knowledge’ and apply it to the case studies of two teachers working in the UK context. Issues of teachers' professional experience, ethnicity, class and gender emerge as significant in examining similarities and differences in the teachers' beliefs, understandings and practices in the area of linking home and school. We end with a discussion of some implications for teacher education and professional development.