This thesis proposal outlines an automated reasoning system for helping a user to navigate through a theory and discover theorems that mathematicians generally find to be interesting. The project focuses on the processes of invention of mathematical concepts, and invention (and verification) of conjectures about concepts by using a scheme-based approach. Mathematical theories usually have infin...

The transition density function of the Wright-Fisher diffusion describes the evolution of population-wide allele frequencies over time. This function has important practical applications in population genetics, but finding an explicit formula under a general diploid selection model has remained a difficult open problem. In this article, we develop a new computational method to tackle this class...

Innovations in biological evolution and in technology have many common features. Some of them involve similar processes, such as trial and error and horizontal information transfer. Others describe analogous outcomes such as multiple independent origins of similar innovations. Yet others display similar temporal patterns such as episodic bursts of change separated by periods of stasis. We revie...

Computer software can be used for artistic design and as a means to facilitate the understanding of mathematical concepts. We discuss some of the issues that arise when we use computers in this way, and some of our own experiences teaching both art and mathematics students using software that can assist in the design and coloring of symmetric images.

General Comments: This is a well-documented and well-written article, however much of it seems to dwell on and define standard mathematical concepts (like deterministic chaos, fixed points and attractors) and seems not appropriate to include in a research publication. Furthermore, a parameter study (as done in this paper) finds different solution style regimes and a linearized stability analysi...

Previous research has found that different presentations of the same concept can result in different patterns of transfer to isomorphic instances of the same concept. Much of this work has framed these effects in terms of advantages and disadvantages of concreteness or abstractness. We note that mathematics is a richly structured field, with deeply interconnected concepts and many distinct aspe...

Abstract. We propose a mathematical derivation of Brinkman’s force for a cloud of particles immersed in an incompressible viscous fluid. Specifically, we consider the Stokes or steady Navier-Stokes equations in a bounded domain Ω ⊂ R for the velocity field u of an incompressible fluid with kinematic viscosity ν and density 1. Brinkman’s force consists of a source term 6πνj where j is the curren...

The term isomorphism literally means equality or sameness (iso) of form (morphism). In mathematics an isomorphism between two systems requires a one-toone correspondence between their elements (that is, each element of one system corresponds to one and only one element of the other system, and conversely), which also preserves structures. Referring to isomorphism as one of the most important an...

In this paper theory from population biology is combined with stochastic differential equations. The aim is to compare previously studied deterministic population models with corresponding stochastic population models. As an illustrative example the models are applied to a classical dataset (Gause, 1934). The first half of the paper gives a brief review of the theoretical background needed in o...

Through reporting on an initiative in South Africa that aimed to provide epistemological access to teachers and learners of mathematics (and science) through translating mathematical concepts into two indigenous languages, this paper argues for the urgent development of mathematical registers in indigenous languages for mathematics and science. The pilot research reported on in this paper indic...