Computability of the Radon-Nikodym derivative

We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the integrable functions on such spaces. For functions f,g on represented sets, f is W-reducible to g if f can be computed by applying the function g at most once. Let RN be the Radon-Nikodym operator on the space under consideration and let EC be the non-computable operator mapping every enumeration of a set of natural numbers to its characteristic function. We prove that for every computable measurable space, RN is W-reducible to EC, and we construct a computable measurable space for which EC is W-reducible to RN

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Computability and dynamical systems

In this paper we explore results that establish a link between dynamical systems and computability theory (not numerical analysis). In the last few decades, computers have increasingly been used as simulation tools for gaining insight into dynamical behavior. However, due to the presence of errors inherent in such numerical simulations, with few exceptions, computers have not been used for the nobler task of proving mathematical results. Nevertheless, there have been some recent developments in the latter direction. Here we introduce some of the ideas and techniques used so far, and suggest some lines of research for further work on this fascinating topic

Systems Perspectives on the Interaction Between Human and Technological Resources

Peer reviewe

Gender differences in mathematics: A discourse analysis

Perspectives dealing with the study of gender and mathematics have failed generally to move beyond the individual/society divide. The contradictory nature of subjectivity and the operation and interpenetration of power and knowledge have not been taken into account. This article is based on the post-structuralist framework. The work of Walkerdine, which highlights the processes within the classroom which allow girls to succeed in mathematics but never actually be successful, is of interest. The methodology used is that of discourse analysis which makes clear both the positionings available to the participants as well as the power relations formed. The sample was drawn from a top-achieving Std 8 Higher Grade class in an affluent Model C school. This represents a theoretically salient sample as the literature points to ‘gender differences’ being most pronounced in the upper levels of mathematics education. The analysis clearly highlights the double-bind within which girls find themselves in the mathematics classroom. The apparent equality of opportunity and non-sexism is counteracted by the positioning of girls as hard working but without natural flair in mathematics. The characteristics that make it possible to achieve in mathematics are ascribed to males. The resistance to this powerful ‘disciplinary technology’ is the invoking of the feminist discourse.This article is affiliated to the Educational Psychology Department at the University of Cape Town and WITS Rural FacilityFull text available on publisher website: http://journals.sagepub.com/doi/abs/10.1177/00812463950250030