The 'algebra as object' analogy: a view from school

Treating algebraic symbols as objects (eg. “‘a’ means ‘apple’”) is a means of introducing elementary simplification of algebra, but causes problems further on. This current school-based research included an examination of texts still in use in the mathematics department, and interviews with mathematics teachers, year 7 pupils and then year 10 pupils asking them how they would explain, “3a + 2a = 5a” to year 7 pupils. Results included the notion that the ‘algebra as object’ analogy can be found in textbooks in current usage, including those recently published. Teachers knew that they were not ‘supposed’ to use the analogy but not always clear why, nevertheless stating methods of teaching consistent with an‘algebra as object’ approach. Year 7 pupils did not explicitly refer to ‘algebra as object’, although some of their responses could be so interpreted. In the main, year 10 pupils used ‘algebra as object’ to explain simplification of algebra, with some complicated attempts to get round the limitations. Further research would look to establish whether the appearance of ‘algebra as object’ in pupils’ thinking between year 7 and 10 is consistent and, if so, where it arises. Implications also are for on-going teacher training with alternatives to introducing such simplification.

Response theory for equilibrium and non-equilibrium statistical mechanics: causality and generalized Kramers-Kronig relations

We consider the general response theory recently proposed by Ruelle for describing the impact of small perturbations to the non-equilibrium steady states resulting from Axiom A dynamical systems. We show that the causality of the response functions entails the possibility of writing a set of Kramers-Kronig (K-K) relations for the corresponding susceptibilities at all orders of nonlinearity. Nonetheless, only a special class of directly observable susceptibilities obey K-K relations. Specific results are provided for the case of arbitrary order harmonic response, which allows for a very comprehensive K-K analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short-time response of the perturbed system. These results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models, and shed light on the very general impact of considering the principle of causality for testing self-consistency: the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey. The theory exposed in the present paper is dual to the time-dependent theory of perturbations to equilibrium states and to non-equilibrium steady states, and has in principle similar range of applicability and limitations. In order to connect the equilibrium and the non equilibrium steady state case, we show how to rewrite the classical response theory by Kubo so that response functions formally identical to those proposed by Ruelle, apart from the measure involved in the phase space integration, are obtained. These results, taking into account the chaotic hypothesis by Gallavotti and Cohen, might be relevant in several fields, including climate research. In particular, whereas the fluctuation-dissipation theorem does not work for non-equilibrium systems, because of the non-equivalence between internal and external fluctuations, K-K relations might be robust tools for the definition of a self-consistent theory of climate change.

Evidence of dispersion relations for the nonlinear response of the Lorenz 63 system

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.

Collision prediction of a moving object within a virtual world

This paper provides a solution for predicting moving/moving and moving/static collisions of objects within a virtual environment. Feasible prediction in real-time virtual worlds can be obtained by encompassing moving objects within a sphere and static objects within a convex polygon. Fast solutions are then attainable by describing the movement of objects parametrically in time as a polynomial.

Keeping the world at hand: rapid visuomotor processing for hand-object interactions

The existence of hand-centred visual processing has long been established in the macaque premotor cortex. These hand-centred mechanisms have been thought to play some general role in the sensory guidance of movements towards objects, or, more recently, in the sensory guidance of object avoidance movements. We suggest that these hand-centred mechanisms play a specific and prominent role in the rapid selection and control of manual actions following sudden changes in the properties of the objects relevant for hand-object interactions. We discuss recent anatomical and physiological evidence from human and non-human primates, which indicates the existence of rapid processing of visual information for hand-object interactions. This new evidence demonstrates how several stages of the hierarchical visual processing system may be bypassed, feeding the motor system with hand-related visual inputs within just 70 ms following a sudden event. This time window is early enough, and this processing rapid enough, to allow the generation and control of rapid hand-centred avoidance and acquisitive actions, for aversive and desired objects, respectively

Conductance of amyloid β based peptide filaments: structure–function relations.

Controlling the morphology of self-assembled peptide nanostructures, particularly those based on amyloid peptides, has been the focus of intense research. In order to exploit these structures in electronic applications, further understanding of their electronic behavior is required. In this work, the role of peptide morphology in determining electronic conduction along self-assembled peptide nanofilament networks is demonstrated. The peptides used in this work were based on the sequence AAKLVFF, which is an extension of a core sequence from the amyloid b peptide. We show that the incorporation of a non-natural amino acid, 2-thienylalanine, instead of phenylalanine improves the obtained conductance with respect to that obtained for a similar structure based on the native sequence, which was not the case for the incorporation of 3-thienylalanine. Furthermore, we demonstrate that the morphology of the self-assembled structures, which can be controlled by the solvent used in the assembly process, strongly affects the conductance, with larger conduction obtained for a morphology of long, straight filaments. Our results demonstrate that, similar to natural systems, the assembly and folding of peptides could be of great importance for optimizing their function as components of electronic devices. Hence, sequence design and assembly conditions can be used to control the performance of peptide based structures in such electronic applications.

Young people’s caring relations and transitions within families affected by HIV

This chapter provides insight into young people’s caring relations and transitions within what is often considered a particularly ‘troubling’ familial context in both the global North and South: living with HIV. I analyse the findings from two qualitative studies of young people’s caring roles in families affected by HIV in the UK, Tanzania and Uganda from the perspective of a feminist ethics of care, emotion work and life course transitions.

Transatlantic Relations: Sharing Ideals and Costs

A study of European relations with the USA and Canada after the end of the Cold War

Haunted by History: Myths in International Relations

A series of studies on interpretations of history and their use in political debates. Includes: Beatrice Heuser & Cyril Buffet: ‘Michel and Marianne’ Beatrice Heuser & Cyril Buffet: ‘Of Myths and Men’, Beatrice Heuser & Cyril Buffet: ‘Historical Myths and the Denial of Change’, Beatrice Heuser: ‘Dunkirk, Dien Bien Phu, and Suez, or why France doesn't trust allies and has learned to love the bomb’.