Beurling zeta functions, generalised primes, and fractal membranes

We study generalised prime systems P (1 < p(1) <= p(2) <= ..., with p(j) is an element of R tending to infinity) and the associated Beurling zeta function zeta p(s) = Pi(infinity)(j=1)(1 - p(j)(-s))(-1). Under appropriate assumptions, we establish various analytic properties of zeta p(s), including its analytic continuation, and we characterise the existence of a suitable generalised functional equation. In particular, we examine the relationship between a counterpart of the Prime Number Theorem (with error term) and the properties of the analytic continuation of zeta p(s). Further we study 'well-behaved' g-prime systems, namely, systems for which both the prime and integer counting function are asymptotically well-behaved. Finally, we show that there exists a natural correspondence between generalised prime systems and suitable orders on N-2. Some of the above results are relevant to the second author's theory of 'fractal membranes', whose spectral partition functions are given by Beurling-type zeta functions, as well as to joint work of that author and R. Nest on zeta functions attached to quasicrystals.

Optimization of apodization functions in terahertz transient spectrometry

A quadratic programming optimization procedure for designing asymmetric apodization windows tailored to the shape of time-domain sample waveforms recorded using a terahertz transient spectrometer is proposed. By artificially degrading the waveforms, the performance of the designed window in both the time and the frequency domains is compared with that of conventional rectangular, triangular (Mertz), and Hamming windows. Examples of window optimization assuming Gaussian functions as the building elements of the apodization window are provided. The formulation is sufficiently general to accommodate other basis functions. (C) 2007 Optical Society of America

Action observation and robotic agents: learning and anthropomorphism

The ‘action observation network’ (AON), which is thought to translate observed actions into motor codes required for their execution, is biologically tuned: it responds more to observation of human, than non-human, movement. This biological specificity has been taken to support the hypothesis that the AON underlies various social functions, such as theory of mind and action understanding, and that, when it is active during observation of non-human agents like humanoid robots, it is a sign of ascription of human mental states to these agents. This review will outline evidence for biological tuning in the AON, examining the features which generate it, and concluding that there is evidence for tuning to both the form and kinematic profile of observed movements, and little evidence for tuning to belief about stimulus identity. It will propose that a likely reason for biological tuning is that human actions, relative to non-biological movements, have been observed more frequently while executing corresponding actions. If the associative hypothesis of the AON is correct, and the network indeed supports social functioning, sensorimotor experience with non-human agents may help us to predict, and therefore interpret, their movements.

Mathematics as a tool in the Life Sciences

Teaching mathematics to students in the biological sciences is often fraught with difficulty. Students often discover mathematics to be a very 'dry' subject in which it is difficult to see the motivation of learning it given its often abstract application. In this paper I advocate the use of mathematical modelling as a method for engaging students in understanding the use of mathematics in helping to solve problems in the Biological Sciences. The concept of mathematics as a laboratory tool is introduced and the importance of presenting students with relevant, real-world examples of applying mathematics in the Biological Sciences is discussed.

Predicting long-term growth in students' mathematics achievement: the unique contributions of motivation and cognitive strategies

This research examined how motivation (perceived control, intrinsic motivation, and extrinsic motivation), cognitive learning strategies (deep and surface strategies), and intelligence jointly predict long-term growth in students' mathematics achievement over 5 years. Using longitudinal data from six annual waves (Grades 5 through 10; Mage = 11.7 years at baseline; N = 3,530), latent growth curve modeling was employed to analyze growth in achievement. Results showed that the initial level of achievement was strongly related to intelligence, with motivation and cognitive strategies explaining additional variance. In contrast, intelligence had no relation with the growth of achievement over years, whereas motivation and learning strategies were predictors of growth. These findings highlight the importance of motivation and learning strategies in facilitating adolescents' development of mathematical competencies.

Diversity in mathematics assessment equals diversity in opinion?

We undertook a study to investigate the views of both students and staff in our department towards assessment in mathematics, as a precursor to considering increasing the diversity of assessment types. In a survey and focus group there was reasonable agreement amongst the students with regards major themes such as mode of assessment. However, this level of agreement was not seen amongst the staff, where discussions regarding diversity in mathematics assessment definitely revealed a difference of opinion. As a consequence, we feel that the greatest barriers to increasing diversity may be with staff, and so more efforts are needed to communicate to staff the advantages and disadvantages, in order to give them greater confidence in trying a range of assessment types.

littleBits go LARGE: making electronics more accessible to people with learning disabilities

The “littleBits go LARGE" project extends littleBits electronic modules, an existing product that is aimed at simplifying electronics for a wide range of audiences. In this project we augment the littleBits modules to make them more accessible to people with learning disabilities. We will demonstrate how we have made the modules easier to handle and manipulate physically, and how we are augmenting the design of the modules to make their functions more obvious and understandable.

Latent learning in end stage renal disease (ESRD)

Cognitive functions such as attention and memory are known to be impaired in End Stage Renal Disease (ESRD), but the sites of the neural changes underlying these impairments are uncertain. Patients and controls took part in a latent learning task, which had previously shown a dissociation between patients with Parkinson’s disease and those with medial temporal damage. ESRD patients (n=24) and age and education-matched controls (n=24) were randomly assigned to either an exposed or unexposed condition. In Phase 1 of the task, participants learned that a cue (word) on the back of a schematic head predicted that the subsequently seen face would be smiling. For the exposed (but not unexposed) condition, an additional (irrelevant) colour cue was shown during presentation. In Phase 2, a different association, between colour and facial expression, was learned. Instructions were the same for each phase: participants had to predict whether the subsequently viewed face was going to be happy or sad. No difference in error rate between the groups was found in Phase 1, suggesting that patients and controls learned at a similar rate. However, in Phase 2, a significant interaction was found between group and condition, with exposed controls performing significantly worse than unexposed (therefore demonstrating learned irrelevance). In contrast, exposed patients made a similar number of errors to unexposed in Phase 2. The pattern of results in ESRD was different from that previously found in Parkinson’s disease, suggesting a different neural origin.