A technique for delivering individualised formative problems and examples

Active learning plays a strong role in mathematics and statistics, and formative problems are vital for developing key problem-solving skills. To keep students engaged and help them master the fundamentals before challenging themselves further, we have developed a system for delivering problems tailored to a student‟s current level of understanding. Specifically, by adapting simple methodology from clinical trials, a framework for delivering existing problems and other illustrative material has been developed, making use of macros in Excel. The problems are assigned a level of difficulty (a „dose‟), and problems are presented to the student in an order depending on their ability, i.e. based on their performance so far on other problems. We demonstrate and discuss the application of the approach with formative examples developed for a first year course on plane coordinate geometry, and also for problems centred on the topic of chi-square tests.

Flavonoids as modulators of memory and learning: molecular interactions resulting in behavioural effects

There is considerable interest in the potential of a group of dietary-derived phytochemicals known as flavonoids in modulating neuronal function and thereby influencing memory, learning and cognitive function. The present review begins by detailing the molecular events that underlie the acquisition and consolidation of new memories in the brain in order to provide a critical background to understanding the impact of flavonoid-rich diets or pure flavonoids on memory. Data suggests that despite limited brain bioavailability, dietary supplementation with flavonoid-rich foods, such as blueberry, green tea and Ginkgo biloba lead to significant reversals of age-related deficits on spatial memory and learning. Furthermore, animal and cellular studies suggest that the mechanisms underpinning their ability to induce improvements in memory are linked to the potential of absorbed flavonoids and their metabolites to interact with and modulate critical signalling pathways, transcription factors and gene and/or protein expression which control memory and learning processes in the hippocampus; the brain structure where spatial learning occurs. Overall, current evidence suggests that human translation of these animal investigations are warranted, as are further studies, to better understand the precise cause-and-effect relationship between flavonoid intake and cognitive outputs.

On discrimination algorithms for ill-posed problems with an application to magnetic tomography

In this paper we explore classification techniques for ill-posed problems. Two classes are linearly separable in some Hilbert space X if they can be separated by a hyperplane. We investigate stable separability, i.e. the case where we have a positive distance between two separating hyperplanes. When the data in the space Y is generated by a compact operator A applied to the system states ∈ X, we will show that in general we do not obtain stable separability in Y even if the problem in X is stably separable. In particular, we show this for the case where a nonlinear classification is generated from a non-convergent family of linear classes in X. We apply our results to the problem of quality control of fuel cells where we classify fuel cells according to their efficiency. We can potentially classify a fuel cell using either some external measured magnetic field or some internal current. However we cannot measure the current directly since we cannot access the fuel cell in operation. The first possibility is to apply discrimination techniques directly to the measured magnetic fields. The second approach first reconstructs currents and then carries out the classification on the current distributions. We show that both approaches need regularization and that the regularized classifications are not equivalent in general. Finally, we investigate a widely used linear classification algorithm Fisher's linear discriminant with respect to its ill-posedness when applied to data generated via a compact integral operator. We show that the method cannot stay stable when the number of measurement points becomes large.

Applications of behavioural economics to tax evasion

The paper reviews recent models that have applied the techniques of behavioural economics to the analysis of the tax compliance choice of an individual taxpayer. The construction of these models is motivated by the failure of the Yitzhaki version of the Allingham–Sandmo model to predict correctly the proportion of taxpayers who will evade and the effect of an increase in the tax rate upon the chosen level of evasion. Recent approaches have applied non-expected utility theory to the compliance decision and have addressed social interaction. The models we describe are able to match the observed extent of evasion and correctly predict the tax effect but do not have the parsimony or precision of the Yitzhaki model.

Cognitive, affective and behavioural characteristics of mothers with anxiety disorders in the context of child anxiety disorder

Parental emotional distress, particularly high maternal anxiety, is one of the most consistent predictors of child anxiety treatment outcome. In order to identify the cognitive, affective and behavioural parenting characteristics of mothers of children with anxiety disorders who themselves have an anxiety disorder, we assessed the expectations and appraisals of 88 mothers of anxious children (44 not anxious (NONANX) and 44 with a current anxiety disorder (ANX)) before and after interacting with their 7-12 year old children. There were no observed differences in anxiety and avoidance among children of ANX and NONANX mothers, but, compared to NONANX mothers, ANX mothers held more negative expectations and differed on observations of intrusiveness, expressed anxiety, warmth and the quality of the relationship. Associations were moderated by the degree to which children expressed anxiety during the tasks. Maternal reported negative emotions during the task significantly mediated the association between maternal anxiety status and the observed quality of the relationship. These findings suggest that maternal anxiety disorder is associated with reduced tolerance of children’s negative emotions. This may interfere with the maintenance of a positive, supportive mother-child interaction under conditions of stress, and as such impede optimum treatment outcomes. The findings identify potential cognitive, affective and behavioural targets to improve treatment outcomes for children with anxiety disorders in the context of a current maternal anxiety disorder.

New results and open problems on Toeplitz operators in Bergman spaces

We discuss some of the recent progress in the field of Toeplitz operators acting on Bergman spaces of the unit disk, formulate some new results, and describe a list of open problems -- concerning boundedness, compactness and Fredholm properties -- which was presented at the conference "Recent Advances in Function Related Operator Theory'' in Puerto Rico in March 2010.

Boundary value problems for the elliptic sine-Gordon equation in a semi-strip

We study boundary value problems posed in a semistrip for the elliptic sine-Gordon equation, which is the paradigm of an elliptic integrable PDE in two variables. We use the method introduced by one of the authors, which provides a substantial generalization of the inverse scattering transform and can be used for the analysis of boundary as opposed to initial-value problems. We first express the solution in terms of a 2 by 2 matrix Riemann-Hilbert problem whose \jump matrix" depends on both the Dirichlet and the Neumann boundary values. For a well posed problem one of these boundary values is an unknown function. This unknown function is characterised in terms of the so-called global relation, but in general this characterisation is nonlinear. We then concentrate on the case that the prescribed boundary conditions are zero along the unbounded sides of a semistrip and constant along the bounded side. This corresponds to a case of the so-called linearisable boundary conditions, however a major difficulty for this problem is the existence of non-integrable singularities of the function q_y at the two corners of the semistrip; these singularities are generated by the discontinuities of the boundary condition at these corners. Motivated by the recent solution of the analogous problem for the modified Helmholtz equation, we introduce an appropriate regularisation which overcomes this difficulty. Furthermore, by mapping the basic Riemann-Hilbert problem to an equivalent modified Riemann-Hilbert problem, we show that the solution can be expressed in terms of a 2 by 2 matrix Riemann-Hilbert problem whose jump matrix depends explicitly on the width of the semistrip L, on the constant value d of the solution along the bounded side, and on the residues at the given poles of a certain spectral function denoted by h. The determination of the function h remains open.

Inverse problems in neural field theory

We study inverse problems in neural field theory, i.e., the construction of synaptic weight kernels yielding a prescribed neural field dynamics. We address the issues of existence, uniqueness, and stability of solutions to the inverse problem for the Amari neural field equation as a special case, and prove that these problems are generally ill-posed. In order to construct solutions to the inverse problem, we first recast the Amari equation into a linear perceptron equation in an infinite-dimensional Banach or Hilbert space. In a second step, we construct sets of biorthogonal function systems allowing the approximation of synaptic weight kernels by a generalized Hebbian learning rule. Numerically, this construction is implemented by the Moore–Penrose pseudoinverse method. We demonstrate the instability of these solutions and use the Tikhonov regularization method for stabilization and to prevent numerical overfitting. We illustrate the stable construction of kernels by means of three instructive examples.