Maximum curves and isolated points of entire functions

Given M(r; f) =maxjzj=r (jf(z)j) , curves belonging to the set of points M = fz : jf(z)j = M(jzj; f)g were de�ned by Hardy to be maximum curves. Clunie asked the question as to whether the set M could also contain isolated points. This paper shows that maximum curves consist of analytic arcs and determines a necessary condition for such curves to intersect. Given two entire functions f1(z) and f2(z), if the maximum curve of f1(z) is the real axis, conditions are found so that the real axis is also a maximum curve for the product function f1(z)f2(z). By means of these results an entire function of in�nite order is constructed for which the set M has an in�nite number of isolated points. A polynomial is also constructed with an isolated point.

A systematic review of the effect of dietary exposure that could be achieved through normal dietary intake on learning and performance of school-aged children of relevance to UK schools

The aim of the present review was to perform a systematic in-depth review of the best evidence from controlled trial studies that have investigated the effects of nutrition, diet and dietary change on learning, education and performance in school-aged children (4-18 years) from the UK and other developed countries. The twenty-nine studies identified for the review examined the effects of breakfast consumption, sugar intake, fish oil and vitamin supplementation and 'good diets'. In summary, the studies included in the present review suggest there is insufficient evidence to identify any effect of nutrition, diet and dietary change on learning, education or performance of school-aged children from the developed world. However, there is emerging evidence for the effects of certain fatty acids which appear to be a function of dose and time. Further research is required in settings of relevance to the UK and must be of high quality, representative of all populations, undertaken for longer durations and use universal validated measures of educational attainment. However, challenges in terms of interpreting the results of such studies within the context of factors such as family and community context, poverty, disease and the rate of individual maturation and neurodevelopment will remain. Whilst the importance of diet in educational attainment remains under investigation, the evidence for promotion of lower-fat, -salt and -sugar diets, high in fruits, vegetables and complex carbohydrates, as well as promotion of physical activity remains unequivocal in terms of health outcomes for all schoolchildren.

A new 14+: vocational diplomas and the future of schools, colleges and universities. An educational and campaigning pamphlet

Written in an accessible and campaigning style, this pamphlet affords a valuable context to the introduction of the first group of specialist diplomas for 14 year olds in September 2008. The diplomas are the latest in a line of failed initiatives that have sought to provide vocational ‘alternatives’ for those young people staying in full-time education and not considered ‘academic’. Rather than developing any useful employment skills, Allen and Ainley argue that their introduction reflects the changing significance of education in the division and social control of learners that now extends from school to college and on to university. Those who are opposed to the current post-14 agenda, must not only put forward radical alternatives to the current curriculum offer but also, the authors argue, address issues of democracy and accountability. To do this, teacher trade unionists must make new types of alliances with local communities and also with their students.