On the reduction theory of binary forms

Cremona developed a reduction theory for binary forms of degree 3 and 4 with integer coefficients, the motivation in the case of quartics being to improve 2-descent algorithms for elliptic curves over Q. In this paper we extend some of these results to forms of higher degree. One application of this is to the study of hyperelliptic curves.

Laying out the future of final-form digital documents

It is just over 20 years since Adobe's PostScript opened a new era in digital documents. PostScript allows most details of rendering to be hidden within the imaging device itself, while providing a rich set of primitives enabling document engineers to think of final-form rendering as being just a sophisticated exercise in computer graphics. The refinement of the PostScript model into PDF has been amazingly successful in creating a near-universal interchange format for complex and graphically rich digital documents but the PDF format itself is neither easy to create nor to amend. In the meantime a whole new world of digital documents has sprung up centred around XML-based technologies. The most widespread example is XHTML (with optional CSS styling) but more recently we have seen Scalable Vector Graphics (SVG) emerge as an XML-based, low-level, rendering language with PostScript-compatible rendering semantics. This paper surveys graphically-rich final-form rendering technologies and asks how flexible they can be in allowing adjustments to be made to final appearance without the need for regenerating a whole page or an entire document. Particular attention is focused on the relative merits of SVG and PDF in this regard and on the desirability, in any document layout language, of being able to manipulate the graphic properties of document components parametrically, and at a level of granularity smaller than an entire page.

Calculation of sample size for stroke trials assessing functional outcome: comparison of binary and ordinal approaches

Background Many acute stroke trials have given neutral results. Sub-optimal statistical analyses may be failing to detect efficacy. Methods which take account of the ordinal nature of functional outcome data are more efficient. We compare sample size calculations for dichotomous and ordinal outcomes for use in stroke trials. Methods Data from stroke trials studying the effects of interventions known to positively or negatively alter functional outcome – Rankin Scale and Barthel Index – were assessed. Sample size was calculated using comparisons of proportions, means, medians (according to Payne), and ordinal data (according to Whitehead). The sample sizes gained from each method were compared using Friedman 2 way ANOVA. Results Fifty-five comparisons (54 173 patients) of active vs. control treatment were assessed. Estimated sample sizes differed significantly depending on the method of calculation (Po00001). The ordering of the methods showed that the ordinal method of Whitehead and comparison of means produced significantly lower sample sizes than the other methods. The ordinal data method on average reduced sample size by 28% (inter-quartile range 14–53%) compared with the comparison of proportions; however, a 22% increase in sample size was seen with the ordinal method for trials assessing thrombolysis. The comparison of medians method of Payne gave the largest sample sizes. Conclusions Choosing an ordinal rather than binary method of analysis allows most trials to be, on average, smaller by approximately 28% for a given statistical power. Smaller trial sample sizes may help by reducing time to completion, complexity, and financial expense. However, ordinal methods may not be optimal for interventions which both improve functional outcome