Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry, and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe

Identifying strategies to maximise recruitment and retention of practices and patients in a multicentre randomised controlled trial of an intervention to optimise secondary prevention for coronary heart disease in primary care

<p>Abstract</p> <p>Background</p> <p>Recruitment and retention of patients and healthcare providers in randomised controlled trials (RCTs) is important in order to determine the effectiveness of interventions. However, failure to achieve recruitment targets is common and reasons why a particular recruitment strategy works for one study and not another remain unclear. We sought to describe a strategy used in a multicentre RCT in primary care, to report researchers' and participants' experiences of its implementation and to inform future strategies to maximise recruitment and retention.</p> <p>Methods</p> <p>In total 48 general practices and 903 patients were recruited from three different areas of Ireland to a RCT of an intervention designed to optimise secondary prevention of coronary heart disease. The recruitment process involved telephoning practices, posting information, visiting practices, identifying potential participants, posting invitations and obtaining consent. Retention involved patients attending reviews and responding to questionnaires and practices facilitating data collection.</p> <p>Results</p> <p>We achieved high retention rates for practices (100%) and for patients (85%) over an 18-month intervention period. Pilot work, knowledge of the setting, awareness of change in staff and organisation amongst participant sites, rapid responses to queries and acknowledgement of practitioners' contributions were identified as being important. Minor variations in protocol and research support helped to meet varied, complex and changing individual needs of practitioners and patients and encouraged retention in the trial. A collaborative relationship between researcher and practice staff which required time to develop was perceived as vital for both recruitment and retention.</p> <p>Conclusion</p> <p>Recruiting and retaining the numbers of practices and patients estimated as required to provide findings with adequate power contributes to increased confidence in the validity and generalisability of RCT results. A continuous dynamic process of monitoring progress within trials and tailoring strategies to particular circumstances, whilst not compromising trial protocols, should allow maximal recruitment and retention.</p> <p>Trial registration</p> <p>ISRCTN24081411</p