Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits

For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case.Comment: LaTeX, 27 pages with 12 figures available from the authors or can be generated from Mathematica instructions at end of the fil

Virtual outreach: economic evaluation of joint teleconsultations for patients referred by their general practitioner for a specialist opinion

Objectives To test the hypotheses that, compared with conventional outpatient consultations, joint teleconsultation (virtual outreach) would incur no increased costs to the NHS, reduce costs to patients, and reduce absences from work by patients and their carers.Design Cost consequences study alongside randomised controlled trial.Setting Two hospitals in London and Shrewsbury and 29 general practices in inner London and Wales.Participants 3170 patients identified; 2094 eligible for inclusion and willing to participate. 1051 randomised to virtual outreach and 1043 to standard outpatient appointments.Main outcome measures NHS costs, patient costs, health status (SF-12), time spent attending index consultation, patient satisfaction.Results Overall six month costs were greater for the virtual outreach consultations (pound724 per patient) than for conventional outpatient appointments (pound625): difference in means pound99 ($162; is not an element of138) (95% confidence interval pound10 to pound187, P=0.03). if the analysis is restricted to resource items deemed "attributable" to the index consultation, six month costs were still greater for virtual outreach: difference in means pound108 (pound73 to pound142, P < 0.0001). In both analyses the index consultation accounted for the excess cost. Savings to patients in terms of costs and time occurred in both centres: difference in mean total patient cost 8 pound (5 pound to 10 pound, P < 0.0001). Loss of productive time was less in the virtual outreach group: difference in mean cost pound11 (pound10 to pound12, P < 0.0001).Condusion The main hypothesis that virtual outreach would be cost neutral is rejected, but the hypotheses that costs to patients and losses in productivity would be lower are supported

Progressive multiple sclerosis: Prospects for disease therapy, repair, and restoration of function

Multiple sclerosis is a major cause of neurological disability, which accrues predominantly during progressive forms of the disease. Although development of multifocal inflammatory lesions is the underlying pathological process in relapsing-remitting multiple sclerosis, the gradual accumulation of disability that characterises progressive multiple sclerosis seems to result more from diffuse immune mechanisms and neurodegeneration. As a result, the 14 anti-inflammatory drugs that have regulatory approval for treatment of relapsing-remitting multiple sclerosis have little or no efficacy in progressive multiple sclerosis without inflammatory lesion activity. Effective therapies for progressive multiple sclerosis that prevent worsening, reverse damage, and restore function are a major unmet need. In this Series paper we summarise the current status of therapy for progressive multiple sclerosis and outline prospects for the future

Variation between countries in the frequency of diabetic ketoacidosis at first presentation of type 1 diabetes in children: a systematic review.

AIMS/HYPOTHESIS: Type 1 diabetes is the most frequent endocrine disease in children, with 65,000 children diagnosed worldwide every year. Up to 80% of these children present with diabetic ketoacidosis (DKA), which is associated with both short-term risks and long-term consequences. This study aimed to characterise the worldwide variation in presentation of type 1 diabetes to inform future interventions to reduce this excess morbidity and mortality. METHODS: This was a systematic review of studies indexed on PubMed, EMBASE, Web of Science, Scopus or CINAHL before March 2011 that included unselected groups of children presenting with new-onset type 1 diabetes, reported the proportion presenting with DKA and used a definition of DKA based on measurement of pH or bicarbonate. RESULTS: Sixty-five studies of cohorts comprising over 29,000 children in 31 countries were included. The frequency of DKA at diagnosis ranged from 12.8% to 80%, with highest frequencies in the United Arab Emirates, Saudi Arabia and Romania, and the lowest in Sweden, the Slovak Republic and Canada. Multivariable modelling showed the frequency of DKA was inversely associated with gross domestic product, latitude and background incidence of type 1 diabetes. CONCLUSIONS/INTERPRETATION: This is the first description of the variation in frequency of DKA at presentation of type 1 diabetes in children across countries. It demonstrates large variations that may, at least in part, be explained by different levels of disease awareness and healthcare provision and suggests ways to decrease the excess morbidity and mortality associated with DKA at diagnosis

Robust analysis of stepped wedge trials using cluster-level summaries within periods.

In stepped-wedge trials (SWTs), the intervention is rolled out in a random order over more than 1 time-period. SWTs are often analysed using mixed-effects models that require strong assumptions and may be inappropriate when the number of clusters is small. We propose a non-parametric within-period method to analyse SWTs. This method estimates the intervention effect by comparing intervention and control conditions in a given period using cluster-level data corresponding to exposure. The within-period intervention effects are combined with an inverse-variance-weighted average, and permutation tests are used. We present an example and, using simulated data, compared the method to (1) a parametric cluster-level within-period method, (2) the most commonly used mixed-effects model, and (3) a more flexible mixed-effects model. We simulated scenarios where period effects were common to all clusters, and when they varied according to a distribution informed by routinely collected health data. The non-parametric within-period method provided unbiased intervention effect estimates with correct confidence-interval coverage for all scenarios. The parametric within-period method produced confidence intervals with low coverage for most scenarios. The mixed-effects models' confidence intervals had low coverage when period effects varied between clusters but had greater power than the non-parametric within-period method when period effects were common to all clusters. The non-parametric within-period method is a robust method for analysing SWT. The method could be used by trial statisticians who want to emphasise that the SWT is a randomised trial, in the common position of being uncertain about whether data will meet the assumptions necessary for mixed-effect models