Convex hulls of random walks

We study the convex hulls of random walks establishing both law of large numbers and weak convergence statements for the perimeter length, diameter and shape of the hull. It should come as no surprise that the case where the random walk has drift, and the zero-drift case behave differently. We make use of several different methods to gain a better insight into each case. Classical results such as Cauchy’s surface area formula, the law of large numbers and the central limit theorem give some preliminary law of large number results. Considering the convergence of the random walk and then using the continuous mapping theorem leads to intuitive results in the case with drift where, under the appropriate scaling, non-zero, deterministic limits exist. In the zero-drift case the random limiting process, Brownian motion, provides insight into the behaviour of such a walk. We add to the literature in this area by establishing tighter bounds on the expected diameter of planar Brownian motion. The Brownian motion process is also useful for proving that the convex hull of the zero-drift random walk has no limiting shape. In the case with drift, a martingale difference method was used by Wade and Xu to prove a central limit theorem for the perimeter length. We use this framework to establish similar results for the diameter of the convex hull. Time-space processes give degenerate results here, so we use some geometric properties to further what is known about the variance of the functionals in this case and to prove a weak convergence statement for the diameter. During the study of the geometrical properties, we show that, only finitely often is there a single face in the convex minorant (or concave majorant) of such a walk

Hazardous waste management in veterinary practice in Ireland

This study reports on a survey conducted in February 2006 of the hazardous waste\ud management practices of Irish veterinary practices. Fifty seven veterinary practices\ud responded to the anonymous postal survey which had been sent to a total of one\ud hundred and fifty practices. The survey examined the veterinary practitioner’s\ud knowledge of the legislation, and the practice management of veterinary cadavers,\ud municipal waste, clinical waste, and specific hazardous waste streams such as sharps,\ud pharmaceutical and chemical wastes. The results showed that veterinary practitioners\ud were not ignorant of the hazardous status of many of their wastes and were aware of the\ud legislation governing waste management in Ireland. Veterinary practices were\ud successfully managing municipal waste streams with many of them recycling certain\ud waste streams. Veterinary cadavers were disposed of through a pet cremation company,\ud returned to the client for burial or sent for rendering to a rendering plant. However there\ud was considerable non-compliance in relation to hazardous wastes. Many practices\ud consigned infectious clinical wastes to municipal bins. 48% of practices did not dispose\ud of waste medicines via hazardous waste contractors. 53% disposed of photochemical\ud waste from radiographic processing down the sink or the toilet. Only 47% of vets felt\ud that they were fully complying with waste regulations. Additional questionnaires to\ud local authorities and hazard waste contractors showed that local authorities were\ud receiving and acting on public complaints about veterinary waste, and as few as 50-60%\ud of the veterinary practices in Ireland avail of the services o f a hazardous waste\ud contractor

High-Throughput Proteomics Detection of Novel Splice Isoforms in Human Platelets

Alternative splicing (AS) is an intrinsic regulatory mechanism of all metazoans. Recent findings suggest that 100% of multiexonic human genes give rise to splice isoforms. AS can be specific to tissue type, environment or developmentally regulated. Splice variants have also been implicated in various diseases including cancer. Detection of these variants will enhance our understanding of the complexity of the human genome and provide disease-specific and prognostic biomarkers. We adopted a proteomics approach to identify exon skip events - the most common form of AS. We constructed a database harboring the peptide sequences derived from all hypothetical exon skip junctions in the human genome. Searching tandem mass spectrometry (MS/MS) data against the database allows the detection of exon skip events, directly at the protein level. Here we describe the application of this approach to human platelets, including the mRNA-based verification of novel splice isoforms of ITGA2, NPEPPS and FH. This methodology is applicable to all new or existing MS/MS datasets

Effects of fluoxetine on functional outcomes after acute stroke (FOCUS): a pragmatic, double-blind, randomised, controlled trial

Background Results of small trials indicate that fluoxetine might improve functional outcomes after stroke. The FOCUS trial aimed to provide a precise estimate of these effects. Methods FOCUS was a pragmatic, multicentre, parallel group, double-blind, randomised, placebo-controlled trial done at 103 hospitals in the UK. Patients were eligible if they were aged 18 years or older, had a clinical stroke diagnosis, were enrolled and randomly assigned between 2 days and 15 days after onset, and had focal neurological deficits. Patients were randomly allocated fluoxetine 20 mg or matching placebo orally once daily for 6 months via a web-based system by use of a minimisation algorithm. The primary outcome was functional status, measured with the modified Rankin Scale (mRS), at 6 months. Patients, carers, health-care staff, and the trial team were masked to treatment allocation. Functional status was assessed at 6 months and 12 months after randomisation. Patients were analysed according to their treatment allocation. This trial is registered with the ISRCTN registry, number ISRCTN83290762. Findings Between Sept 10, 2012, and March 31, 2017, 3127 patients were recruited. 1564 patients were allocated fluoxetine and 1563 allocated placebo. mRS data at 6 months were available for 1553 (99·3%) patients in each treatment group. The distribution across mRS categories at 6 months was similar in the fluoxetine and placebo groups (common odds ratio adjusted for minimisation variables 0·951 [95% CI 0·839–1·079]; p=0·439). Patients allocated fluoxetine were less likely than those allocated placebo to develop new depression by 6 months (210 [13·43%] patients vs 269 [17·21%]; difference 3·78% [95% CI 1·26–6·30]; p=0·0033), but they had more bone fractures (45 [2·88%] vs 23 [1·47%]; difference 1·41% [95% CI 0·38–2·43]; p=0·0070). There were no significant differences in any other event at 6 or 12 months. Interpretation Fluoxetine 20 mg given daily for 6 months after acute stroke does not seem to improve functional outcomes. Although the treatment reduced the occurrence of depression, it increased the frequency of bone fractures. These results do not support the routine use of fluoxetine either for the prevention of post-stroke depression or to promote recovery of function. Funding UK Stroke Association and NIHR Health Technology Assessment Programme

On the expected diameter of planar Brownian motion

Known results show that the diameter d1 of the trace of planar Brownian motion run for unit time satisfies 1.595≤Ed1≤2.507. This note improves these bounds to 1.601≤Ed1≤2.355. Simulations suggest that Ed1≈1.99

The convex hull of a planar random walk: perimeter, diameter, and shape

We study the convex hull of the first n steps of a planar random walk, and present large-n asymptotic results on its perimeter length Ln, diameter Dn, and shape. In the case where the walk has a non-zero mean drift, we show that Ln=Dn ! 2 a.s., and give distributional limit theorems and variance asymptotics for Dn, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then lim infn!1 Ln=Dn = 2 and lim supn!1 Ln=Dn = , a.s. Among the tools that we use is a zero-one law for convex hulls of random walks