Cyclic projectors and separation theorems in idempotent convex geometry

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring it is known that one can separate a vector from a closed subsemimodule that does not contain it. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.Comment: 20 pages, 1 figur

A cluster randomised trial of a telephone-based intervention for parents to increase fruit and vegetable consumption in their 3- to 5-year-old children: study protocol

Background: Inadequate fruit and vegetable consumption in childhood increases the risk of developing chronic disease. Despite this, a substantial proportion of children in developed nations, including Australia, do not consume sufficient quantities of fruits and vegetables. Parents are influential in the development of dietary habits of young children but often lack the necessary knowledge and skills to promote healthy eating in their children. The aim of this study is to assess the efficacy of a telephone-based intervention for parents to increase the fruit and vegetable consumption of their 3- to 5-year-old children. Methods/Design: The study, conducted in the Hunter region of New South Wales, Australia, employs a cluster randomised controlled trial design. Two hundred parents from 15 randomly selected preschools will be randomised to receive the intervention, which consists of print resources and four weekly 30-minute telephone support calls delivered by trained telephone interviewers. The calls will assist parents to increase the availability and accessibility of fruit and vegetables in the home, create supportive family eating routines and role-model fruit and vegetable consumption. A further two hundred parents will be randomly allocated to the control group and will receive printed nutrition information only. The primary outcome of the trial will be the change in the child's consumption of fruit and vegetables as measured by the fruit and vegetable subscale of the Children's Dietary Questionnaire. Pre-intervention and post-intervention parent surveys will be administered over the telephone. Baseline surveys will occur one to two weeks prior to intervention delivery, with follow-up data collection calls occurring two, six, 12 and 18 months following baseline data collection. Discussion: If effective, this telephone-based intervention may represent a promising public health strategy to increase fruit and vegetable consumption in childhood and reduce the risk of subsequent chronic disease. Trial registration: Australian Clinical Trials Registry ACTRN12609000820202

Medical Therapies for Uterine Fibroids - A Systematic Review and Network Meta-Analysis of Randomised Controlled Trials

BACKGROUND: Uterine fibroids are common, often symptomatic and a third of women need repeated time off work. Consequently 25% to 50% of women with fibroids receive surgical treatment, namely myomectomy or hysterectomy. Hysterectomy is the definitive treatment as fibroids are hormone dependent and frequently recurrent. Medical treatment aims to control symptoms in order to replace or delay surgery. This may improve the outcome of surgery and prevent recurrence. PURPOSE: To determine whether any medical treatment can be recommended in the treatment of women with fibroids about to undergo surgery and in those for whom surgery is not planned based on currently available evidence. STUDY SELECTION: Two authors independently identified randomised controlled trials (RCT) of all pharmacological treatments aimed at the treatment of fibroids from a list of references obtained by formal search of MEDLINE, EMBASE, Cochrane library, Science Citation Index, and ClinicalTrials.gov until December 2013. DATA EXTRACTION: Two authors independently extracted data from identified studies. DATA SYNTHESIS: A Bayesian network meta-analysis was performed following the National Institute for Health and Care Excellence-Decision Support Unit guidelines. Odds ratios, rate ratios, or mean differences with 95% credible intervals (CrI) were calculated. RESULTS AND LIMITATIONS: A total of 75 RCT met the inclusion criteria, 47 of which were included in the network meta-analysis. The overall quality of evidence was very low. The network meta-analysis showed differing results for different outcomes. CONCLUSIONS: There is currently insufficient evidence to recommend any medical treatment in the management of fibroids. Certain treatments have future promise however further, well designed RCTs are needed

Bezlotoxumab for Prevention of Recurrent Clostridium difficile Infection

BACKGROUND Clostridium difficile is the most common cause of infectious diarrhea in hospitalized patients. Recurrences are common after antibiotic therapy. Actoxumab and bezlotoxumab are human monoclonal antibodies against C. difficile toxins A and B, respectively. METHODS We conducted two double-blind, randomized, placebo-controlled, phase 3 trials, MODIFY I and MODIFY II, involving 2655 adults receiving oral standard-of-care antibiotics for primary or recurrent C. difficile infection. Participants received an infusion of bezlotoxumab (10 mg per kilogram of body weight), actoxumab plus bezlotoxumab (10 mg per kilogram each), or placebo; actoxumab alone (10 mg per kilogram) was given in MODIFY I but discontinued after a planned interim analysis. The primary end point was recurrent infection (new episode after initial clinical cure) within 12 weeks after infusion in the modified intention-to-treat population. RESULTS In both trials, the rate of recurrent C. difficile infection was significantly lower with bezlotoxumab alone than with placebo (MODIFY I: 17% [67 of 386] vs. 28% [109 of 395]; adjusted difference, −10.1 percentage points; 95% confidence interval [CI], −15.9 to −4.3; P<0.001; MODIFY II: 16% [62 of 395] vs. 26% [97 of 378]; adjusted difference, −9.9 percentage points; 95% CI, −15.5 to −4.3; P<0.001) and was significantly lower with actoxumab plus bezlotoxumab than with placebo (MODIFY I: 16% [61 of 383] vs. 28% [109 of 395]; adjusted difference, −11.6 percentage points; 95% CI, −17.4 to −5.9; P<0.001; MODIFY II: 15% [58 of 390] vs. 26% [97 of 378]; adjusted difference, −10.7 percentage points; 95% CI, −16.4 to −5.1; P<0.001). In prespecified subgroup analyses (combined data set), rates of recurrent infection were lower in both groups that received bezlotoxumab than in the placebo group in subpopulations at high risk for recurrent infection or for an adverse outcome. The rates of initial clinical cure were 80% with bezlotoxumab alone, 73% with actoxumab plus bezlotoxumab, and 80% with placebo; the rates of sustained cure (initial clinical cure without recurrent infection in 12 weeks) were 64%, 58%, and 54%, respectively. The rates of adverse events were similar among these groups; the most common events were diarrhea and nausea. CONCLUSIONS Among participants receiving antibiotic treatment for primary or recurrent C. difficile infection, bezlotoxumab was associated with a substantially lower rate of recurrent infection than placebo and had a safety profile similar to that of placebo. The addition of actoxumab did not improve efficacy. (Funded by Merck; MODIFY I and MODIFY II ClinicalTrials.gov numbers, NCT01241552 and NCT01513239.