On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators

We consider one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. It is well known that such operators are generically conjugate to normal operators via a similarity transformation. Motivated by recent interests in quasi-Hermitian Hamiltonians in quantum mechanics, we study properties of the transformations and similar operators in detail. In the case of parity and time reversal boundary conditions, we establish closed integral-type formulae for the similarity transformations, derive a non-local self-adjoint operator similar to the Schrödinger operator and also find the associated “charge conjugation” operator, which plays the role of fundamental symmetry in a Krein-space reformulation of the problem.

GRAIL gravity field determination using the Celestial Mechanics Approach

The NASA mission GRAIL (Gravity Recovery and Interior Laboratory) inherited its concept from the GRACE (Gravity Recovery and Climate Experiment) mission to determine the gravity field of the Moon. We present lunar gravity fields based on the data of GRAIL’s primary mission phase. Gravity field recovery is realized in the framework of the Celestial Mechanics Approach, using a development version of the Bernese GNSS Software along with Ka-band range-rate data series as observations and the GNI1B positions provided by NASA JPL as pseudo-observations. By comparing our results with the official level-2 GRAIL gravity field models we show that the lunar gravity field can be recovered with a high quality by adapting the Celestial Mechanics Approach, even when using pre-GRAIL gravity field models as a priori fields and when replacing sophisticated models of non-gravitational accelerations by appropriately spaced pseudo-stochastic pulses (i.e., instantaneous velocity changes). We present and evaluate two lunar gravity field solutions up to degree and order 200 – AIUB-GRL200A and AIUB-GRL200B. While the first solution uses no gravity field information beyond degree 200, the second is obtained by using the official GRAIL field GRGM900C up to degree and order 660 as a priori information. This reduces the omission errors and demonstrates the potential quality of our solution if we resolved the gravity field to higher degree.

A generalisation of the fractional Brownian field based on non-Euclidean norms

We explore a generalisation of the L´evy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial pth mean body and the polar projection transforms.

Bevacizumab continuation versus no continuation after first-line chemotherapy plus bevacizumab in patients with metastatic colorectal cancer: a randomized phase III non-inferiority trial (SAKK 41/06)

BACKGROUND Chemotherapy plus bevacizumab is a standard option for first-line treatment in metastatic colorectal cancer (mCRC) patients. We assessed whether no continuation is non-inferior to continuation of bevacizumab after completing first-line chemotherapy. PATIENTS AND METHODS In an open-label, phase III multicentre trial, patients with mCRC without disease progression after 4-6 months of standard first-line chemotherapy plus bevacizumab were randomly assigned to continuing bevacizumab at a standard dose or no treatment. CT scans were done every 6 weeks until disease progression. The primary end point was time to progression (TTP). A non-inferiority limit for hazard ratio (HR) of 0.727 was chosen to detect a difference in TTP of 6 weeks or less, with a one-sided significance level of 10% and a statistical power of 85%. RESULTS The intention-to-treat population comprised 262 patients: median follow-up was 36.7 months. The median TTP was 4.1 [95% confidence interval (CI) 3.1-5.4] months for bevacizumab continuation versus 2.9 (95% CI 2.8-3.8) months for no continuation; HR 0.74 (95% CI 0.58-0.96). Non-inferiority could not be demonstrated. The median overall survival was 25.4 months for bevacizumab continuation versus 23.8 months (HR 0.83; 95% CI 0.63-1.1; P = 0.2) for no continuation. Severe adverse events were uncommon in the bevacizumab continuation arm. Costs for bevacizumab continuation were estimated to be ∼30,000 USD per patient. CONCLUSIONS Non-inferiority could not be demonstrated for treatment holidays versus continuing bevacizumab monotheray, after 4-6 months of standard first-line chemotherapy plus bevacizumab. Based on no impact on overall survival and increased treatment costs, bevacizumab as a single agent is of no meaningful therapeutic value. More efficient treatment approaches are needed to maintain control of stabilized disease following induction therapy. CLINICAL TRIAL REGISTRATION ClinicalTrials.gov, number NCT00544700.

Non-surgical treatment of pain associated with posterior tibial tendon dysfunction: study protocol for a randomised clinical trial

BACKGROUND Symptoms associated with pes planovalgus or flatfeet occur frequently, even though some people with a flatfoot deformity remain asymptomatic. Pes planovalgus is proposed to be associated with foot/ankle pain and poor function. Concurrently, the multifactorial weakness of the tibialis posterior muscle and its tendon can lead to a flattening of the longitudinal arch of the foot. Those affected can experience functional impairment and pain. Less severe cases at an early stage are eligible for non-surgical treatment and foot orthoses are considered to be the first line approach. Furthermore, strengthening of arch and ankle stabilising muscles are thought to contribute to active compensation of the deformity leading to stress relief of soft tissue structures. There is only limited evidence concerning the numerous therapy approaches, and so far, no data are available showing functional benefits that accompany these interventions. METHODS After clinical diagnosis and clarification of inclusion criteria (e.g., age 40-70, current complaint of foot and ankle pain more than three months, posterior tibial tendon dysfunction stage I & II, longitudinal arch flattening verified by radiography), sixty participants with posterior tibial tendon dysfunction associated complaints will be included in the study and will be randomly assigned to one of three different intervention groups: (i) foot orthoses only (FOO), (ii) foot orthoses and eccentric exercise (FOE), or (iii) sham foot orthoses only (FOS). Participants in the FOO and FOE groups will be allocated individualised foot orthoses, the latter combined with eccentric exercise for ankle stabilisation and strengthening of the tibialis posterior muscle. Participants in the FOS group will be allocated sham foot orthoses only. During the intervention period of 12 weeks, all participants will be encouraged to follow an educational program for dosed foot load management (e.g., to stop activity if they experience increasing pain). Functional impairment will be evaluated pre- and post-intervention by the Foot Function Index. Further outcome measures include the Pain Disability Index, Visual Analogue Scale for pain, SF-12, kinematic data from 3D-movement analysis and neuromuscular activity during level and downstairs walking. Measuring outcomes pre- and post-intervention will allow the calculation of intervention effects by 3×3 Analysis of Variance (ANOVA) with repeated measures. DISCUSSION The purpose of this randomised trial is to evaluate the therapeutic benefit of three different non-surgical treatment regimens in participants with posterior tibial tendon dysfunction and accompanying pes planovalgus. Furthermore, the analysis of changes in gait mechanics and neuromuscular control will contribute to an enhanced understanding of functional changes and eventually optimise conservative management strategies for these patients. TRIAL REGISTRATION ClinicalTrials.gov Protocol Registration System: ClinicalTrials.gov ID NCT01839669.

Effectiveness of non-steroidal anti-inflammatory drugs for the treatment of pain in knee and hip osteoarthritis: a network meta-analysis.

BACKGROUND Non-steroidal anti-inflammatory drugs (NSAIDs) are the backbone of osteoarthritis pain management. We aimed to assess the effectiveness of different preparations and doses of NSAIDs on osteoarthritis pain in a network meta-analysis. METHODS For this network meta-analysis, we considered randomised trials comparing any of the following interventions: NSAIDs, paracetamol, or placebo, for the treatment of osteoarthritis pain. We searched the Cochrane Central Register of Controlled Trials (CENTRAL) and the reference lists of relevant articles for trials published between Jan 1, 1980, and Feb 24, 2015, with at least 100 patients per group. The prespecified primary and secondary outcomes were pain and physical function, and were extracted in duplicate for up to seven timepoints after the start of treatment. We used an extension of multivariable Bayesian random effects models for mixed multiple treatment comparisons with a random effect at the level of trials. For the primary analysis, a random walk of first order was used to account for multiple follow-up outcome data within a trial. Preparations that used different total daily dose were considered separately in the analysis. To assess a potential dose-response relation, we used preparation-specific covariates assuming linearity on log relative dose. FINDINGS We identified 8973 manuscripts from our search, of which 74 randomised trials with a total of 58 556 patients were included in this analysis. 23 nodes concerning seven different NSAIDs or paracetamol with specific daily dose of administration or placebo were considered. All preparations, irrespective of dose, improved point estimates of pain symptoms when compared with placebo. For six interventions (diclofenac 150 mg/day, etoricoxib 30 mg/day, 60 mg/day, and 90 mg/day, and rofecoxib 25 mg/day and 50 mg/day), the probability that the difference to placebo is at or below a prespecified minimum clinically important effect for pain reduction (effect size [ES] -0·37) was at least 95%. Among maximally approved daily doses, diclofenac 150 mg/day (ES -0·57, 95% credibility interval [CrI] -0·69 to -0·46) and etoricoxib 60 mg/day (ES -0·58, -0·73 to -0·43) had the highest probability to be the best intervention, both with 100% probability to reach the minimum clinically important difference. Treatment effects increased as drug dose increased, but corresponding tests for a linear dose effect were significant only for celecoxib (p=0·030), diclofenac (p=0·031), and naproxen (p=0·026). We found no evidence that treatment effects varied over the duration of treatment. Model fit was good, and between-trial heterogeneity and inconsistency were low in all analyses. All trials were deemed to have a low risk of bias for blinding of patients. Effect estimates did not change in sensitivity analyses with two additional statistical models and accounting for methodological quality criteria in meta-regression analysis. INTERPRETATION On the basis of the available data, we see no role for single-agent paracetamol for the treatment of patients with osteoarthritis irrespective of dose. We provide sound evidence that diclofenac 150 mg/day is the most effective NSAID available at present, in terms of improving both pain and function. Nevertheless, in view of the safety profile of these drugs, physicians need to consider our results together with all known safety information when selecting the preparation and dose for individual patients. FUNDING Swiss National Science Foundation (grant number 405340-104762) and Arco Foundation, Switzerland.