Some error estimates for the finite volume element method for a parabolic problem

We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work \cite{clt11} for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in $L_2$ to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric

Optimal control of number squeezing in trapped Bose-Einstein condensates

We theoretically analyze atom interferometry based on trapped ultracold atoms, and employ optimal control theory in order to optimize number squeezing and condensate trapping. In our simulations, we consider a setup where the confinement potential is transformed from a single to a double well, which allows to split the condensate. To avoid in the ensuing phase-accumulation stage of the interferometer dephasing due to the nonlinear atom-atom interactions, the atom number fluctuations between the two wells should be sufficiently low. We show that low number fluctuations (high number squeezing) can be obtained by optimized splitting protocols. Two types of solutions are found: in the Josephson regime we find an oscillatory tunnel control and a parametric amplification of number squeezing, while in the Fock regime squeezing is obtained solely due to the nonlinear coupling, which is transformed to number squeezing by peaked tunnel pulses. We study splitting and squeezing within the frameworks of a generic two-mode model, which allows us to study the basic physical mechanisms, and the multi-configurational time dependent Hartree for bosons method, which allows for a microscopic modeling of the splitting dynamics in realistic experiments. Both models give similar results, thus highlighting the general nature of these two solution schemes. We finally analyze our results in the context of atom interferometry.Comment: 17 pages, 21 figures, minor correction

Re-ranking Permutation-Based Candidate Sets with the n-Simplex Projection

In the realm of metric search, the permutation-based approaches have shown very good performance in indexing and supporting approximate search on large databases. These methods embed the metric objects into a permutation space where candidate results to a given query can be efficiently identified. Typically, to achieve high effectiveness, the permutation-based result set is refined by directly comparing each candidate object to the query one. Therefore, one drawback of these approaches is that the original dataset needs to be stored and then accessed during the refining step. We propose a refining approach based on a metric embedding, called n-Simplex projection, that can be used on metric spaces meeting the n-point property. The n-Simplex projection provides upper- and lower-bounds of the actual distance, derived using the distances between the data objects and a finite set of pivots. We propose to reuse the distances computed for building the data permutations to derive these bounds and we show how to use them to improve the permutation-based results. Our approach is particularly advantageous for all the cases in which the traditional refining step is too costly, e.g. very large dataset or very expensive metric function

Patellofemoral pain syndrome (PFPS): a systematic review of anatomy and potential risk factors

Patellofemoral Pain Syndrome (PFPS), a common cause of anterior knee pain, is successfully treated in over 2/3 of patients through rehabilitation protocols designed to reduce pain and return function to the individual. Applying preventive medicine strategies, the majority of cases of PFPS may be avoided if a pre-diagnosis can be made by clinician or certified athletic trainer testing the current researched potential risk factors during a Preparticipation Screening Evaluation (PPSE). We provide a detailed and comprehensive review of the soft tissue, arterial system, and innervation to the patellofemoral joint in order to supply the clinician with the knowledge required to assess the anatomy and make recommendations to patients identified as potentially at risk. The purpose of this article is to review knee anatomy and the literature regarding potential risk factors associated with patellofemoral pain syndrome and prehabilitation strategies. A comprehensive review of knee anatomy will present the relationships of arterial collateralization, innervations, and soft tissue alignment to the possible multifactoral mechanism involved in PFPS, while attempting to advocate future use of different treatments aimed at non-soft tissue causes of PFPS

Interpreting outcome following foot surgery in people with rheumatoid arthritis

BACKGROUND: Foot surgery is common in RA but the current lack of understanding of how patients interpret outcomes inhibits evaluation of procedures in clinical and research settings. This study aimed to explore which factors are important to people with RA when they evaluate the outcome of foot and ankle surgery. METHODS AND RESULTS: Semi structured interviews with 11 RA participants who had mixed experiences of foot surgery were conducted and analysed using thematic analysis. Responses showed that while participants interpreted surgical outcome in respect to a multitude of factors, five major themes emerged: functional ability, participation, appearance of feet and footwear, surgeons' opinion, and pain. Participants interpreted levels of physical function in light of other aspects of their disease, reflecting on relative change from their preoperative state more than absolute levels of ability. Appearance was important to almost all participants: physical appearance, foot shape, and footwear were closely interlinked, yet participants saw these as distinct concepts and frequently entered into a defensive repertoire, feeling the need to justify that their perception of outcome was not about cosmesis. Surgeons' post-operative evaluation of the procedure was highly influential and made a lasting impression, irrespective of how the outcome compared to the participants' initial goals. Whilst pain was important to almost all participants, it had the greatest impact upon them when it interfered with their ability to undertake valued activities. CONCLUSIONS: People with RA interpret the outcome of foot surgery using multiple interrelated factors, particularly functional ability, appearance and surgeons' appraisal of the procedure. While pain was often noted, this appeared less important than anticipated. These factors can help clinicians in discussing surgical options in patients

Effectiveness of trigger point dry needling for plantar heel pain: study protocol for a randomised controlled trial

<p>Abstract</p> <p>Background</p> <p>Plantar heel pain (plantar fasciitis) is a common and disabling condition, which has a detrimental impact on health-related quality of life. Despite the high prevalence of plantar heel pain, the optimal treatment for this disorder remains unclear. Consequently, an alternative therapy such as dry needling is increasingly being used as an adjunctive treatment by health practitioners. Only two trials have investigated the effectiveness of dry needling for plantar heel pain, however both trials were of a low methodological quality. This manuscript describes the design of a randomised controlled trial to evaluate the effectiveness of dry needling for plantar heel pain.</p> <p>Methods</p> <p>Eighty community-dwelling men and woman aged over 18 years with plantar heel pain (who satisfy the inclusion and exclusion criteria) will be recruited. Eligible participants with plantar heel pain will be randomised to receive either one of two interventions, (i) real dry needling or (ii) sham dry needling. The protocol (including needling details and treatment regimen) was formulated by general consensus (using the Delphi research method) using 30 experts worldwide that commonly use dry needling for plantar heel pain. Primary outcome measures will be the pain subscale of the Foot Health Status Questionnaire and "first step" pain as measured on a visual analogue scale. The secondary outcome measures will be health related quality of life (assessed using the Short Form-36 questionnaire - Version Two) and depression, anxiety and stress (assessed using the Depression, Anxiety and Stress Scale - short version). Primary outcome measures will be performed at baseline, 2, 4, 6 and 12 weeks and secondary outcome measures will be performed at baseline, 6 and 12 weeks. Data will be analysed using the intention to treat principle.</p> <p>Conclusion</p> <p>This study is the first randomised controlled trial to evaluate the effectiveness of dry needling for plantar heel pain. The trial will be reported in accordance with the Consolidated Standards of Reporting Trials and the Standards for Reporting Interventions in Clinical Trials of Acupuncture guidelines. The findings from this trial will provide evidence for the effectiveness of trigger point dry needling for plantar heel pain.</p> <p>Trial registration</p> <p>Australian New Zealand 'Clinical Trials Registry'. <a href="http://www.anzctr.org.au/ACTRN12610000611022.aspx">ACTRN12610000611022</a>.</p

Explicit methods for stiff stochastic differential equations

Multiscale differential equations arise in the modeling of many important problems in the science and engineering. Numerical solvers for such problems have been extensively studied in the deterministic case. Here, we discuss numerical methods for (mean-square stable) stiff stochastic differential equations. Standard explicit methods, as for example the Euler-Maruyama method, face severe stepsize restriction when applied to stiff problems. Fully implicit methods are usually not appropriate for stochastic problems and semi-implicit methods (implicit in the deterministic part) involve the solution of possibly large linear systems at each time-step. In this paper, we present a recent generalization of explicit stabilized methods, known as Chebyshev methods, to stochastic problems. These methods have much better (mean-square) stability properties than standard explicit methods. We discuss the construction of this new class of methods and illustrate their performance on various problems involving stochastic ordinary and partial differential equations