Gaussian elimination as an iterative algorithm

Gaussian elimination (GE) for solving an $n \times n$ linear system of equations $Ax=b$ is the archetypical direct method of numerical linear algebra, as opposed to iterative. In this note we want to point out that GE has an iterative side too

An extension of Chebfun to two dimensions

An object-oriented MATLAB system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2” objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented

A cluster randomised controlled trial of a pharmacist-led collaborative intervention to improve statin prescribing and attainment of cholesterol targets in primary care

Background: Small trials with short term follow up suggest pharmacists’ interventions targeted at healthcare professionals can improve prescribing. In comparison with clinical guidance, contemporary statin prescribing is sub-optimal and achievement of cholesterol targets falls short of accepted standards, for patients with atherosclerotic vascular disease who are at highest absolute risk and who stand to obtain greatest benefit. We hypothesised that a pharmacist-led complex intervention delivered to doctors and nurses in primary care, would improve statin prescribing and achievement of cholesterol targets for incident and prevalent patients with vascular disease, beyond one year.<p></p> Methods: We allocated general practices to a 12-month Statin Outreach Support (SOS) intervention or usual care. SOS was delivered by one of 11 pharmacists who had received additional training. SOS comprised academic detailing and practical support to identify patients with vascular disease who were not prescribed a statin at optimal dose or did not have cholesterol at target, followed by individualised recommendations for changes to management. The primary outcome was the proportion of patients achieving cholesterol targets. Secondary outcomes were: the proportion of patients prescribed simvastatin 40 mg with target cholesterol achieved; cholesterol levels; prescribing of simvastatin 40 mg; prescribing of any statin and the proportion of patients with cholesterol tested. Outcomes were assessed after an average of 1.7 years (range 1.4–2.2 years), and practice level simvastatin 40 mg prescribing was assessed after 10 years.<p></p> Findings: We randomised 31 practices (72 General Practitioners (GPs), 40 nurses). Prior to randomisation a subset of eligible patients were identified to characterise practices; 40% had cholesterol levels below the target threshold. Improvements in data collection procedures allowed identification of all eligible patients (n = 7586) at follow up. Patients in practices allocated to SOS were significantly more likely to have cholesterol at target (69.5% vs 63.5%; OR 1.11, CI 1.00–1.23; p = 0.043) as a result of improved simvastatin prescribing. Subgroup analysis showed the primary outcome was achieved by prevalent but not incident patients. Statistically significant improvements occurred in all secondary outcomes for prevalent patients and all but one secondary outcome (the proportion of patients with cholesterol tested) for incident patients. SOS practices prescribed more simvastatin 40 mg than usual care practices, up to 10 years later.<p></p> Interpretation: Through a combination of educational and organisational support, a general practice based pharmacist led collaborative intervention can improve statin prescribing and achievement of cholesterol targets in a high-risk primary care based population

Neuromuscular fatigue, muscle temperature and hypoxia: an integrative approach.

Real world exposures to physiologically and/or psychologically stressful environments are often multifactorial. For example, high-altitude typically combines exposure to hypobaric hypoxia, solar radiation and cold ambient temperatures, while sea level thermal stress is often combined with supplementary or transient stressors such as rain, solar radiation and wind. In such complex environments, the effect of one stressor on performance may be subject to change, simply due to the presence of another independent stressor. Such differential influences can occur in three basic forms; additive, antagonistic and synergistic, each term defining a fundamental concept of inter-parameter interactions. As well as the natural occurrence of stressors in combination, understanding interactions is fundamental to experimentally modelling how multiple physiological strains integrate in their influence on or regulation of - exercise intensity. In this thesis the current literature on neuromuscular fatigue and the influence of thermal and hypoxic stress is reviewed (Chapter 1). This is followed by an outline of the methodological developments used in the subsequent experiments (Chapter 2). In the first experimental study (Chapter 3) a novel approach was adopted to investigate the combined effect of muscle cooling and hypoxia on neuromuscular fatigue in humans. The results showed that the neuromuscular system s maximal force generating capacity declined by 8.1 and 13.9% during independent cold and hypoxic stress compared to control. Force generation decreased by 21.4% during combined hypoxic-cold compared to control, closely matching the additive value of hypoxia and cold individually (22%). This was also reflected in the measurement of mechanical fatigue (electromechanical ratio), demonstrating an additive response during combined hypoxic-cold. From this study, it was concluded that when moderate hypoxia and cold environmental temperatures are combined during low intensity exercise, the level of fatigue increases additively with no interaction between these stressors. Before conducting a more complex investigation on combined stressors, a better understanding of the role of muscle temperature on central fatigue - i.e. voluntary muscle activation via the afferent signalling pathways was sought. The focus of Chapter 4 was to quantify the relationship between muscle temperature and voluntary muscle activation (central fatigue) across a wide range of temperatures. The primary finding was that different muscle temperatures can induce significant changes in voluntary activation (0.5% reduction per-degree-centigrade increase in muscle temperature) when neural drive is sustained for a prolonged effort (e.g. 120-s); however this effect is not exhibited during efforts that are brief in duration (e.g. 3-s). To further explore this finding, Chapter 5 investigated the effect of metaboreceptive feedback at two different muscle temperatures, using post-exercise muscle ischemia, on voluntary activation of a remote muscle group. The results showed that at the same perceived mental effort, peripheral limb discomfort was significantly higher with increasing muscle temperature (2% increase per-degree-centigrade increase). However any influence of increased muscle temperature on leg muscle metaboreceptive feedback did not appear to inhibit voluntary muscle activation - i.e. central control - of a remote muscle group, as represented by an equal force output and voluntary activation in the thermoneutral, contralateral leg. In Chapter 6, the psycho-sensory effects of changes in muscle temperature on central fatigue during dynamic exercise were investigated. During sustained dynamic exercise, fatigue development appeared to occur at a faster rate in hot muscle (4% increase per-degree-centigrade increase) leading to a nullification of the beneficial effects of increased muscle temperature on peak power output after a period of ~60-s maximal exercise. In support of previous studies using isometric exercise (Chapter 4 and 6), participants reported significantly higher muscular pain and discomfort in hot muscle compared to cooler muscle during dynamic exercise (2 and 1% increase per-degree-centigrade increase respectively), however this did not result in a lower power output. From Chapters 4, 5 and 6 it was concluded that in addition to faster rates of metabolite accumulation due to cardiovascular strain, it is possible that a direct sensitisation of the metaboreceptive group III and IV muscle afferents occurs in warmer muscle. This likely contributes to the reduction in voluntary muscle activation during exercise in the heat, while it may attenuate central fatigue in the cold. It was also interpreted that muscle afferents may have a similar signalling role to cutaneous sensory afferents; the latter of which are recognised for their role in providing thermal feedback to the cognitive-behavioural centres of the brain and aiding exercise regulation under thermal stress. The impact of body core and active muscle temperature on voluntary muscle activation represented a similar ratio (5 to 1 respectively) to the temperature manipulated (single leg) to non-temperature manipulated mass (rest of body) in Chapters 4, 5 and 6. This indicates that voluntary muscle activation may also be regulated based on a central meta-representation of total body heat content i.e. the summed firing rates of all activated thermoreceptors in the brain, skin, muscle, viscera and spine. Building on the initial findings of Chapter 3, Chapter 7 investigated the causative factors behind the expression of different interaction types during exposure to multi-stressor environments. This was achieved by studying the interaction between thermal stress and hypoxia on the rate of peripheral and central fatigue development during a high intensity bout of knee extension exercise to exhaustion. The results showed that during combined exposure to moderate hypoxia and mild cold, the reductions in time to exhaustion were additive of the relative effects of hypoxia and cold independently. This differs from the findings in Chapter 3, in which fatigue was additive of the absolute effects of cold and hypoxia. In contrast, combining moderate hypoxia with severe heat stress resulted in a significant antagonistic interaction on both the absolute and relative reductions in time to exhaustion i.e. the combined effect being significantly less than the sum of the individual effects. Based on the results in Chapter 7, a quantitative paradigm for understanding of systematic integration of multifactorial stressors was proposed. This is, that the interaction type between stressors is influenced by the impact magnitude of the individual stressors effect on exercise capacity, whereby the greater the stressors impact, the greater the probability that one stressor will be cancelled out by the other. This is the first study to experimentally model the overarching principles characterising the presence of simultaneous physiological strains, suggesting multifactorial integration be subject to the worst strain takes precedence when the individual strains are severe

Waveform Transition Graphs: a designer-friendly formalism for asynchronous behaviours

The paper proposes a new formal model for describing asynchronous behaviours involving the interplay of causality, concurrency and choice. The model is called Waveform Transition Graphs. Its main aim is simplifying the learning process for industrial engineers in accessing powerful synthesis tools provided for Signal Transition Graphs by sacrificing some of the expressive power of the latter. This formalism is developed based on feedback from engineers of Dialog Semiconductor.Peer ReviewedPostprint (author's final draft

A Sensitivity Matrix Methodology for Inverse Problem Formulation

We propose an algorithm to select parameter subset combinations that can be estimated using an ordinary least-squares (OLS) inverse problem formulation with a given data set. First, the algorithm selects the parameter combinations that correspond to sensitivity matrices with full rank. Second, the algorithm involves uncertainty quantification by using the inverse of the Fisher Information Matrix. Nominal values of parameters are used to construct synthetic data sets, and explore the effects of removing certain parameters from those to be estimated using OLS procedures. We quantify these effects in a score for a vector parameter defined using the norm of the vector of standard errors for components of estimates divided by the estimates. In some cases the method leads to reduction of the standard error for a parameter to less than 1% of the estimate

Continuous analogues of matrix factorizations

Analogues of QR, LU, SVD, and Cholesky factorizations are proposed for problems in which the usual discrete matrix is replaced by a “quasimatrix,” continuous in one dimension, or a “cmatrix,” continuous in both dimensions. Applications include Chebfun and similar computations involving functions of one or two variables. Two challenges arise: the generalization of the notions of triangular structure and row and column pivoting to continuous variables (required in all cases except the SVD), and the convergence of the infinite series that define the cmatrix factorizations. The generalizations of the factorizations work out neatly, but mathematical questions remain about convergence of the series. For example, our theorem about existence of an LU factorization of a cmatrix (a convergent infinite series) requires the cmatrix to be analytic in a “stadium” region of the complex plane

Parameter Estimation and Uncertainty Quantication for an Epidemic Model

We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number (R0 )—an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of R0. Situations are highlighted in which this correlation allows R0 to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts

The Guide to Community Preventive Services A Public Health Imperative

Community Guide) promises to be a substantial and necessary tool in collective efforts to improve the public health. This supplement to the American Journal of Preventive Medicine (AJPM) is important, not only for its content, but for the example provided of this approach in practice and policymaking. Painstaking and meticulous methodology yielded extensive reviews of evidence relevant to the reduction of injuries to motor vehicle occupants by increasing child safety seat use, increasing safety belt use, and reducing alcohol-impaired driving. The evidence is then weighed, with a specified protocol, to determine if recommendations can be formulated. 1–3 Akin to the Guide to Clinical Preventive Services, initially issued in 1989 by the U.S. Preventive Services Task Force and aimed at prevention for the individual patient, this new guide steers an evidence-based course through the broader ocean of population-based prevention. 4,5 Although great progress has been made, as this supplement demonstrates, the future holds significant challenges for this undertaking

Comment on: Subjective thermal strain impairs endurance performance in a temperate environment

Comment on: Subjective thermal strain impairs endurance performance in a temperate environmen