A spectral method for elliptic equations: the Dirichlet problem

An elliptic partial differential equation Lu=f with a zero Dirichlet boundary condition is converted to an equivalent elliptic equation on the unit ball. A spectral Galerkin method is applied to the reformulated problem, using multivariate polynomials as the approximants. For a smooth boundary and smooth problem parameter functions, the method is proven to converge faster than any power of 1/n with n the degree of the approximate Galerkin solution. Examples in two and three variables are given as numerical illustrations. Empirically, the condition number of the associated linear system increases like O(N), with N the order of the linear system.Comment: This is latex with the standard article style, produced using Scientific Workplace in a portable format. The paper is 22 pages in length with 8 figure

A Spectral Method for Elliptic Equations: The Neumann Problem

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving an elliptic partial differential equation $-\Delta u+\gamma u=f$ over $\Omega$ with a Neumann boundary condition. The problem is converted to an equivalent elliptic problem over the unit ball $B$, and then a spectral Galerkin method is used to create a convergent sequence of multivariate polynomials $u_{n}$ of degree $\leq n$ that is convergent to $u$. The transformation from $\Omega$ to $B$ requires a special analytical calculation for its implementation. With sufficiently smooth problem parameters, the method is shown to be rapidly convergent. For $u\in C^{\infty}(\overline{\Omega})$ and assuming $\partial\Omega$ is a $C^{\infty}$ boundary, the convergence of $\Vert u-u_{n}\Vert_{H^{1}}$ to zero is faster than any power of $1/n$. Numerical examples in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ show experimentally an exponential rate of convergence.Comment: 23 pages, 11 figure

Taking a Proposal Seriously : Orientations to Agenda and Agency in Support Workers’ Responses to Client Proposals

While joint decision-making is regularly launched by a proposal, it is the recipients’ responses that crucially influence the proposal outcome. This chapter examines how support workers respond to the proposals made by clients during rehabilitation group meetings at the Clubhouse. Drawing on a collection of 180 client-initiated proposal sequences, the paper describes two dilemmas that the support workers face when seeking to take client proposals “seriously.” The first concerns the meeting’s agenda and consists of a tension between providing recognition for the individual client and encouraging collective participation. The second dilemma has to do with agency and consists of a tension between focusing on the client as the originator of the proposal and avoiding treating him or her alone accountable for it. The analysis of these dilemmas contributes to a deeper understanding of group decision-making, in general, while these findings have specific relevance in mental health rehabilitation.While joint decision-making is regularly launched by a proposal, it is the recipients’ responses that crucially influence the proposal outcome. This chapter examines how support workers respond to the proposals made by clients during rehabilitation group meetings at the Clubhouse. Drawing on a collection of 180 client-initiated proposal sequences, the paper describes two dilemmas that the support workers face when seeking to take client proposals “seriously.” The first concerns the meeting’s agenda and consists of a tension between providing recognition for the individual client and encouraging collective participation. The second dilemma has to do with agency and consists of a tension between focusing on the client as the originator of the proposal and avoiding treating him or her alone accountable for it. The analysis of these dilemmas contributes to a deeper understanding of group decision-making, in general, while these findings have specific relevance in mental health rehabilitation.Peer reviewe

Measuring inequality: tools and an illustration

BACKGROUND: This paper examines an aspect of the problem of measuring inequality in health services. The measures that are commonly applied can be misleading because such measures obscure the difficulty in obtaining a complete ranking of distributions. The nature of the social welfare function underlying these measures is important. The overall object is to demonstrate that varying implications for the welfare of society result from inequality measures. METHOD: Various tools for measuring a distribution are applied to some illustrative data on four distributions about mental health services. Although these data refer to this one aspect of health, the exercise is of broader relevance than mental health. The summary measures of dispersion conventionally used in empirical work are applied to the data here, such as the standard deviation, the coefficient of variation, the relative mean deviation and the Gini coefficient. Other, less commonly used measures also are applied, such as Theil's Index of Entropy, Atkinson's Measure (using two differing assumptions about the inequality aversion parameter). Lorenz curves are also drawn for these distributions. RESULTS: Distributions are shown to have differing rankings (in terms of which is more equal than another), depending on which measure is applied. CONCLUSION: The scope and content of the literature from the past decade about health inequalities and inequities suggest that the economic literature from the past 100 years about inequality and inequity may have been overlooked, generally speaking, in the health inequalities and inequity literature. An understanding of economic theory and economic method, partly introduced in this article, is helpful in analysing health inequality and inequity

"This does my head in". Ethnographic study of self-management by people with diabetes

<p>Abstract</p> <p>Background</p> <p>Self-management is rarely studied 'in the wild'. We sought to produce a richer understanding of how people live with diabetes and why self-management is challenging for some.</p> <p>Method</p> <p>Ethnographic study supplemented with background documents on social context. We studied a socio-economically and ethnically diverse UK population. We sampled 30 people with diabetes (15 type 1, 15 type 2) by snowballing from patient groups, community contacts and NHS clinics. Participants (aged 5-88, from a range of ethnic and socio-economic groups) were shadowed at home and in the community for 2-4 periods of several hours (total 88 visits, 230 hours); interviewed (sometimes with a family member or carer) about their self-management efforts and support needs; and taken out for a meal. Detailed field notes were made and annotated. Data analysis was informed by structuration theory, which assumes that individuals' actions and choices depend on their dispositions and capabilities, which in turn are shaped and constrained (though not entirely determined) by wider social structures.</p> <p>Results</p> <p>Self-management comprised both practical and cognitive tasks (e.g. self-monitoring, menu planning, medication adjustment) and socio-emotional ones (e.g. coping with illness, managing relatives' input, negotiating access to services or resources). Self-management was hard work, and was enabled or constrained by economic, material and socio-cultural conditions within the family, workplace and community. Some people managed their diabetes skilfully and flexibly, drawing on personal capabilities, family and social networks and the healthcare system. For others, capacity to self-manage (including overcoming economic and socio-cultural constraints) was limited by co-morbidity, cognitive ability, psychological factors (e.g. under-confidence, denial) and social capital. The consequences of self-management efforts strongly influenced people's capacity and motivation to continue them.</p> <p>Conclusion</p> <p>Self-management of diabetes is physically, intellectually, emotionally and socially demanding. Non-engagement with self-management may make sense in the context of low personal resources (e.g. health literacy, resilience) and overwhelming personal, family and social circumstances. Success of self-management as a policy solution will be affected by interacting influences at three levels: [a] at micro level by individuals' dispositions and capabilities; [b] at meso level by roles, relationships and material conditions within the family and in the workplace, school and healthcare organisation; and [c] at macro level by prevailing economic conditions, cultural norms and expectations, and the underpinning logic of the healthcare system. We propose that the research agenda on living with diabetes be extended and the political economy of self-management systematically studied.</p

Establishing a library of resources to help people understand key concepts in assessing treatment claims—The “Critical thinking and Appraisal Resource Library” (CARL)

Background People are frequently confronted with untrustworthy claims about the effects of treatments. Uncritical acceptance of these claims can lead to poor, and sometimes dangerous, treatment decisions, and wasted time and money. Resources to help people learn to think critically about treatment claims are scarce, and they are widely scattered. Furthermore, very few learning-resources have been assessed to see if they improve knowledge and behavior. Objectives Our objectives were to develop the Critical thinking and Appraisal Resource Library (CARL). This library was to be in the form of a database containing learning resources for those who are responsible for encouraging critical thinking about treatment claims, and was to be made available online. We wished to include resources for groups we identified as ‘intermediaries’ of knowledge, i.e. teachers of schoolchildren, undergraduates and graduates, for example those teaching evidence-based medicine, or those communicating treatment claims to the public. In selecting resources, we wished to draw particular attention to those resources that had been formally evaluated, for example, by the creators of the resource or independent research groups. Methods CARL was populated with learning-resources identified from a variety of sources—two previously developed but unmaintained inventories; systematic reviews of learning-interventions; online and database searches; and recommendations by members of the project group and its advisors. The learning-resources in CARL were organised by ‘Key Concepts’ needed to judge the trustworthiness of treatment claims, and were made available online by the James Lind Initiative in Testing Treatments interactive (TTi) English (www.testingtreatments.org/category/learning-resources).TTi English also incorporated the database of Key Concepts and the Claim Evaluation Tools developed through the Informed Healthcare Choices (IHC) project (informedhealthchoices.org). Results We have created a database of resources called CARL, which currently contains over 500 open-access learning-resources in a variety of formats: text, audio, video, webpages, cartoons, and lesson materials. These are aimed primarily at ‘Intermediaries’, that is, ‘teachers’, ‘communicators’, ‘advisors’, ‘researchers’, as well as for independent ‘learners’. The resources included in CARL are currently accessible at www.testingtreatments.org/category/learning-resources Conclusions We hope that ready access to CARL will help to promote the critical thinking about treatment claims, needed to help improve healthcare choices

Periodic density functional theory calculations of bulk and the (010) surface of goethite

<p>Abstract</p> <p>Background</p> <p>Goethite is a common and reactive mineral in the environment. The transport of contaminants and anaerobic respiration of microbes are significantly affected by adsorption and reduction reactions involving goethite. An understanding of the mineral-water interface of goethite is critical for determining the molecular-scale mechanisms of adsorption and reduction reactions. In this study, periodic density functional theory (DFT) calculations were performed on the mineral goethite and its (010) surface, using the Vienna <it>Ab Initio </it>Simulation Package (VASP).</p> <p>Results</p> <p>Calculations of the bulk mineral structure accurately reproduced the observed crystal structure and vibrational frequencies, suggesting that this computational methodology was suitable for modeling the goethite-water interface. Energy-minimized structures of bare, hydrated (one H₂O layer) and solvated (three H₂O layers) (010) surfaces were calculated for 1 × 1 and 3 × 3 unit cell slabs. A good correlation between the calculated and observed vibrational frequencies was found for the 1 × 1 solvated surface. However, differences between the 1 × 1 and 3 × 3 slab calculations indicated that larger models may be necessary to simulate the relaxation of water at the interface. Comparison of two hydrated surfaces with molecularly and dissociatively adsorbed H₂O showed a significantly lower potential energy for the former.</p> <p>Conclusion</p> <p>Surface Fe-O and (Fe)O-H bond lengths are reported that may be useful in surface complexation models (SCM) of the goethite (010) surface. These bond lengths were found to change significantly as a function of solvation (i.e., addition of two extra H₂O layers above the surface), indicating that this parameter should be carefully considered in future SCM studies of metal oxide-water interfaces.</p

Teaching sociology to undergraduate medical students

Understanding the social basis of health and medicine and the contexts of clinical care are essential components of good medical practice. This includes the ways in which social factors such as class, ethnicity and gender influence health outcomes and how people experience health, illness and healthcare. In our Guide we describe what sociology is and what it brings to medicine, beginning with the nature of the ‘sociological imagination’. Sociological theory and methods are reviewed in order to explain and illustrate the role of sociology in the context of undergraduate medical education. Reference is made to A Core Curriculum for Sociology in UK Undergraduate Medical Education by Collett et al. (2016). Teaching and student learning are discussed in terms of organisation and delivery, with an emphasis on practice. Sections are also included on assessment, evaluation, opportunities and challenges and the value of a ‘community of practice’ for sociology teachers in medical education. <br/