A Personal Perspective on the History of the Numerical Analysis of Fredholm Integral Equations of the Second Kind

This is a personal perspective on the development of numerical methods for solving Fredholm integral equations of the second kind, discussing work being done principally during the 1950s and 1960s. The principal types of numerical methods being studied were projection methods (Galerkin, collocation) and Nyström methods. During the 1950s and 1960s, functional analysis became the framework for the analysis of numerical methods for solving integral equations, and this in‡uenced the questions being asked. This paper looks at the history of the analyses being done at that time.

Namaste Care in the home setting: Developing initial realist explanatory theories and uncovering unintended outcomes.

Introduction The End-of-Life Namaste Care Program for People with Dementia, challenges the misconception that people with dementia are a ‘shell’; it provides a holistic approach using the five senses, which can provide positive ways of communicating and emotional responses. It is proposed Namaste Care can improve communication and the relationships families and friends have with the person with dementia. Previously used in care homes, this study is the first to explore the pioneering use of Namaste Care in people’s own homes. Objective To develop initial programme theories detailing if, how and under which circumstances Namaste Care works when implemented at home. Design A qualitative realist approach following the RAMESES II guidelines was employed to understand not only whether Namaste Care has positive outcomes, but also how these are generated, for whom they happen and in which circumstances. Setting A hospice in the North East of England, operating in the community, through volunteers. Participants Programme theories were developed from three focus groups with volunteers implementing Namaste Care (n=8; n=8; n=11) and eight interviews with family carers (n=8). Results Four refined explanatory theories are presented: increasing engagement, respite for family carers, importance of matched volunteers and increasing social interaction. It was identified that while Namaste Care achieved some of the same goals in the home setting as it does in the care home setting, it could also function in a different way that promoted socialisation. Conclusions Namaste Care provides holistic and personalised care to people with both moderate and advanced dementia, improving engagement and reducing social isolation. In the present study carers often chose to use Namaste Care sessions as respite. This was often linked to their frustration of the unavoidable dominance of task-focussed care in daily life. Individualised Namaste Care activities thus led to positive outcomes for both those with dementia and their carers

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

The influence of substrate roughness, patterning, curvature, and compliance in peeling problems

NMP is supported by the European Commission under the Graphene FET Flagship (WP14 'Polymer composites' No. 604391) and FET Proactive 'Neurofibres' grant No. 732344. FB is supported by 'Neurofibres' grant No. 732344

Improved Placental Parameter Estimation Using Data-Driven Bayesian Modelling

The placenta plays a key contribution to successful pregnancy outcome. New MR imaging techniques are able to reveal intricate details about placental structure and function and measure placental blood flow and feto-placental oxygenation. Placental diffusion-weighted MRI is however challenging due to maternal breathing motion and poor signal-to-noise ratio making motion correction important for subsequent quantitative analysis. In this work, we (i) introduce an iterative model-based registration technique which incorporates a placenta-specific model into the motion correction process and (ii) describe a new technique making use of a Bayesian shrinkage prior to obtain robust estimates of individual and population trends in parameters. Our results suggest that the proposed registration method improves alignment of placental data and that the Bayesian fitting technique allows the estimation of voxel-level placenta flow parameters and the population trend in each parameter with gestational age (GA). We report gestational age dependent differences in vascular compartments and fetal oxygen saturation values observed across 9 normally grown pregnancies between 25–34 weeks gestational age and show qualitatively improved parameter mapping and more precise longitudinal fitting. Fetal oxygen saturation ( FO2 ) is observed to decrease at FO2=−3.6(GAweeks)+190.2(%) . This technique provides a robust framework for analysing longitudinal changes in both normal and pathological placental function