Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions

In this paper, a space fractional di®usion equation (SFDE) with non- homogeneous boundary conditions on a bounded domain is considered. A new matrix transfer technique (MTT) for solving the SFDE is proposed. The method is based on a matrix representation of the fractional-in-space operator and the novelty of this approach is that a standard discretisation of the operator leads to a system of linear ODEs with the matrix raised to the same fractional power. Analytic solutions of the SFDE are derived. Finally, some numerical results are given to demonstrate that the MTT is a computationally e±cient and accurate method for solving SFDE.

'We just make the pictures...?' How work is portrayed in children's feature length films

This article explores how adult paid work is portrayed in 'family' feature length films. The study extends previous critical media literature which has overwhelmingly focused on depictions of gender and violence, exploring the visual content of films that is relevant to adult employment. Forty-two G/PG films were analyzed for relevant themes. Consistent with the exploratory nature of the research, themes emerged inductively from the films' content. Results reveal six major themes: males are more visible in adult work roles than women; the division of labour remains gendered; work and home are not mutually exclusive domains; organizational authority and power is wielded in punitive ways; there are avenues to better employment prospects; and status/money is paramount. The findings of the study reflect a range of subject matters related to occupational characteristics and work-related communication and interactions which are typically viewed by children in contemporary society.

Using Trauma Injury Severity Score (TRISS) variables to predict length of hospital stay following trauma in New Zealand

Aim – To develop and assess the predictive capabilities of a statistical model that relates routinely collected Trauma Injury Severity Score (TRISS) variables to length of hospital stay (LOS) in survivors of traumatic injury. Method – Retrospective cohort study of adults who sustained a serious traumatic injury, and who survived until discharge from Auckland City, Middlemore, Waikato, or North Shore Hospitals between 2002 and 2006. Cubic-root transformed LOS was analysed using two-level mixed-effects regression models. Results – 1498 eligible patients were identified, 1446 (97%) injured from a blunt mechanism and 52 (3%) from a penetrating mechanism. For blunt mechanism trauma, 1096 (76%) were male, average age was 37 years (range: 15-94 years), and LOS and TRISS score information was available for 1362 patients. Spearman’s correlation and the median absolute prediction error between LOS and the original TRISS model was ρ=0.31 and 10.8 days, respectively, and between LOS and the final multivariable two-level mixed-effects regression model was ρ=0.38 and 6.0 days, respectively. Insufficient data were available for the analysis of penetrating mechanism models. Conclusions – Neither the original TRISS model nor the refined model has sufficient ability to accurately or reliably predict LOS. Additional predictor variables for LOS and other indicators for morbidity need to be considered.

Using a longitudinal model to estimate the effect of methicillin-resistant staphylococcus aureus infection on length of stay in an intensive care unit

Healthcare-associated methicillin-resistant Staphylococcus aureus(MRSA) infection may cause increased hospital stay or, sometimes, death. Quantifying this effect is complicated because it is a time-dependent exposure: infection may prolong hospital stay, while longer stays increase the risk of infection. We overcome these problems by using a multinomial longitudinal model for estimating the daily probability of death and discharge. We then extend the basic model to estimate how the effect of MRSA infection varies over time, and to quantify the number of excess ICU days due to infection. We find that infection decreases the relative risk of discharge (relative risk ratio = 0.68, 95% credible interval: 0.54, 0.82), but is only indirectly associated with increased mortality. An infection on the first day of admission resulted in a mean extra stay of 0.3 days (95% CI: 0.1, 0.5) for a patient with an APACHE II score of 10, and 1.2 days (95% CI: 0.5, 2.0) for a patient with an APACHE II score of 30. The decrease in the relative risk of discharge remained fairly constant with day of MRSA infection, but was slightly stronger closer to the start of infection. These results confirm the importance of MRSA infection in increasing ICU stay, but suggest that previous work may have systematically overestimated the effect size.

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.