Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

How to progress forward in improving safety outcomes in the work-related driving field : a review of work-related driving research and development of an intervention framework [Extended abstract]

Road crashes are now the most common cause of work-related injury, death and absence in a number of countries. Given the impact of workrelated driving crashes on social and economic aspects of business and the community, workrelated road safety and risk management has received increasing attention in recent years. However, limited academic research has progressed on improving safety within the work-related driving sector. The aim of this paper is to present a review of work-related driving safety research to date, and provide an intervention framework for the future development and implementation of workrelated driving safety intervention strategies.

Disclosing donation decisions : the role of organ donor prototypes in an extended theory of planned behaviour

This study explored the role of donor prototype evaluations (perceptions of the typical organ donor) in organ donation communication decisions using an extended theory of planned behaviour (TPB) model. The model incorporated attitude, subjective norm, perceived behavioural control, moral norm, self-identity, and donor prototype evaluations to predict intentions to record consent on an organ donor register and discuss the organ donation decision with significant others. Participants completed surveys assessing the extended TPB constructs related to registering (n = 359) and discussing (n = 282). Results supported a role for donor prototype evaluations in predicting discussing intentions only. Both extended TPB structural equation models were a good fit to the data, accounting for 74 and 76% of the variance in registering and discussing intentions, respectively. Participants’ self-reported discussing behaviour (but not registering behaviour given low numbers of behavioural performers) was assessed 4 weeks later, with discussing intention as the only significant predictor of behaviour (Nagelkerke R2 = 0.11). These findings highlight the impact of people's perceptions of a typical donor on their decisions to discuss their organ donation preference, assisting our understanding of the factors influencing individuals' communication processes in efforts to bridge the gap between organ supply and demand.

Graduate Attribute Mapping With the Extended CDIO Framework

The CDIO (Conceive-Design-Implement-Operate) Initiative has been globally recognised as an enabler for engineering education reform. With the CDIO process, the CDIO Standards and the CDIO Syllabus, many scholarly contributions have been made around cultural change, curriculum reform and learning environments. In the Australasian region, reform is gaining significant momentum within the engineering education community, the profession, and higher education institutions. This paper presents the CDIO Syllabus cast into the Australian context by mapping it to the Engineers Australia Graduate Attributes, the Washington Accord Graduate Attributes and the Queensland University of Technology Graduate Capabilities. Furthermore, in recognition that many secondary schools and technical training institutions offer introductory engineering technology subjects, this paper presents an extended self-rating framework suited for recognising developing levels of proficiency at a preparatory level. A demonstrator mapping tool has been created to demonstrate the application of this extended graduate attribute mapping framework as a precursor to an integrated curriculum information model.

An extended CDIO syllabus framework with preparatory engineering proficiencies

The CDIO Initiative has been globally recognised as an enabler for engineering education reform. With the CDIO process, the CDIO Standards and the CDIO Syllabus, many scholarly contributions have been made around cultural change, curriculum reform and learning environments. In the Australasian region, reform is gaining significant momentum within the engineering education community, the profession, and higher education institutions. This paper presents the CDIO Syllabus cast into the Australian context by mapping it to the Engineers Australia Graduate Attributes, the Washington Accord Graduate Attributes and the Queensland University of Technology Graduate Capabilities. Furthermore, in recognition that many secondary schools and technical training institutions offer introductory engineering technology subjects, this paper presents an extended self-rating framework suited for recognising developing levels of proficiency at a preparatory level. The framework is consistent with conventional application to undergraduate programs and professional practice, but adapted for the preparatory context. As with the original CDIO framework with proficiency levels, this extended framework is informed by Bloom’s Educational Objectives. A proficiency evaluation of Queensland Study Authority’s Engineering Technology senior syllabus is demonstrated indicating proficiency levels embedded within this secondary school subject within a preparatory scope. Through this extended CDIO framework, students and faculty have greater awareness and access to tools to promote (i) student engagement in their own graduate capability development, (ii) faculty engagement in course and program design, through greater transparency and utility of the continuum of graduate capability development with associate levels of proficiency, and the context in which they exist in terms of pre-tertiary engineering studies; and (iii) course maintenance and quality audit methodology for the purpose of continuous improvement processes and program accreditation.

The role of self-efficacy in dental patients’ brushing and flossing: Testing an extended Health Belief Model

Objective: In an effort to examine the decreasing oral health trend of Australian dental patients, the Health Belief Model (HBM) was utilised to understand the beliefs underlying brushing and flossing self-care. The HBM states that perception of severity and susceptibility to inaction and an estimate of the barriers and benefits of behavioural performance influences people’s health behaviours. Self-efficacy, confidence in one’s ability to perform oral self-care, was also examined. Methods: In dental waiting rooms, a community sample (N = 92) of dental patients completed a questionnaire assessing HBM variables and self-efficacy, as well as their performance of the oral hygiene behaviours of brushing and flossing. Results: Partial support only was found for the HBM with barriers emerging as the sole HBM factor influencing brushing and flossing behaviours. Self-efficacy significantly predicted both oral hygiene behaviours also. Conclusion: Support was found for the control factors, specifically a consideration of barriers and self-efficacy, in the context of understanding dental patients’ oral hygiene decisions. Practice implications: Dental professionals should encourage patients’ self-confidence to brush and floss at recommended levels and discuss strategies that combat barriers to performance, rather than emphasising the risks of inaction or the benefits of oral self-care.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Closed-form design equations for decoupling networks of small arrays

Small element spacing in compact arrays results in strong mutual coupling between the array elements. A decoupling network consisting of reactive cross-coupling elements can alleviate problems associated with the coupling. Closed-form design equations for the decoupling networks of symmetrical arrays with two or three elements are presented.