A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Adams-Type Methods for the Numerical Solution of Stochastic Ordinary Differential Equations

Stochastic differential equations (SDEs) arise fi om physical systems where the parameters describing the system can only be estimated or are subject to noise. There has been much work done recently on developing numerical methods for solving SDEs. This paper will focus on stability issues and variable stepsize implementation techniques for numerically solving SDEs effectively. (C) 2000 Elsevier Science B.V. All rights reserved.

High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula

In recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations (SODEs), as SODEs are often more appropriate than their deterministic counterparts in many modelling situations. However, unlike the deterministic case numerical methods for SODEs are considerably less sophisticated due to the difficulty in representing the (possibly large number of) random variable approximations to the stochastic integrals. Although Burrage and Burrage [High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Applied Numerical Mathematics 22 (1996) 81-101] were able to construct strong local order 1.5 stochastic Runge-Kutta methods for certain cases, it is known that all extant stochastic Runge-Kutta methods suffer an order reduction down to strong order 0.5 if there is non-commutativity between the functions associated with the multiple Wiener processes. This order reduction down to that of the Euler-Maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques. An additional difficulty in solving SODEs arises even in the Linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple Wiener processes. Thus in this present paper first the work of Magnus [On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7 (1954) 649-673] (applied to deterministic non-commutative Linear problems) will be applied to non-commutative linear SODEs and methods of strong order 1.5 for arbitrary, linear, non-commutative SODE systems will be constructed - hence giving an accurate approximation to the general linear problem. Secondly, for general nonlinear non-commutative systems with an arbitrary number (d) of Wiener processes it is shown that strong local order I Runge-Kutta methods with d + 1 stages can be constructed by evaluated a set of Lie brackets as well as the standard function evaluations. A method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of Wiener processes. Finally some numerical results are presented which illustrate the efficacy of these approaches. (C) 1999 Elsevier Science B.V. All rights reserved.

General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems

In many modeling situations in which parameter values can only be estimated or are subject to noise, the appropriate mathematical representation is a stochastic ordinary differential equation (SODE). However, unlike the deterministic case in which there are suites of sophisticated numerical methods, numerical methods for SODEs are much less sophisticated. Until a recent paper by K. Burrage and P.M. Burrage (1996), the highest strong order of a stochastic Runge-Kutta method was one. But K. Burrage and P.M. Burrage (1996) showed that by including additional random variable terms representing approximations to the higher order Stratonovich (or Ito) integrals, higher order methods could be constructed. However, this analysis applied only to the one Wiener process case. In this paper, it will be shown that in the multiple Wiener process case all known stochastic Runge-Kutta methods can suffer a severe order reduction if there is non-commutativity between the functions associated with the Wiener processes. Importantly, however, it is also suggested how this order can be repaired if certain commutator operators are included in the Runge-Kutta formulation. (C) 1998 Elsevier Science B.V. and IMACS. All rights reserved.

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if an s-stage SRK contains Stratonovich integrals up to order p then the strong order of the SRK cannot exceed min{(p + 1)/2, (s - 1)/2), p greater than or equal to 2, s greater than or equal to 3 or 1 if p = 1.

Spatial and temporal foraging patterns of Queensland fruit fly, Bactrocera tryoni(Froggatt) (Diptera: Tephritidae), for protein and implications for management

Fruit flies require protein for reproductive development and actively feed upon protein sources in the field. Liquid protein baits mixed with insecticide are used routinely to manage pest fruit flies, such as Bactrocera tryoni (Froggatt). However, there are still some gaps in the underpinning science required to improve the efficacy of bait spray technology. The spatial and temporal foraging behaviour of B. tryoni in response to protein was investigated in the field. A series of linked trials using either wild flies in the open field or laboratory-reared flies in field cages and a netted orchard were undertaken using nectarines and guavas. Key questions investigated were the fly's response to protein relative to: height of protein within the canopy, fruiting status of the tree, time of day, season and size of the experimental arena. Canopy height had a significant response on B. tryoni foraging, with more flies foraging on protein in the mid to upper canopy. Fruiting status also had a significant effect on foraging, with most flies responding to protein when applied to fruiting hosts. B. tryoni demonstrated a repeatable diurnal response pattern to protein, with the peak response being between 12:00–16:00 h. Season showed significant but unpredictable effects on fruit fly response to protein in the subtropical environment where the work was undertaken. Relative humidity, but not temperature or rainfall, was positively correlated with protein response. The number of B. tryoni responding to protein decreased dramatically as the spatial scale increased from field cage through to the open field. Based on these results, it is recommend that, to be most effective, protein bait sprays should be applied to the mid to upper canopies of fruiting hosts. Overall, the results show that the protein used, an industry standard, has very low attractancy to B. tryoni and that further work is urgently needed to develop more volatile protein baits.

Netflix : a house of cards or the new HBO?

'House of Cards' was a $100 million, 13-episode TV series starring Kevin Spacey and directed by David Fincher, that premiered in 2013 exclusively online, available only to Netflix subscribers. For over a year, Netflix had received much media attention for both its rapid international expansion and for its financial woes resulting in a plummeting share price and had some financial analysts postulating that it might not even survive. This article asks how can a company that teetering on the brink spend $100 million commissioning a TV series, let alone push ahead with global expansion? It asks is Netflix’s business model sustainable? And if it is, what does its entry into original programming mean for the future of television?

Language affects length of stay in emergency departments in Queensland public hospitals

BACKGROUND: A long length of stay (LOS) in the emergency department (ED) associated with overcrowding has been found to adversely affect the quality of ED care. The objective of this study is to determine whether patients who speak a language other than English at home have a longer LOS in EDs compared to those whose speak only English at home. METHODS: A secondary data analysis of a Queensland state-wide hospital EDs dataset (Emergency Department Information System) was conducted for the period, 1 January 2008 to 31 December 2010. RESULTS: The interpreter requirement was the highest among Vietnamese speakers (23.1%) followed by Chinese (19.8%) and Arabic speakers (18.7%). There were significant differences in the distributions of the departure statuses among the language groups (Chi-squared=3236.88, P<0.001). Compared with English speakers, the Beta coeffi cient for the LOS in the EDs measured in minutes was among Vietnamese, 26.3 (95%CI: 22.1–30.5); Arabic, 10.3 (95%CI: 7.3–13.2); Spanish, 9.4 (95%CI: 7.1–11.7); Chinese, 8.6 (95%CI: 2.6–14.6); Hindi, 4.0 (95%CI: 2.2–5.7); Italian, 3.5 (95%CI: 1.6–5.4); and German, 2.7 (95%CI: 1.0–4.4). The fi nal regression model explained 17% of the variability in LOS. CONCLUSION: There is a close relationship between the language spoken at home and the LOS at EDs, indicating that language could be an important predictor of prolonged LOS in EDs and improving language services might reduce LOS and ease overcrowding in EDs in Queensland's public hospitals.

Predicting vitamin D deficiency in older Australian adults

OBJECTIVE There has been a dramatic increase in vitamin D testing in Australia in recent years, prompting calls for targeted testing. We sought to develop a model to identify people most at risk of vitamin D deficiency. DESIGN AND PARTICIPANTS This is a cross-sectional study of 644 60- to 84-year-old participants, 95% of whom were Caucasian, who took part in a pilot randomized controlled trial of vitamin D supplementation. MEASUREMENTS Baseline 25(OH)D was measured using the Diasorin Liaison platform. Vitamin D insufficiency and deficiency were defined using 50 and 25 nmol/l as cut-points, respectively. A questionnaire was used to obtain information on demographic characteristics and lifestyle factors. We used multivariate logistic regression to predict low vitamin D and calculated the net benefit of using the model compared with 'test-all' and 'test-none' strategies. RESULTS The mean serum 25(OH)D was 42 (SD 14) nmol/1. Seventy-five per cent of participants were vitamin D insufficient and 10% deficient. Serum 25(OH)D was positively correlated with time outdoors, physical activity, vitamin D intake and ambient UVR, and inversely correlated with age, BMI and poor self-reported health status. These predictors explained approximately 21% of the variance in serum 25(OH)D. The area under the ROC curve predicting vitamin D deficiency was 0·82. Net benefit for the prediction model was higher than that for the 'test-all' strategy at all probability thresholds and higher than the 'test-none' strategy for probabilities up to 60%. CONCLUSION Our model could predict vitamin D deficiency with reasonable accuracy, but it needs to be validated in other populations before being implemented.

Spectrum sensing optimisation for dynamic primary user signal

We propose a multi-layer spectrum sensing optimisation algorithm to maximise sensing efficiency by computing the optimal sensing and transmission durations for a fast changing, dynamic primary user. Dynamic primary user traffic is modelled as a random process, where the primary user changes states during both the sensing period and transmission period to reflect a more realistic scenario. Furthermore, we formulate joint constraints to correctly reflect interference to the primary user and lost opportunity of the secondary user during the transmission period. Finally, we implement a novel duty cycle based detector that is optimised with respect to PU traffic to accurately detect primary user activity during the sensing period. Simulation results show that unlike currently used detection models, the proposed algorithm can jointly optimise the sensing and transmission durations to simultaneously satisfy the optimisation constraints for the considered primary user traffic.

Numerical techniques for simulating a fractional mathematical model of epidermal wound healing

A number of mathematical models investigating certain aspects of the complicated process of wound healing are reported in the literature in recent years. However, effective numerical methods and supporting error analysis for the fractional equations which describe the process of wound healing are still limited. In this paper, we consider the numerical simulation of a fractional mathematical model of epidermal wound healing (FMM-EWH), which is based on the coupled advection-diffusion equations for cell and chemical concentration in a polar coordinate system. The space fractional derivatives are defined in the Left and Right Riemann-Liouville sense. Fractional orders in the advection and diffusion terms belong to the intervals (0, 1) or (1, 2], respectively. Some numerical techniques will be used. Firstly, the coupled advection-diffusion equations are decoupled to a single space fractional advection-diffusion equation in a polar coordinate system. Secondly, we propose a new implicit difference method for simulating this equation by using the equivalent of Riemann-Liouville and Grünwald-Letnikov fractional derivative definitions. Thirdly, its stability and convergence are discussed, respectively. Finally, some numerical results are given to demonstrate the theoretical analysis.

Higher order thinking in chemistry curriculum and its assessment

Higher-order thinking has featured persistently in the reform agenda for science education. The intended curriculum in various countries sets out aspirational statements for the levels of higher-order thinking to be attained by students. This study reports the extent to which chemistry examinations from four Australian states align and facilitate the intended higher-order thinking skills stipulated in curriculum documents. Through content analysis, the curriculum goals were identified for each state and compared to the nature of question items in the corresponding examinations. Categories of higher-order thinking were adapted from the OECD’s PISA Science test to analyze question items. There was considerable variation in the extent to which the examinations from the states supported the curriculum intent of developing and assessing higher-order thinking. Generally, examinations that used a marks-based system tended to emphasize lower-order thinking, with a greater distribution of marks allocated for lower-order thinking questions. Examinations associated with a criterion-referenced examination tended to award greater credit for higher-order thinking questions. The level of complexity of chemistry was another factor that limited the extent to which examination questions supported higher-order thinking. Implications from these findings are drawn for the authorities responsible for designing curriculum and assessment procedures and for teachers.

The economics of awards, status and performance: the John Bates Clark Medal and the Fellowship of the Econometric Society

This thesis investigates whether receiving an important award in academia raises recipients’ subsequent research productivity and status compared to a synthetic control group of non-recipient scholars with similar previous research performance. It examines the case of being awarded the John Bates Clark Medal and becoming a Fellow of the Econometric Society finding evidence of positive incentive and status effects that raise both productivity and citation levels.

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments.

The getting of wisdom : the future of PM university education in Australia

The dynamic and complex nature of project management (PM) in Australia provides exciting opportunities for universities to that are willing to actively engage with their corporate partners and other key stakeholders to develop program and course offerings that simultaneously address the needs of students, employers, and other stakeholders and further the current body of PM knowledge and research. This article identifies key challenges and opportunities for the future direction of PM university education. This draws on descriptions of successful program models, examining teaching on PM skills on generic business and engineering degrees, dedicated graduate masters programs in PM, doctoral research programs in PM, and forms of effective collaboration between industry and academia.