Existence of travelling wave solutions for a model of tumour invasion

The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed.

Sing & Grow : the co-existence of evaluation research and clinical practice in an early intervention music therapy project

Sing & Grow is a short term early intervention music therapy program for at risk families. Sing & Grow uses music to strengthen parent-child relationships by increasing positive parent-child interactions, assisting parents to bond with their children, and extending the repertoire of parents’ skills in relating to their child through interactive . Both the Australian and New Zealand governments are looking for evidence based research to highlight the effectiveness of funded programs in early childhood. As a government funded program, independent evaluation is a requirement of the delivery of the service. This paper explains the process involved in setting up and managing this large scale evaluation from engaging the evaluators and designing the project, to the data gathering stage. It describes the various challenges encountered and concludes that a highly collaborative and communicative partnership bet en researchers and clinicians is essential to ensure data can be gathered with minimal disturbance to clinical music therapy practice.

Real or imagined? A study exploring the existence of common method variance effects in road safety research

Common method variance (CMV) has received little attention within the field of road safety research despite a heavy reliance on self-report data. Two surveys were completed by 214 motorists over a two-month period, allowing associations between social desirability and key road safety variables and relationships between scales across the two survey waves to be examined. Social desirability was found to have a strong negative correlation with the Driver Behaviour Questionnaire (DBQ) sub-scales as well as age, but not with crashes and offences. Drivers who scored higher on the social desirability scale were also less likely to report aberrant driving behaviours as measured by the DBQ. Controlling for social desirability did not substantially alter the predictive relationship between the DBQ and the crash and offences variables. The strength of the correlations within and between the two waves were also compared with the results strongly suggesting that effects associated with CMV were present. Identification of CMV would be enhanced by the replication of this study with a larger sample size and comparing self-report data with official sources.

Once more into the mire, dear friends : determining the existence of a duty of care in negligence

There were signs in the 1997 High Court decision in Hill v Van Erp that the different members of the bench were beginning to move in the same direction when it came to the tort equivalent of the search for the Holy Grail, a common approach to the determination of the existence of a duty of care in negligence. However, the court's subsequent decision in Perre v Apand signaled a slide back to uncertainty with the seven judges favouring five different approaches. This Note examines those five approaches in the search for guidance for those at the "coalface" - litigants, their legal advisers and trial judges.

Numerical solutions for thin film flow down the outside and inside of a vertical cylinder

We consider a model for thin film flow down the outside and inside of a vertical cylinder. Our focus is to study the effect that the curvature of the cylinder has on the gravity-driven instability of the advancing contact line and to simulate the resulting fingering patterns that form due to this instability. The governing partial differential equation is fourth order with a nonlinear degenerate diffusion term that represents the stabilising effect of surface tension. We present numerical solutions obtained by implementing an efficient alternating direction implicit scheme. When compared to the problem of flow down a vertical plane, we find that increasing substrate curvature tends to increase the fingering instability for flow down the outside of the cylinder, whereas flow down the inside of the cylinder substrate curvature has the opposite effect. Further, we demonstrate the existence of nontrivial travelling wave solutions which describe fingering patterns that propagate down the inside of a cylinder at constant speed without changing form. These solutions are perfectly analogous to those found previously for thin film flow down an inclined plane.

Novel solutions for a model of wound healing angiogenesis

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions.

Effective removal of zinc (II) from aqueous solutions by tricalcium aluminate (C3A)

Recently, studies have identified high zinc levels in various environmental resources, and excessive intake of zinc has long been considered to be harmful to human health. The aim of this research was to investigate the effectiveness of tricalcium aluminate (C3A) as a removal agent of zinc from aqueous solution. Inductively coupled plasma-atomic emission spectrometer (ICP-AES), X-ray diffraction (XRD) and scanning electron microscopy (SEM) have been used to characterize such removal behavior. The effects of various factors such as pH influence, temperature and contact time were investigated. The adsorption capacity of C3A for Zn2+ was computed to be up to 13.73 mmol g−1, and the highest zinc removal capacity was obtained when the initial pH of Zn(NO3)2 solution was between 6.0 and 7.0, with temperature around 308 K. The XRD analysis showed that the resultant products were ZnAl-LDHs. Combined with the analysis of solution component, it was proved the existence of both precipitation and cation exchange in the removal process. From the experimental results, it was clear that C3A could be potentially used as a cost-effective material for the removal of zinc in aqueous environment.

Flow of a micropolar fluid bounded by a stretching sheet

We consider boundary layer flow of a micropolar fluid driven by a porous stretching sheet. A similarity solution is defined, and numerical solutions using Runge-Kutta and quasilinearisation schemes are obtained. A perturbation analysis is also used to derive analytic solutions to first order in the perturbing parameter. The resulting closed form solutions involve relatively complex expressions, and the analysis is made more tractable by a combination of offline and online work using a computational algebra system (CAS). For this combined numerical and analytic approach, the perturbation analysis yields a number of benefits with regard to the numerical work. The existence of a closed form solution helps to discriminate between acceptable and spurious numerical solutions. Also, the expressions obtained from the perturbation work can provide an accurate description of the solution for ranges of parameters where the numerical approaches considered here prove computationally more difficult.

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.

An examination of the applicability of hydrotalcite for removing oxalate anions from Bayer process solutions

Hydrotalcite and thermally activated hydrotalcites were examined for their potential as methods for the removal of oxalate anions from Bayer Process liquors. Hydrotalcite was prepared and characterised by a number of methods, including X-ray diffraction, thermogravimetric analysis, nitrogen adsorption analysis and vibrational spectroscopy. Thermally activated hydrotalcites were prepared by a low temperature method and characterised using X-ray diffraction, nitrogen adsorption analysis and vibrational spectroscopy. Oxalate intercalated hydrotalcite was prepared by two methods and analysed with X-ray diffraction and for the first time thermogravimetric analysis, Raman spectroscopy and infrared emission spectroscopy. The adsorption of oxalate anions by hydrotalcite and thermally activated hydrotalcite was tested in a range of solutions using both batch and kinetic adsorption models.

Pulse dynamics in a three-component system : existence analysis

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Numerical methods for strong solutions of stochastic differential equations : an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.