The African Institute for Mathematical Sciences

Its mission is to promote Mathematics and Science in Africa and to provide a focal point for Mathematics university training in Africa. It offers scholarships for up to 50 students to come and study for a period of nine months. Of the 50 students, about 15 positions are reserved for females. In the 2006/2007 intake there were over 250 applicants. The students are housed and fed and their return travel from their home town is fully funded. Lecturers also stay at AIMS and share their meals with the students, so that a rapport quickly develops. The students are away from their families and friends for nine months and are absolutely committed to the discipline of Mathematics. When they first arrive, some of them have little ability in English but since all tuition is in English they quickly learn. Some find the transitions difficult but they all support one another and at the end of their time their English skills are very good. The students do a series of subjects that last for about three weeks each, consisting of 30 contact hours, as well as a thesis/project. Each course has a number of assignments associated with it and these get evaluated. AIMS has seven or eight teaching assistants who help with the tutorials, marking, advice, and who are a vital component of AIMS.

Australian learning and teaching council projects in the mathematical sciences : a retrospective

Since 2004, the Australian Learning and Teaching Council (ALTC) and its predecessor, the Carrick Institute for Learning and Teaching in Higher Education, have funded numerous teaching and educational research-based projects in the Mathematical Sciences. In light of the Commonwealth Government’s decision to close the ALTC in 2011, it is appropriate to take account of the ALTCs input into the Mathe- matical Sciences in higher education. Here we present an overview of ALTC projects in the Mathematical Sciences, as well as report on the contributions they have made to the Discipline.

A geometrical analysis of trajectory design for underwater vehicles

Designing trajectories for a submerged rigid body motivates this paper. Two approaches are addressed: the time optimal approach and the motion planning ap- proach using concatenation of kinematic motions. We focus on the structure of singular extremals and their relation to the existence of rank-one kinematic reduc- tions; thereby linking the optimization problem to the inherent geometric frame- work. Using these kinematic reductions, we provide a solution to the motion plan- ning problem in the under-actuated scenario, or equivalently, in the case of actuator failures. We finish the paper comparing a time optimal trajectory to one formed by concatenation of pure motions.

Coexistence of stable spots and fronts in a three-component FitzHugh-Nagumo system

We investigate regions of bistability between different travelling and stationary structures in a planar singularly-perturbed three-component reaction-diffusion system that arises in the context of gas discharge systems. In previous work, we delineated the existence and stabil-ity regions of stationary localized spots in this system. Here, we complement this analysis by establishing the stability regions of planar travelling fronts and stationary stripes. Taken together, these results imply that stable fronts and spots can coexist in three-component systems. Numerical simulations indicate that the stable fronts never move towards stable spots but instead move away from them.

Some practical issues in the design and analysis of computer experiments

Deterministic computer simulations of physical experiments are now common techniques in science and engineering. Often, physical experiments are too time consuming, expensive or impossible to conduct. Complex computer models or codes, rather than physical experiments lead to the study of computer experiments, which are used to investigate many scientific phenomena of this nature. A computer experiment consists of a number of runs of the computer code with different input choices. The Design and Analysis of Computer Experiments is a rapidly growing technique in statistical experimental design. This thesis investigates some practical issues in the design and analysis of computer experiments and attempts to answer some of the questions faced by experimenters using computer experiments. In particular, the question of the number of computer experiments and how they should be augmented is studied and attention is given to when the response is a function over time.

New constructions of anonymous membership broadcasting schemes

An anonymous membership broadcast scheme is a method in which a sender broadcasts the secret identity of one out of a set of n receivers, in such a way that only the right receiver knows that he is the intended receiver, while the others can not determine any information about this identity (except that they know that they are not the intended ones). In a w-anonymous membership broadcast scheme no coalition of up to w receivers, not containing the selected receiver, is able to determine any information about the identity of the selected receiver. We present two new constructions of w-anonymous membership broadcast schemes. The first construction is based on error-correcting codes and we show that there exist schemes that allow a flexible choice of w while keeping the complexities for broadcast communication, user storage and required randomness polynomial in log n,. The second construction is based on the concept of collision-free arrays, which is introduced in this paper. The construction results in more flexible schemes, allowing trade-offs between different complexities.

Constructing commutative semifields of square order

The projection construction has been used to construct semifields of odd characteristic using a field and a twisted semifield [Commutative semi-fields from projection mappings, Designs, Codes and Cryptography, 61 (2011), 187{196]. We generalize this idea to a projection construction using two twisted semifields to construct semifields of odd characteristic. Planar functions and semifields have a strong connection so this also constructs new planar functions.

A mathematical model of chlamydial infection incorporating movement of chlamydial particles

We present a spatiotemporal mathematical model of chlamydial infection, host immune response and spatial movement of infectious particles. The re- sulting partial differential equations model both the dynamics of the infection and changes in infection profile observed spatially along the length of the host genital tract. This model advances previous chlamydia modelling by incorporating spatial change, which we also demonstrate to be essential when the timescale for movement of infectious particles is equal to, or shorter than, the developmental cycle timescale. Numerical solutions and model analysis are carried out, and we present a hypothesis regarding the potential for treatment and prevention of infection by increasing chlamydial particle motility.

Clinical strategies for the alleviation of contractures from a predictive mathematical model of dermal repair

Hypertrophic scars arise when there is an overproduction of collagen during wound healing. These are often associated with poor regulation of the rate of programmed cell death(apoptosis) of the cells synthesizing the collagen or by an exuberant inflammatory response that prolongs collagen production and increases wound contraction. Severe contractures that occur, for example, after a deep burn can cause loss of function especially if the wound is over a joint such as the elbow or knee. Recently, we have developed a morphoelastic mathematical model for dermal repair that incorporates the chemical, cellular and mechanical aspects of dermal wound healing. Using this model, we examine pathological scarring in dermal repair by first assuming a smaller than usual apoptotic rate for myofibroblasts, and then considering a prolonged inflammatory response, in an attempt to determine a possible optimal intervention strategy to promote normal repair, or terminate the fibrotic scarring response. Our model predicts that in both cases it is best to apply the intervention strategy early in the wound healing response. Further, the earlier an intervention is made, the less aggressive the intervention required. Finally, if intervention is conducted at a late time during healing, a significant intervention is required; however, there is a threshold concentration of the drug or therapy applied, above which minimal further improvement to wound repair is obtained.

Mathematical modelling of tumour growth and interaction with host tissue and the immune system

In this thesis, three mathematical models describing the growth of solid tumour incorporating the host tissue and the immune system response are developed and investigated. The initial model describes the dynamics of the growing tumour and immune response before being extended in the second model by introducing a time-varying dendritic cell-based treatment strategy. Finally, in the third model, we present a mathematical model of a growing tumour using a hybrid cellular automata. These models can provide information to pre-experimental work to assist in designing more effective and efficient laboratory experiments related to tumour growth and interactions with the immune system and immunotherapy.

Mathematical investigation of the interactions between the inflammatory response and mechanical aspects of dermal wound repair

Dermal wound repair involves complex interactions between cells, cytokines and mechanics to close injuries to the skin. In particular, we investigate the contribution of fibroblasts, myofibroblasts, TGFβ, collagen and local tissue mechanics to wound repair in the human dermis. We develop a morphoelastic model where a realistic representation of tissue mechanics is key, and a fibrocontractive model that involves a reasonable approximation to the true kinetics of the important bioactive species. We use each of these descriptions to elucidate the mechanisms that generate pathologies such as hypertrophic scars, contractures and keloids. We find that for hypertrophic scar and contracture development, factors regulating the myofibroblast phenotype are critical, with heightened myofibroblast activation, reduced myofibroblast apoptosis or prolonged inflammation all predicted as mediators for scar hypertrophy and contractures. Prevention of these pathologies is predicted when myofibroblast apoptosis is induced, myofibroblast activation is blocked or TGFβ is neutralised. To investigate keloid invasion, we develop a caricature representation of the fibrocontractive model and find that TGFβ spread is the driving factor behind keloid growth. Blocking activation of TGFβ is found to cause keloid regression. Thus, we recommend myofibroblasts and TGFβ as targets for clinicians when developing intervention strategies for prevention and cure of fibrotic scars.

A comparison of mathematical models for phase separation in high-rate LiFePO4 cathodes

We construct a two-scale mathematical model for modern, high-rate LiFePO4cathodes. We attempt to validate against experimental data using two forms of the phase-field model developed recently to represent the concentration of Li+ in nano-sized LiFePO4crystals. We also compare this with the shrinking-core based model we developed previously. Validating against high-rate experimental data, in which electronic and electrolytic resistances have been reduced is an excellent test of the validity of the crystal-scale model used to represent the phase-change that may occur in LiFePO4material. We obtain poor fits with the shrinking-core based model, even with fitting based on “effective” parameter values. Surprisingly, using the more sophisticated phase-field models on the crystal-scale results in poorer fits, though a significant parameter regime could not be investigated due to numerical difficulties. Separate to the fits obtained, using phase-field based models embedded in a two-scale cathodic model results in “many-particle” effects consistent with those reported recently.

Mathematical modeling of the cancer cell’s control circuitry : paving the way to individualized therapeutic strategies

Cancer is a disease of signal transduction in which the dysregulation of the network of intracellular and extracellular signaling cascades is sufficient to thwart the cells finely-tuned biochemical control mechanisms. A keen interest in the mathematical modeling of cell signaling networks and the regulation of signal transduction has emerged in recent years, and has produced a glimmer of insight into the sophisticated feedback control and network regulation operating within cells. In this review, we present an overview of published theoretical studies on the control aspects of signal transduction, emphasizing the role and importance of mechanisms such as ‘ultrasensitivity’ and feedback loops. We emphasize that these exquisite and often subtle control strategies represent the key to orchestrating ‘simple’ signaling behaviors within the complex intracellular network, while regulating the trade-off between sensitivity and robustness to internal and external perturbations. Through a consideration of these apparent paradoxes, we explore how the basic homeostasis of the intracellular signaling network, in the face of carcinogenesis, can lead to neoplastic progression rather than cell death. A simple mathematical model is presented, furnishing a vivid illustration of how ‘control-oriented’ models of the deranged signaling networks in cancer cells may enucleate improved treatment strategies, including patient-tailored combination therapies, with the potential for reduced toxicity and more robust and potent antitumor activity.