On the order of the fractional Laplacian in determining the spatio-temporal evolution of a space-fractional model of cardiac electrophysiology

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made.

Characterization of anomalous relaxation using the time-fractional Bloch equation and multiple echo T2 *-weighted magnetic resonance imaging at 7T

PURPOSE To study the utility of fractional calculus in modeling gradient-recalled echo MRI signal decay in the normal human brain. METHODS We solved analytically the extended time-fractional Bloch equations resulting in five model parameters, namely, the amplitude, relaxation rate, order of the time-fractional derivative, frequency shift, and constant offset. Voxel-level temporal fitting of the MRI signal was performed using the classical monoexponential model, a previously developed anomalous relaxation model, and using our extended time-fractional relaxation model. Nine brain regions segmented from multiple echo gradient-recalled echo 7 Tesla MRI data acquired from five participants were then used to investigate the characteristics of the extended time-fractional model parameters. RESULTS We found that the extended time-fractional model is able to fit the experimental data with smaller mean squared error than the classical monoexponential relaxation model and the anomalous relaxation model, which do not account for frequency shift. CONCLUSIONS We were able to fit multiple echo time MRI data with high accuracy using the developed model. Parameters of the model likely capture information on microstructural and susceptibility-induced changes in the human brain.

SCOOT CBN

SCOOT is a hybrid event combining the web, mobile devices, public displays and cultural artifacts across multiple public parks and museums in an effort to increase the perceived and actual access to cultural participation by everyday people. The research field is locative game design and the context was the re-invigoration of public sites as a means for exposing the underlying histories of sites and events. The key question was how to use game play technologies and processes within everyday places in ways that best promote playful and culturally meaningful experiences whilst shifting the loci of control away from commercial and governmental powers. The research methodology was primarily practice led underpinned by ethnographic and action research methods. In 2004 SCOOT established itself as a national leader in the field by demonstrating innovative methods for stimulating rich interactions across diverse urban places using technically-augmented game play. Despite creating a sophisticated range of software and communication tools SCOOT most dramatically highlighted the role of the ubiquitous mobile phone in facilitating socially beneficial experiences. Through working closely with the SCOOT team, collaborating organisations developed important new knowledge around the potential of new technologies and processes for motivating, sustaining and reinvigorating public engagement. Since 2004, SCOOT has been awarded $600,00 in competitive and community funding as well as countless in kind support from partner organisations such as Arts Victoria, National Gallery of Victoria, Melbourne Museum, Australian Centre for the Moving Image, Federation Square, Art Centre of Victoria, The State Library of Victoria, Brisbane River Festival, State Library of Queensland, Brisbane Maritime Museum, Queensland University of Technology, and Victoria University.

The function of property in creating an effective legal regime for trade in environmental offsets

Market-based environmental regulation is becoming increasingly common within international and national frameworks. Environmental offset and trading regimes are part of the market-based instrument revolution. This paper proposes that environmental market mechanisms could be used to introduce an ethic of land holder responsibility. In order for market based regimes to attract sufficient levels of stakeholder engagement, participants within such scheme require an incentive to participate and furthermore need to feel a sense of security about investing in such processes. A sense of security is often associated with property based interests. This paper explores the property related issues connected with environmental offset and trading scheme initiatives. Relevant property-related considerations include land tenure considerations, public versus private management of land choices, characteristics and powers associated with property interests, theories defining property and the recognition of legal proprietal interests. The Biodiversity Banking Scheme in New South Wales is then examined as a case study followed by a critique on the role of environmental markets.

Predatory police : the roles of ethics and networks as mediating factors

The institution the police force has been established to protect citizens and their property from harm and predatory opportunism. However, there have been occasions when the very people assigned to protect become part of the predatory force against society. Predatory policing occurs when the police use their powers to extort money in the form of bribes. While, the concept is receiving attention in Europe but there have not been any direct studies in Australia. To overcome this research deficit and determine the extent, if any, of predatory policing in Australia data is interrogated from four police corruption inquiries in the Australian states of Queensland, New South Wales, Victoria and Western Australia. In addition, it examines the role of the type of networks used by corrupt police officers. The synthesis and application of public corruption and network literatures to the predatory policing domain provides new and relevant insights to assist those responsible for the administration of our institutions of justice. The paper concludes with a framework, drawn from the first stage of the project, to assist in the conceptualisation and monitoring of this public problem.

Effect of inaccuracies in anthropometric data and linked-segment inverse dynamic modeling on kinetics of gait in persons with partial foot amputation

The accuracy of data derived from linked-segment models depends on how well the system has been represented. Previous investigations describing the gait of persons with partial foot amputation did not account for the unique anthropometry of the residuum or the inclusion of a prosthesis and footwear in the model and, as such, are likely to have underestimated the magnitude of the peak joint moments and powers. This investigation determined the effect of inaccuracies in the anthropometric input data on the kinetics of gait. Toward this end, a geometric model was developed and validated to estimate body segment parameters of various intact and partial feet. These data were then incorporated into customized linked-segment models, and the kinetic data were compared with that obtained from conventional models. Results indicate that accurate modeling increased the magnitude of the peak hip and knee joint moments and powers during terminal swing. Conventional inverse dynamic models are sufficiently accurate for research questions relating to stance phase. More accurate models that account for the anthropometry of the residuum, prosthesis, and footwear better reflect the work of the hip extensors and knee flexors to decelerate the limb during terminal swing phase.

Alternative schooling programs for at risk youth : three case studies

This thesis develops a critical realist explanatory critique of alternative schooling programs for youth at risk taking place at three case study sites. Throughout the thesis the author pursues the question, \Are alternative provisions of schooling working academically and socially for youth at risk?. The academic lens targets literacy learning and associated pedagogies. Social outcomes are posited as positive social behaviours and continued engagement in learning. A four phased analysis, drawing on critical realism, interpretive and subject specific theories is used to elicit explanations for the research question. An overall framework is a critical realist methodology as set out by Danermark, Ekstrom, Jakobsen and Karlsson (2002, p. 129). Consequently phase one describes the phenomena of alternative schooling programs taking place at three case study sites. This is reported first as staff narratives that are resolved into imaginable historical causal components of \generative events., \prior schooling structures., \models of alternative schooling., \purpose., \individual agency., and \relations with linked community organisations.. Then transcendental questions are posed about each component using retroduction to uncover structures, underlying mechanisms and powers, and individual agency. In the second phase the researcher uses modified grounded theory methodology to theoretically redescribe causal categories related to a \needed different teaching and administrative approach. that emerged from the previous critique. A transcendental question is then applied to this redescription. The research phenomena are again theoretically redescribed in the third phase, this time using three theoretically based constructs associated with literacy and literacy pedagogies; the NRS, the 4 Resources Model, and Productive Pedagogies. This redescription is again questioned in terms of its core or \necessary. components. The fourth phase makes an explanatory critique by comparing and critiquing all previous explanations, recontextualising them in a wider macro reality of alternative schooling. Through this critical realist explanatory critiquing process, a response emerges not only to whether alternative provisions of schooling are working, but also how they are working, and how they are not working, with realistically based implications for future improvement.

Bifocal lens control of myopia progression in children

This research investigated underlying issues that were critical to the success of the bifocal trial and comprised of three studies. The first study evaluated if Chinese-Canadian children were suitable subjects for the bifocal trial. The high prevalence of myopia in Chinese children suggests that genetic input plays a role in myopia development, but the rapid increase in prevalence over the last few decades indicates environmental factors are also important. Since this bifocal trial was conducted in Canada, this work aimed to determine whether Chinese children who had migrated to Canada would still have high myopia prevalence and a high rate of myopia progression. The second study determined the optimal bifocal lens power for myopia treatment and the effect of incorporating base-in prism into the bifocal. In the majority of published myopia control studies, the power of the prescribed near addition was usually predetermined in the belief that the near addition would always help to improve the near focus. In fact, the effect of near addition on the accommodative error might be quite different even for individuals in which the same magnitude of accommodation lag had been measured. Therefore, this work was necessary to guide the selection of bifocal and prism powers most suitable for the subsequent bifocal trial. The third study, the ultimate goal of this research, was to conduct a longitudinal clinical trial to determine if bifocals and prismatic bifocals could control myopia progression in children.

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 method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

Stochastic modelling of financial time series with memory and multifractal scaling

Financial processes may possess long memory and their probability densities may display heavy tails. Many models have been developed to deal with this tail behaviour, which reflects the jumps in the sample paths. On the other hand, the presence of long memory, which contradicts the efficient market hypothesis, is still an issue for further debates. These difficulties present challenges with the problems of memory detection and modelling the co-presence of long memory and heavy tails. This PhD project aims to respond to these challenges. The first part aims to detect memory in a large number of financial time series on stock prices and exchange rates using their scaling properties. Since financial time series often exhibit stochastic trends, a common form of nonstationarity, strong trends in the data can lead to false detection of memory. We will take advantage of a technique known as multifractal detrended fluctuation analysis (MF-DFA) that can systematically eliminate trends of different orders. This method is based on the identification of scaling of the q-th-order moments and is a generalisation of the standard detrended fluctuation analysis (DFA) which uses only the second moment; that is, q = 2. We also consider the rescaled range R/S analysis and the periodogram method to detect memory in financial time series and compare their results with the MF-DFA. An interesting finding is that short memory is detected for stock prices of the American Stock Exchange (AMEX) and long memory is found present in the time series of two exchange rates, namely the French franc and the Deutsche mark. Electricity price series of the five states of Australia are also found to possess long memory. For these electricity price series, heavy tails are also pronounced in their probability densities. The second part of the thesis develops models to represent short-memory and longmemory financial processes as detected in Part I. These models take the form of continuous-time AR(∞) -type equations whose kernel is the Laplace transform of a finite Borel measure. By imposing appropriate conditions on this measure, short memory or long memory in the dynamics of the solution will result. A specific form of the models, which has a good MA(∞) -type representation, is presented for the short memory case. Parameter estimation of this type of models is performed via least squares, and the models are applied to the stock prices in the AMEX, which have been established in Part I to possess short memory. By selecting the kernel in the continuous-time AR(∞) -type equations to have the form of Riemann-Liouville fractional derivative, we obtain a fractional stochastic differential equation driven by Brownian motion. This type of equations is used to represent financial processes with long memory, whose dynamics is described by the fractional derivative in the equation. These models are estimated via quasi-likelihood, namely via a continuoustime version of the Gauss-Whittle method. The models are applied to the exchange rates and the electricity prices of Part I with the aim of confirming their possible long-range dependence established by MF-DFA. The third part of the thesis provides an application of the results established in Parts I and II to characterise and classify financial markets. We will pay attention to the New York Stock Exchange (NYSE), the American Stock Exchange (AMEX), the NASDAQ Stock Exchange (NASDAQ) and the Toronto Stock Exchange (TSX). The parameters from MF-DFA and those of the short-memory AR(∞) -type models will be employed in this classification. We propose the Fisher discriminant algorithm to find a classifier in the two and three-dimensional spaces of data sets and then provide cross-validation to verify discriminant accuracies. This classification is useful for understanding and predicting the behaviour of different processes within the same market. The fourth part of the thesis investigates the heavy-tailed behaviour of financial processes which may also possess long memory. We consider fractional stochastic differential equations driven by stable noise to model financial processes such as electricity prices. The long memory of electricity prices is represented by a fractional derivative, while the stable noise input models their non-Gaussianity via the tails of their probability density. A method using the empirical densities and MF-DFA will be provided to estimate all the parameters of the model and simulate sample paths of the equation. The method is then applied to analyse daily spot prices for five states of Australia. Comparison with the results obtained from the R/S analysis, periodogram method and MF-DFA are provided. The results from fractional SDEs agree with those from MF-DFA, which are based on multifractal scaling, while those from the periodograms, which are based on the second order, seem to underestimate the long memory dynamics of the process. This highlights the need and usefulness of fractal methods in modelling non-Gaussian financial processes with long memory.