On the zeros of the Abelian integrals for a class of Liénard systems

A planar polynomial differential system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for the general nonlinear dynamical systems. In this paper, we investigated a class of Liénard systems of the form x'=y, y'=f(x)+y g(x) with deg f=5 and deg g=4. We proved that the related elliptic integrals of the Liénard systems have at most three zeros including multiple zeros, which implies that the number of limit cycles bifurcated from the periodic orbits of the unperturbed system is less than or equal to 3.

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.

Learning the Kernel Matrix with Semi-Definite Programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space -- classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semi-definite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -- using the labelled part of the data one can learn an embedding also for the unlabelled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method to learn the 2-norm soft margin parameter in support vector machines, solving another important open problem. Finally, the novel approach presented in the paper is supported by positive empirical results.

An example of cognitive obstacles in advanced integration: the case of scalar line integrals

Cognitive obstacles that arise in the teaching and learning of scalar line integrals, derived from cognitive aids provided to students when first learning about integration of single variable functions are described. A discussion of how and why the obstacles cause students problems is presented and possible strategies to overcome the obstacles are outlined.

Random projections on manifolds of symmetric positive definite matrices for image classification

Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent spaces, or (2) embedding into Reproducing Kernel Hilbert Spaces (RKHS). While embedding into tangent spaces allows the use of existing Euclidean-based learning algorithms, manifold shape is only approximated which can cause loss of discriminatory information. The RKHS approach retains more of the manifold structure, but may require non-trivial effort to kernelise Euclidean-based learning algorithms. In contrast to the above approaches, in this paper we offer a novel solution that allows SPD matrices to be used with unmodified Euclidean-based learning algorithms, with the true manifold shape well-preserved. Specifically, we propose to project SPD matrices using a set of random projection hyperplanes over RKHS into a random projection space, which leads to representing each matrix as a vector of projection coefficients. Experiments on face recognition, person re-identification and texture classification show that the proposed approach outperforms several recent methods, such as Tensor Sparse Coding, Histogram Plus Epitome, Riemannian Locality Preserving Projection and Relational Divergence Classification.

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Lyprinol : is it a useful anti-inflammatory agent?

The New Zealand green lipped mussel preparation Lyprinol is available without a prescription from a supermarket, pharmacy or Web. The Food and Drug Administration have recently warned Lyprinol USA about their extravagant anti-inflammatory claims for Lyprinol appearing on the web. These claims are put to thorough review. Lyprinol does have anti-inflammatory mechanisms, and has anti-inflammatory effects in some animal models of inflammation. Lyprinol may have benefits in dogs with arthritis. There are design problems with the clinical trials of Lyprinol in humans as an anti-inflammatory agent in osteoarthritis and rheumatoid arthritis, making it difficult to give a definite answer to how effective Lyprinol is in these conditions, but any benefit is small. Lyprinol also has a small benefit in atopic allergy. As anti-inflammatory agents, there is little to choose between Lyprinol and fish oil. No adverse effects have been reported with Lyprinol. Thus, although it is difficult to conclude whether Lyprinol does much good, it can be concluded that Lyprinol probably does no major harm.

Meniscal repair in the elite athlete : results of 45 repairs with a minimum 5-year follow-up

Background The preservation of meniscal tissue is important to protect joint surfaces. Purpose We have an aggressive approach to meniscal repair, including repairing tears other than those classically suited to repair. Here we present the medium- to long-term outcome of meniscal repair (inside-out) in elite athletes. Study Design Case series; Level of evidence, 4. Methods Forty-two elite athletes underwent 45 meniscal repairs. All repairs were performed using an arthroscopically assisted inside-out technique. Eighty-three percent of these athletes had ACL reconstruction at the same time. Patients returned a completed questionnaire (including Lysholm and International Knee Documentation Committee [IKDC] scores). Mean follow-up was 8.5 years. Failure was defined by patients developing symptoms of joint line pain and/or locking or swelling requiring repeat arthroscopy and partial meniscectomy. Results The average Lysholm and subjective IKDC scores were 89.6 and 85.4, respectively. Eighty-one percent of patients returned to their main sport and most to a similar level at a mean time of 10.4 months after repair, reflecting the high level of ACL reconstruction in this group. We identified 11 definite failures, 10 medial and 1 lateral meniscus, that required excision; this represents a 24% failure rate. We identified 1 further patient who had possible failed repairs, giving a worst-case failure rate of 26.7% at a mean of 42 months after surgery. However, 7 of these failures were associated with a further injury. Therefore, the atraumatic failure rate was 11%. Age and size and location of the tears were not associated with a higher failure rate. Medial meniscal repairs were significantly more likely to fail than lateral meniscal repairs, with a failure rate of 36.4% and 5.6%, respectively (P < .05). Conclusion Meniscal repair and healing are possible, and most elite athletes can return to their preinjury level of activity.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Quality of child abuse information for measurement and surveillance : linking hospital and child welfare data

National estimates of the prevalence of child abuse-related injuries are obtained from a variety of sectors including welfare, justice, and health resulting in inconsistent estimates across sectors. The International Classification of Diseases (ICD) is used as the international standard for categorising health data and aggregating data for statistical purposes, though there has been limited validation of the quality, completeness or concordance of these data with other sectors. This research study examined the quality of documentation and coding of child abuse recorded in hospital records in Queensland and the concordance of these data with child welfare records. A retrospective medical record review was used to examine the clinical documentation of over 1000 hospitalised injured children from 20 hospitals in Queensland. A data linkage methodology was used to link these records with records in the child welfare database. Cases were sampled from three sub-groups according to the presence of target ICD codes: Definite abuse, Possible abuse, unintentional injury. Less than 2% of cases coded as being unintentional were recoded after review as being possible abuse, and only 5% of cases coded as possible abuse cases were reclassified as unintentional, though there was greater variation in the classification of cases as definite abuse compared to possible abuse. Concordance of health data with child welfare data varied across patient subgroups. This study will inform the development of strategies to improve the quality, consistency and concordance of information between health and welfare agencies to ensure adequate system responses to children at risk of abuse.

Quality of child abuse information in emergency department injury surveillance data : linking hospital and child welfare data

Emergency departments (EDs) are often the first point of contact with an abused child. Despite legal mandate, the reporting of definite or suspected abusive injury to child safety authorities by ED clinicians varies due to a number of factors including training, access to child safety professionals, departmental culture and a fear of ‘getting it wrong’. This study examined the quality of documentation and coding of child abuse captured by ED based injury surveillance data and ED medical records in the state of Queensland and the concordance of these data with child welfare records. A retrospective medical record review was used to examine the clinical documentation of almost 1000 injured children included in the Queensland Injury Surveillance Unit database (QISU) from 10 hospitals in urban and rural centres. Independent experts re-coded the records based on their review of the notes. A data linkage methodology was then used to link these records with records in the state government’s child welfare database. Cases were sampled from three sub-groups according to the surveillance intent codes: Maltreatment by parent, Undetermined and Unintentional injury. Only 0.1% of cases coded as unintentional injury were recoded to maltreatment by parent, while 1.2% of cases coded as maltreatment by parent were reclassified as unintentional and 5% of cases where the intent was undetermined by the triage nurse were recoded as maltreatment by parent. Quality of documentation varied across type of hospital (tertiary referral centre, children’s, urban, regional and remote). Concordance of health data with child welfare data varied across patient subgroups. Outcomes from this research will guide initiatives to improve the quality of intentional child injury surveillance systems.

Good results for early treatment of clinically isolated syndrome prior to multiple sclerosis with interferon beta-1b and glatiramer

Background: The first sign of developing multiple sclerosis is a clinically isolated syndrome that resembles a multiple sclerosis relapse. Objective/methods: The objective was to review the clinical trials of two medicines in clinically isolated syndromes (interferon β and glatiramer acetate) to determine whether they prevent progression to definite multiple sclerosis. Results: In the BENEFIT trial, after 2 years, 45% of subjects in the placebo group developed clinically definite multiple sclerosis, and the rate was lower in the interferon β-1b group. Then all subjects were offered interferon β-1b, and the original interferon β-1b group became the early treatment group, and the placebo group became the delayed treatment group. After 5 years, the number of subjects with clinical definite multiple sclerosis remained lower in the early treatment than late treatment group. In the PreCISe trial, after 2 years, the time for 25% of the subjects to convert to definite multiple sclerosis was prolonged in the glatiramer group. Conclusions: Interferon β-1b and glatiramer acetate slow the progression of clinically isolated syndromes to definite multiple sclerosis. However, it is not known whether this early treatment slows the progression to the physical disabilities experienced in multiple sclerosis.

Reliability of routinely collected hospital data for child maltreatment surveillance

Background: Internationally, research on child maltreatment-related injuries has been hampered by a lack of available routinely collected health data to identify cases, examine causes, identify risk factors and explore health outcomes. Routinely collected hospital separation data coded using the International Classification of Diseases and Related Health Problems (ICD) system provide an internationally standardised data source for classifying and aggregating diseases, injuries, causes of injuries and related health conditions for statistical purposes. However, there has been limited research to examine the reliability of these data for child maltreatment surveillance purposes. This study examined the reliability of coding of child maltreatment in Queensland, Australia. Methods: A retrospective medical record review and recoding methodology was used to assess the reliability of coding of child maltreatment. A stratified sample of hospitals across Queensland was selected for this study, and a stratified random sample of cases was selected from within those hospitals. Results: In 3.6% of cases the coders disagreed on whether any maltreatment code could be assigned (definite or possible) versus no maltreatment being assigned (unintentional injury), giving a sensitivity of 0.982 and specificity of 0.948. The review of these cases where discrepancies existed revealed that all cases had some indications of risk documented in the records. 15.5% of cases originally assigned a definite or possible maltreatment code, were recoded to a more or less definite strata. In terms of the number and type of maltreatment codes assigned, the auditor assigned a greater number of maltreatment types based on the medical documentation than the original coder assigned (22% of the auditor coded cases had more than one maltreatment type assigned compared to only 6% of the original coded data). The maltreatment types which were the most ‘under-coded’ by the original coder were psychological abuse and neglect. Cases coded with a sexual abuse code showed the highest level of reliability. Conclusion: Given the increasing international attention being given to improving the uniformity of reporting of child-maltreatment related injuries and the emphasis on the better utilisation of routinely collected health data, this study provides an estimate of the reliability of maltreatment-specific ICD-10-AM codes assigned in an inpatient setting.

Listening in to others : in between noise and silence

Since the launch of the ‘Clean Delhi, Green Delhi’ campaign in 2003, slums have become a significant social and political issue in India’s capital city. Through this campaign, the state, in collaboration with Delhi’s middle class through the ‘Bhagidari system’ (literally translated as ‘participatory system’), aims to transform Delhi into a ‘world-class city’ that offers a sanitised, aesthetically appealing urban experience to its citizens and Western visitors. In 2007, Delhi won the bid to host the 2010 Commonwealth Games; since then, this agenda has acquired an urgent, almost violent, impetus to transform Delhi into an environmentally friendly, aesthetically appealing and ‘truly international city’. Slums and slum-dwellers, with their ‘filth, dirt, and noise’, have no place in this imagined city. The violence inflicted upon slum-dwellers, including the denial of their judicial rights, is justified on these accounts. In addition, the juridical discourse since 2000 has ‘re-problematised slums as ‘nuisance’. The rising antagonism of the middle-classes against the poor, supported by the state’s ambition to have a ‘world-class city’, has allowed a new rhetoric to situate the slums in the city. These representations articulate slums as homogenised spaces of experience and identity. The ‘illegal’ status of slum-dwellers, as encroachers upon public space, is stretched to involve ‘social, cultural, and moral’ decadence and depravity. This thesis is an ethnographic exploration of everyday life in a prominent slum settlement in Delhi. It sensually examines the social, cultural and political materiality of slums, and the relationship of slums with the middle class. In doing so, it highlights the politics of sensorial ordering of slums as ‘filthy, dirty, and noisy’ by the middle classes to calcify their position as ‘others’ in order to further segregate, exclude and discriminate the slums. The ethnographic experience in the slums, however, highlights a complex sensorial ordering and politics of its own. Not only are the interactions between diverse communities in slums highly restricted and sensually ordained, but the middle class is identified as a sensual ‘other’, and its sensual practices prohibited. This is significant in two ways. First, it highlights the multiplicity of social, cultural experience and engagement in the slums, thereby challenging its homogenised representation. Second, the ethnographic exploration allowed me to frame a distinct sense of self amongst the slums, which is denied in mainstream discourses, and allowed me to identify the slums’ own ’others’, middle class being one of them. This thesis highlights sound – its production, performances and articulations – as an act with social, cultural, and political implications and manifestations. ‘Noise’ can be understood as a political construct to identify ‘others’ – and both slum-dwellers and the middle classes identify different sonic practices as noise to situate the ‘other’ sonically. It is within this context that this thesis frames the position of Listener and Hearer, which corresponds to their social-political positions. These positions can be, and are, resisted and circumvented through sonic practices. For instance, amplification tactics in the Karimnagar slums, which are understood as ‘uncultured, callous activities to just create more noise’ by the slums’ middle-class neighbours, also serve definite purposes in shaping and navigating the space through the slums’ soundscapes, asserting a presence that is otherwise denied. Such tactics allow the residents to define their sonic territories and scope of sonic performances; they are significant in terms of exerting one’s position, territory and identity, and they are very important in subverting hierarchies. The residents of the Karimnagar slums have to negotiate many social, cultural, moral and political prejudices in their everyday lives. Their identity is constantly under scrutiny and threat. However, the sonic cultures and practices in the Karimnagar slums allow their residents to exert a definite sonic presence – which the middle class has to hear. The articulation of noise and silence is an act manifesting, referencing and resisting social, cultural, and political power and hierarchies.