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.

Impact of music therapy to promote positive parenting and child development

The effectiveness of a 10-week group music therapy program for marginalized parents and their children aged 0–5 years was examined. Musical activities were used to promote positive parent–child relationships and children’s behavioral, communicative and social development. Participants were 358 parents and children from families facing social disadvantage, young parents or parents of a child with a disability. Significant improvements were found for therapist-observed parent and child behaviors, and parent-reported irritable parenting, educational activities in the home, parent mental health and child communication and social play skills. This study provides evidence of the potential effectiveness of music therapy for early intervention.

A general approach to control a positive buck-boost converter to achieve robustness against input voltage fluctuations and load changes

A Positive Buck-Boost converter is a known DC-DC converter which may be controlled to act as Buck or Boost converter with same polarity of the input voltage. This converter has four switching states which include all the switching states of the above mentioned DC-DC converters. In addition there is one switching state which provides a degree of freedom for the positive Buck-Boost converter in comparison to the Buck, Boost, and inverting Buck-Boost converters. In other words the Positive Buck-Boost Converter shows a higher level of flexibility for its inductor current control compared to the other DC-DC converters. In this paper this extra degree of freedom is utilised to increase the robustness against input voltage fluctuations and load changes. To address this capacity of the positive Buck-Boost converter, two different control strategies are proposed which control the inductor current and output voltage against any fluctuations in input voltage and load changes. Mathematical analysis for dynamic and steady state conditions are presented in this paper and simulation results verify the proposed method.

A new pulsed power supply topology based on positive buck-boost converters concept

Improving efficiency and flexibility in pulsed power supply technologies is the most substantial concern of pulsed power systems specifically with regard to plasma generation. Recently, the improvement of pulsed power supply has become of greater concern due to the extension of pulsed power applications to environmental and industrial areas. With this respect, a current source based topology is proposed in this paper as a pulsed power supply which gives the possibility of power flow control during load supplying mode. The main contribution in this configuration is utilization of low-medium voltage semiconductor switches for high voltage generation. A number of switch-diode-capacitor units are designated at the output of topology to exchange the current source energy into voltage form and generate a pulsed power with sufficient voltage magnitude and stress. Simulations carried out in Matlab/SIMULINK platform as well as experimental tests on a prototype setup have verified the capability of this topology in performing desired duties. Being efficient and flexible are the main advantages of this topology.

Positive relationships among collaboration for innovation, past innovation abandonment and future product introduction in manufacturing SMEs

The importance of collaboration for firm level innovation has been well established but much of the research focuses on large firms, with little research on small and medium enterprises. This paper investigates the links between product innovation and external collaboration and between future product innovation and past abandonment in small and medium sized firms, analysing data from 449 manufacturing firms, collected through the Australian Business Longitudinal Database. Our findings indicate firms that sought ideas or solutions from external network such as suppliers, or business partners reported higher level of new product introduction than firms that did not have any external collaboration. Further, firms with past abandonment experiences reported higher levels of new product introduction than firms that did not have such experience. Additionally, the findings indicated that firms with external collaboration were more likely to introduce new products even if they had previously experienced abandonment of a product innovation than firms without external collaboration. Implications, limitations and future research are outlined.

Positive design and construction of mechanisms for the sustainable development in microenterprises in Africa

Enterprise development and its contribution to societal and economic outcomes are well known. However, limited research into microenterprises and the practices of microfinance and microcredit in developing countries has been carried out. This chapter presents the findings of research based on six years of engagement with the microentrepreneurs of Beira in Mozambique and suggests a model for responsible and sustainable support for enterprise development in developing economies. Building on semistructured interviews, observation, and participatory action research, this research project articulates a new approach supportive of enterprise development, as a process of cocreation with local people and based on sustainability principles. These findings are part of a longitudinal study of the successes and failures of small enterprises and their impact on social and economic activity.

Positive correlation of male and female Bactrocera tryoni (Froggatt) (Diptera: Tephritidae) catches in orange-ammonia traps

Populations of the Queensland fruit fly, Bactrocera tryoni, are routinely monitored using cue-lure, a male-only attractant. Such monitoring provides no information about females and there is little information available to show if male and female B. tryoni numbers are correlated in the field. Using a data set of 1 148 weekly clearances of orange-ammonia baited traps, which catch both males and females, the correlation between male and female numbers was tested for 48 weeks of the year (four weeks each month) and for the combined data set. Weekly male and female trap catches were almost entirely highly correlated, regardless of mean population size or time of year. For the whole year, the correlation between male and female numbers was r = 0.722, significant at p<0.001. Results suggest that changes in the number if male B. tryoni, as detected through cue-lure sampling, will reflect changes in numbers of female B. tryoni.

How to tell the difference between a positive image and a stereotype: reading Priscilla, Queen of the Desert

This paper intervenes in critical discussions about the representation of homosexuality. Rejecting the ‘manifest content’ of films, it turns to cultural history to map those public discourses which close down the ways in which films can be discussed. With relation to The Adventures of Priscilla, Queen of the Desert, it examines discussions of the film in Australian newspapers (both queer and mainstream) and finds that while there is disagreement about the interpretation to be made of the film, the terms within which those interpretations can be made are quite rigid. A matrix based on similarity, difference and value provides a series of positions and a vocabulary (transgression, assimilation, positive images and stereotypes) through which to make sense of this film. The article suggests that this matrix, and the idea that similarity and difference provide a suitable axis for making sense of homosexual identity, are problematic in discussing homosexual representation.

Mining positive and negative patterns for relevance feature discovery

It is a big challenge to guarantee the quality of discovered relevance features in text documents for describing user preferences because of the large number of terms, patterns, and noise. Most existing popular text mining and classification methods have adopted term-based approaches. However, they have all suffered from the problems of polysemy and synonymy. Over the years, people have often held the hypothesis that pattern-based methods should perform better than term-based ones in describing user preferences, but many experiments do not support this hypothesis. The innovative technique presented in paper makes a breakthrough for this difficulty. This technique discovers both positive and negative patterns in text documents as higher level features in order to accurately weight low-level features (terms) based on their specificity and their distributions in the higher level features. Substantial experiments using this technique on Reuters Corpus Volume 1 and TREC topics show that the proposed approach significantly outperforms both the state-of-the-art term-based methods underpinned by Okapi BM25, Rocchio or Support Vector Machine and pattern based methods on precision, recall and F measures.