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.

Heart versus mind : the functions of emotional and cognitive loyalty

While there is substantial research on attitudinal and behavioral loyalty, the deconstruction of attitudinal loyalty into its two key components – emotional and cognitive loyalty – has been largely ignored. Despite the existence of managerial strategies aimed at increasing each of these two components, there is little academic research to support these managerial efforts. This paper seeks to advance the understanding of emotional and cognitive brand loyalty by examining the psychological function that these dimensions of brand loyalty perform for the consumer. We employ Katz’s (1960) four functions of attitudes (utilitarian, knowledge, value-expression, ego-defence) to investigate this question. Surveys using a convenience sample were completed by 268 consumers in two metropolitan cities on a variety of goods, services and durable products. The relationship between the functions and dimensions of loyalty were examined using MANOVA. The results show that both the utilitarian and knowledge functions of loyalty are significantly positively related to cognitive loyalty while the ego-defensive function of loyalty is significantly positively related to emotional loyalty. The results for the value-expressive function of loyalty were nonsignificant.

The old face of 'new' social networks : the republic of letters as a virtual community

This paper explores the idea of virtual participation through the historical example of the republic of letters in early modern Europe (circa 1500-1800). By reflecting on the construction of virtuality in a historical context, and more specifically in a pre-digital environment, this paper calls attention to accusations of technological determinism in ongoing research concerning the affordances of the Internet and related media of communication. It argues that ‘the virtual’ is not synonymous with ‘the digital’ and suggests that, in order to articulate what is novel about modern technologies, we must first understand the social interactions underpinning the relationships which are facilitated through those technologies. By analysing the construction of virtuality in a pre-digital environment, this paper thus offers a baseline from which scholars might consider what is different about the modes of interaction and communication being engaged in via modern media.

Pre-digital Virtuality: Early Modern Scholars and the Republic of Letters

This chapter explores the idea of virtual participation through the historical example of the republic of letters in early modern Europe (circa 1500-1800). By reflecting on the construction of virtuality in a historical context, and more specifically in a pre-digital environment, it calls attention to accusations of technological determinism in ongoing research concerning the affordances of the Internet and related media of communication. It argues that ‘the virtual’ is not synonymous with ‘the digital’ and suggests that, in order to articulate what is novel about modern technologies, we must first understand the social interactions underpinning the relationships which are facilitated through those technologies. By analysing the construction of virtuality in a pre-digital environment, this chapter thus offers a baseline from which scholars might consider what is different about the modes of interaction and communication being engaged in via modern media.

Review of Collaboration and Interdisciplinarity in the Republic of Letters : Essays in Honour of Richard G. Maber, edited by Paul Scott.

This volume brings together a number of essays that seek to explore the nature of early modern scholarship, ostensibly with special regard to the themes of interdisciplinarity and collaboration. As one might expect, the essays thus cover a gamut of topics – political manoeuvring, philosophical debates, gift-giving and dramatic performance – and each study is important and useful in its own right. As a whole, however, this collection serves more as a starting point for an exploration of its themes, than as an authoritative overview of the subject at hand.

Principles of motor learning in ecological dynamics : a comment on functions of learning and the acquisition of motor skills (with reference to sport)

This paper provides a commentary on the contribution by Dr Chow who questioned whether the functions of learning are general across all categories of tasks or whether there are some task-particular aspects to the functions of learning in relation to task type. Specifically, they queried whether principles and practice for the acquisition of sport skills are different than what they are for musical, industrial, military and human factors skills. In this commentary we argue that ecological dynamics contains general principles of motor learning that can be instantiated in specific performance contexts to underpin learning design. In this proposal, we highlight the importance of conducting skill acquisition research in sport, rather than relying on empirical outcomes of research from a variety of different performance contexts. Here we discuss how task constraints of different performance contexts (sport, industry, military, music) provide different specific information sources that individuals use to couple their actions when performing and acquiring skills. We conclude by suggesting that his relationship between performance task constraints and learning processes might help explain the traditional emphasis on performance curves and performance outcomes to infer motor learning.

Laboratory measurement of hydraulic conductivity functions of two unsaturated sandy soils during drying and wetting processes

The importance of applying unsaturated soil mechanics to geotechnical engineering design has been well understood. However, the consumption of time and the necessity for a specific laboratory testing apparatus when measuring unsaturated soil properties have limited the application of unsaturated soil mechanics theories in practice. Although methods for predicting unsaturated soil properties have been developed, the verification of these methods for a wide range of soil types is required in order to increase the confidence of practicing engineers in using these methods. In this study, a new permeameter was developed to measure the hydraulic conductivity of unsaturated soils using the steady-state method and directly measured suction (negative pore-water pressure) values. The apparatus is instrumented with two tensiometers for the direct measurement of suction during the tests. The apparatus can be used to obtain the hydraulic conductivity function of sandy soil over a low suction range (0-10 kPa). Firstly, the repeatability of the unsaturated hydraulic conductivity measurement, using the new permeameter, was verified by conducting tests on two identical sandy soil specimens and obtaining similar results. The hydraulic conductivity functions of the two sandy soils were then measured during the drying and wetting processes of the soils. A significant hysteresis was observed when the hydraulic conductivity was plotted against the suction. However, the hysteresis effects were not apparent when the conductivity was plotted against the volumetric water content. Furthermore, the measured unsaturated hydraulic conductivity functions were compared with predictions using three different predictive methods that are widely incorporated into numerical software. The results suggest that these predictive methods are capable of capturing the measured behavior with reasonable agreement.

Identifying Core Functions of Asset Management

It is widely acknowledged that effective asset management requires an interdisciplinary approach, in which synergies should exist between traditional disciplines such as: accounting, engineering, finance, humanities, logistics, and information systems technologies. Asset management is also an important, yet complex business practice. Business process modelling is proposed as an approach to manage the complexity of asset management through the modelling of asset management processes. A sound foundation for the systematic application and analysis of business process modelling in asset management is, however, yet to be developed. Fundamentally, a business process consists of activities (termed functions), events/states, and control flow logic. As both events/states and control flow logic are somewhat dependent on the functions themselves, it is a logical step to first identify the functions within a process. This research addresses the current gap in knowledge by developing a method to identify functions common to various industry types (termed core functions). This lays the foundation to extract such functions, so as to identify both commonalities and variation points in asset management processes. This method describes the use of a manual text mining and a taxonomy approach. An example is presented.

Expression and in vitro functions of the ghrelin axis in endometrial cancer

Ghrelin is a peptide hormone produced in the stomach and a range of other tissues, where it has endocrine, paracrine and autocrine roles in both normal and disease states. Ghrelin has been shown to be an important growth factor for a number of tumours, including prostate and breast cancers. In this study, we examined the expression of the ghrelin axis (ghrelin and its receptor, the growth hormone secretagogue receptor, GHSR) in endometrial cancer. Ghrelin is expressed in a range of endometrial cancer tissues, while its cognate receptor, GHSR1a, is expressed in a small subset of normal and cancer tissues. Low to moderately invasive endometrial cancer cell lines were examined by RT-PCR and immunoblotting, demonstrating that ghrelin axis mRNA and protein expression correlate with differentiation status of Ishikawa, HEC1B and KLE endometrial cancer cell lines. Moreover, treatment with ghrelin potently stimulated cell proliferation and inhibited cell death. Taken together, these data indicate that ghrelin promotes the progression of endometrial cancer cells in vitro, and may contribute to endometrial cancer pathogenesis and represent a novel treatment target.

Homogeneous bent functions of degree n in 2n variables do not exist for n>3

We prove that homogeneous bent functions f:GF(2)^2n --> GF(2) of degree n do not exist for n>3. Consequently homogeneous bent functions must have degree 3.

Independent functions of yeast Pcf11p in pre-mRNA 3' end processing and in transcription termination

Pcf11p, an essential subunit of the yeast cleavage factor IA, is required for pre‐mRNA 3′ end processing, binds to the C‐terminal domain (CTD) of the largest subunit of RNA polymerase II (RNAP II) and is involved in transcription termination. We show that the conserved CTD interaction domain (CID) of Pcf11p is essential for cell viability. Interestingly, the CTD binding and 3′ end processing activities of Pcf11p can be functionally uncoupled from each other and provided by distinct Pcf11p fragments in trans. Impaired CTD binding did not affect the 3′ end processing activity of Pcf11p and a deficiency of Pcf11p in 3′ end processing did not prevent CTD binding. Transcriptional run‐on analysis with the CYC1 gene revealed that loss of cleavage activity did not correlate with a defect in transcription termination, whereas loss of CTD binding did. We conclude that Pcf11p is a bifunctional protein and that transcript cleavage is not an obligatory step prior to RNAP II termination.

Critical linguistics in the early years : exploring functions of language through sophisticated picture books and process drama strategies

In this chapter we describe a critical fairytales unit taught to 4.5 to 5.5 year olds in a context of intensifying pressure to raise literacy achievement. The unit was infused with lessons on reinterpreted fairytales followed by process drama activities built around a sophisticated picture book, Beware of the Bears (MacDonald, 2004). The latter entailed a text analytic approach to critical literacy derived from systemic functional linguistics (Halliday, 1978; Halliday & Matthiessen, 2004). This approach provides a way of analysing how words and discourse are used to represent the world in a particular way and shape reader relations with the author in a particular field (Janks, 2010).

Exploring the post-genomic world: Differing explanatory and manipulatory functions of post-genomic sciences

Richard Lewontin proposed that the ability of a scientific field to create a narrative for public understanding garners it social relevance. This article applies Lewontin's conceptual framework of the functions of science (manipulatory and explanatory) to compare and explain the current differences in perceived societal relevance of genetics/genomics and proteomics. We provide three examples to illustrate the social relevance and strong cultural narrative of genetics/genomics for which no counterpart exists for proteomics. We argue that the major difference between genetics/genomics and proteomics is that genomics has a strong explanatory function, due to the strong cultural narrative of heredity. Based on qualitative interviews and observations of proteomics conferences, we suggest that the nature of proteins, lack of public understanding, and theoretical complexity exacerbates this difference for proteomics. Lewontin's framework suggests that social scientists may find that omics sciences affect social relations in different ways than past analyses of genetics.

Interface control of surface photochemical reactivity in ultrathin epitaxial ferroelectric films

Asymmetrical electrical boundary conditions in (001)-oriented Pb(Zr 0.2TiO0.8)O3 (PZT) epitaxial ultrathin ferroelectric films are exploited to control surface photochemical reactivity determined by the sign of the surface polarization charge. It is shown that the preferential orientation of polarization in the as-grown PZT layer can be manipulated by choosing an appropriate type of bottom electrode material. PZT films deposited on the SrRuO3 electrodes exhibit preferential upward polarization (C) whilst the same films grown on the (La,Sr)CoO 3-electrodes are polarized downward (C-). Photochemical activity of the PZT surfaces with different surface polarization charges has been tested by studying deposition of silver nanoparticles from AgNO3 solution under UV irradiation. PZT surfaces with preferential C orientation possess a more active surface for metal reduction than their C- counterparts, evidenced by large differences in the concentration of deposited silver nanoparticles. This effect is attributed to band bending at the bottom interface which varies depending on the difference in work functions of PZT and electrode materials.

Examining the predictive value of combining the theory of planned behaviour and the volunteer functions inventory

The functions of the volunteer functions inventory were combined with the constructs of the theory of planned behaviour (i.e., attitudes, subjective norms, and perceived behavioural control) to establish whether a stronger, single explanatory model prevailed. Undertaken in the context of episodic, skilled volunteering by individuals who were retired or approaching retirement (N = 186), the research advances on prior studies which either examined the predictive capacity of each model independently or compared their explanatory value. Using hierarchical regression analysis, the functions of the volunteer functions inventory (when controlling for demographic variables) explained an additional 7.0% of variability in individuals’ willingness to volunteer over and above that accounted for by the theory of planned behaviour. Significant predictors in the final model included attitudes, subjective norms and perceived behavioural control from the theory of planned behaviour and the understanding function from the volunteer functions inventory. It is proposed that the items comprising the understanding function may represent a deeper psychological construct (e.g., self-actualisation) not accounted for by the theory of planned behaviour. The findings highlight the potential benefit of combining these two prominent models in terms of improving understanding of volunteerism and providing a single parsimonious model for raising rates of this important behaviour.