Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

IL-18 inhibits growth of murine orthotopic prostate carcinomas via both adaptive and innate immune mechanisms

Interleukin(IL)-18 is a pleiotrophic cytokine with functions in immune modulation, angiogenesis and bone metabolism. In this study, the potential of IL-18 as an immunotherapy for prostate cancer (PCa) was examined using the murine model of prostate carcinoma, RM1 and a bone metastatic variant RM1(BM)/B4H7-luc. RM1 and RM1(BM)/B4H7-luc cells were stably transfected to express bioactive IL-18. These cells were implanted into syngeneic immunocompetent mice, with or without an IL-18-neutralising antibody (αIL-18, SK113AE4). IL-18 significantly inhibited the growth of both subcutaneous and orthotopic RM1 tumors and the IL-18 neutralizing antibody abrogated the tumor growth-inhibition. In vivo neutralization of interferon-gamma (IFN-γ) completely eliminated the anti-tumor effects of IL-18 confirming an essential role of IFN-γ as a down-stream mediator of the anti-tumor activity of IL-18. Tumors from mice in which IL-18 and/or IFN-γ was neutralized contained significantly fewer CD4+ and CD8+ T cells than those with functional IL-18. The essential role of adaptive immunity was demonstrated as tumors grew more rapidly in RAG1−/− mice or in mice depleted of CD4+ and/or CD8+ cells than in normal mice. The tumors in RAG1−/− mice were also significantly smaller when IL-18 was present, indicating that innate immune mechanisms are involved. IL-18 also induced an increase in tumor infiltration of macrophages and neutrophils but not NK cells. In other experiments, direct injection of recombinant IL-18 into established tumors also inhibited tumor growth, which was associated with an increase in intratumoral macrophages, but not T cells. These results suggest that local IL-18 in the tumor environment can significantly potentiate anti-tumor immunity in the prostate and clearly demonstrate that this effect is mediated by innate and adaptive immune mechanisms.

Four years on – are the Gazelles still running? A longitudinal study of firm performance after a period of rapid growth

Gazelles, or very rapidly growing firms, are important because they contribute disproportionately to economic growth. There is a concern that some of these firms pursue growth too aggressively resulting in lower subsequent performance. We investigate the relationship between growth and subsequent profitability for gazelle firms, and how this is moderated by firm strategy. Previous empirical research regarding the growth-profitability relationship for firms in general is rather inconclusive, with only one study specifically investigating gazelle firms. Likewise, there are theoretical arguments both for and against growth leading to profitability that equally apply to gazelle firms. Further, while contingency theory might suggest the relationship depends on the firm’s strategy, earlier studies have not investigated this relationship. We address these questions using longitudinal data (seven years) for a sample of 964 Danish Gazelle firms. Our study finds a clear positive relationship between growth and subsequent profitability among gazelle firms. Moreover, this relationship is stronger for firms pursuing a broad market strategy rather than a focus or niche strategy. An important managerial implication is that the growth strategy should be clearly integrated with the general strategic orientation of the firm.

Behavioural changes add validity to the construct of posttraumatic growth

The research aimed to identify positive behavioural changes that people may make as a result of negotiating the aftermath of a traumatic experience, thereby extending the current cognitive model of posttraumatic growth (PTG). It was hypothesised that significant others would corroborate survivor’s cognitive and behavioural reports of PTG. The sample comprised 176 participants; 88 trauma survivors and 88 significant others. University students accounted for 64% of the sample and 36% were from the broader community. Approximately one third were male. All participants completed the Posttraumatic Growth Inventory [PTGI] and open ended questions regarding behavioural changes. PTGI scores in the survivor sample were corroborated by the significant others with only the Appreciation of Life factor of the PTGI differing between the two groups (e.g., total PTGI scores between groups explained 33.64% of variance). Nearly all of the survivors also reported positive changes in their behaviour and these changes were also corroborated by the significant others. Results provide validation of the posttraumatic growth construct and the PTGI as an instrument of measurement. Findings may also influence therapeutic practice for example, the potential usefulness of corroborating others.

A qualitative approach to assessing the validity of the posttraumatic growth inventory

The Posttraumatic Growth Inventory (PTGI; Tedeschi & Calhoun, 1996) is the most commonly used measure of positive psychological change that can result from negotiating a traumatic experience. Whilst the PTGI has strong internal reliability, validity studies are still sparse. The presented research details trauma survivors’ understanding of items comprising the PTGI in order to qualitatively assess content validity. Participants were 14 trauma survivors who completed the PTGI and participated in a semi-structured interview. Thematic Analysis was conducted on participant’s transcribed interviews. One latent theme was identified reflecting that questions were consistently understood. A relationship was found between the constituent themes identified and the five factors of the PTGI. Participants answered the PTGI statements in a way that is consistent with the purpose of the instrument with only a small discrepancy found when some participants used the PTGI scale to indicate when a decrease in an element of the inventory had been experienced. Overall results supported the content validity of the PTGI.

Psychosocial experience of cancer : surpassing mere survival and recognising the potential for posttraumatic growth as well as distress.

The publication of the fourth edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-IV; American Psychiatric Association, 1994) introduced the notion that a life-threatening illness can be a stressor and catalyst for Posttraumatic Stress Disorder (PTSD). Since then a solid body of research has been established investigating the post-diagnosis experience of cancer. These studies have identified a number of short and long-term life changes resulting from a diagnosis of cancer and associated treatments. In this chapter, we discuss the psychosocial response to the cancer experience and the potential for cancer-related distress. Cancer can represent a life-threatening diagnosis that may be associated with aggressive treatments and result in physical and psychological changes. The potential for future trauma through the lasting effects of the disease and treatment, and the possibility of recurrence, can be a source of continued psychological distress. In addition to the documented adverse repercussions of cancer, we also outline the recent shift that has occurred in the psycho-oncology literature regarding positive life change or posttraumatic growth that is commonly reported after a diagnosis of cancer. Adopting a salutogenic framework acknowledges that the cancer experience is a dynamic psychosocial process with both negative and positive repercussions. Next, we describe the situational and individual factors that are associated with posttraumatic growth and the types of positive life change that are prevalent in this context. Finally, we discuss the implications of this research in a therapeutic context and the directions of future posttraumatic growth research with cancer survivors. This chapter will present both quantitative and qualitative research that indicates the potential for personal growth from adversity rather than just mere survival and return to pre-diagnosis functioning. It is important to emphasise however, that the presence of growth and prevalence of resilience does not negate the extremely distressing nature of a cancer diagnosis for the patient and their families and the suffering that can accompany treatment regimes. Indeed, it will be explained that for growth to occur, the experience must be one that quite literally shatters previously held schemas in order to act as a catalyst for change.

Hormone-dependent bacterial growth persistence and biofilm formation - A pilot study investigating human follicular fluid collected during IVF cycles

Human follicular fluid, considered sterile, is aspirated as part of an in vitro fertilization (IVF) cycle. However, it is easily contaminated by the trans-vaginal collection route and little information exists in its potential to support the growth of microorganisms. The objectives of this study were to determine whether human follicular fluid can support bacterial growth over time, whether the steroid hormones estradiol and progesterone (present at high levels within follicular fluid) contribute to the in vitro growth of bacterial species, and whether species isolated from follicular fluid form biofilms. We found that bacteria in follicular fluid could persist for at least 28 weeks in vitro and that the steroid hormones stimulated the growth of some bacterial species, specifically Lactobacillus spp., Bifidobacterium spp. Streptococcus spp. and E. coli. Several species, Lactobacillus spp., Propionibacterium spp., and Streptococcus spp., formed biofilms when incubated in native follicular fluids in vitro (18/24, 75%). We conclude that bacteria aspirated along with follicular fluid during IVF cycles demonstrate a persistent pattern of growth. This discovery is important since it can offer a new avenue for investigation in infertile couples.

A comment on "Inherited Trust and Growth"

Algan and Cahuc (2010) argue that “inherited trust” is a key factor in explaining growth rates across countries. They derive a measure of inherited trust by linking respondents’ “home countries: in the United States General Social Survey (1972-2004) and the 2000 wave of the World Values Survey. Algan and Cahuc then estimate trust levels for people born before 1910 (inherited trust in 1935) and afterwards (inherited trust in 2000). They show a strong link between economic growth rates and inherited trust. We do not challenge this result, but we do argue that: (1) The 2000 World Values Survey has many anomalous results; (2) the estimates for inherited trust in 1935 are mostly based upon tiny samples for most ethnic heritage groups in the General Social Survey; and (3) Algan and Cahuc’s findings are based upon two-tailed rather than one-tailed tests. We reestimate their model using the more reliable waves of the World Values Survey and find much weaker relationships between inherited trust in 1935 and trust in the home country. We also suggest caution in the overall measure of inherited trust in 1935.

Mentoring as professional development: ‘growth for both’ mentor and mentee

Teachers need professional development to keep current with teaching practices, although costs for extensive professional development can be prohibitive across an education system. Mentoring provides one way for embedding cost-effective professional development. This mixed-method study includes surveying mentor teachers (n = 101) on a five-part Likert scale and interviews with experienced mentors (n = 10) to investigate professional development for mentors as a result of the mentoring process. Quantitative data were analysed through a pedagogical knowledge framework and qualitative data were collated into themes. Survey data showed that although mentoring of pedagogical knowledge was variable, mentoring pedagogical knowledge practices occurs with the majority of mentors, which requires mentors to evaluate and articulate teaching practices. Qualitative data showed that mentoring acted as professional development and led towards enhancing communication skills, developing leadership roles (problem-solving and building capacity) and advancing pedagogical knowledge. Providing professional development to teachers on mentoring can help to build capacity in two ways: quality mentoring of preservice teachers through explicit mentoring practices, and reflecting and deconstructing teaching practices for mentors’ own pedagogical advancements.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation

The method of lines is a standard method for advancing the solution of partial differential equations (PDEs) in time. In one sense, the method applies equally well to space-fractional PDEs as it does to integer-order PDEs. However, there is a significant challenge when solving space-fractional PDEs in this way, owing to the non-local nature of the fractional derivatives. Each equation in the resulting semi-discrete system involves contributions from every spatial node in the domain. This has important consequences for the efficiency of the numerical solver, especially when the system is large. First, the Jacobian matrix of the system is dense, and hence methods that avoid the need to form and factorise this matrix are preferred. Second, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. In this paper, we show how an effective preconditioner is essential for improving the efficiency of the method of lines for solving a quite general two-sided, nonlinear space-fractional diffusion equation. A key contribution is to show, how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Organic and acquisitive growth : re-examining, testing and extending Penrose’s growth theory

The Theory of the Growth of The Firm by Edith Penrose, first published in 1959, is a seminal contribution to the field of management. Penrose's intention was to create a theory of firm growth which was logically consistent and empirically tractable (Buckley and Casson, 2007). Much attention, however, has been focused on her unintended contribution to the resource-based view (henceforth RBV) (e.g. Kor and Mahoney, 2004; Lockett and Thompson, 2004) rather than her firm growth theory. We feel that this is unfortunate because despite a rapidly growing body of empirical work, conceptual advancement in growth studies has been limited (Davidsson and Wiklund, 2000; Davidsson et ai., 2006; Delmar, 1997; Storey, 1994). The growth literature frequently references Penrose's work, but little explicit testing of her ideas has been undertaken. This is surprising given that Penrose's work remains the most comprehensive theory of growth to date. One explanation is that she did not formality present her arguments, favouring verbal exposition over formalized models (Lockett, 2005; Lockett and Thompson, 2004). However, the central propositions and conclusions of her theory can be operationalized and empirically tested.

Layered silicates in Chondritic Porous Aggregate W7029*A : a case of primary growth

An important subset of extraterrestrial particles that reach the Earth's stratosphere include the so-called Chondritic Porous Aggregates (CPA's) [1-3]. In general, CPA's have a fluffy morphology and consist of numerous (>104)subparticles that are often <100A in size [4]. Mineral species in CPA's include Mg-rich pyroxene and olivine, Fe- and (Fe,Ni)-sulphides, taenite, Fe,Ni-carbides, magnetite, Ti-metal, a Bi-phase (metal or oxide), and variable amounts of carbonaceous material [1, 5-7]. Hydrated silicates are rare in CPA's and are limited to aggregates that have not been severely altered (thermo-metamorphosed) during atmospheric entry [8]. The presence of hydrated silicates in one cosmic dust particle was established by X-ray diffraction [2] and has been inferred in others by infra-red spectroscopy [8]. If CPA's are cometary, their mineralogy and morphology suggest that at least two episodes of aggregation occurred and that variations in porosity may be related to local differences in ice-to-dust ratio [3].

Growth of C36-films on diamond surface through molecular dynamics simulation

In this paper, the initial stage of films assembled by energetic C36 fullerenes on diamond (001)–(2 × 1) surface at low-temperature was investigated by molecular dynamics simulation using the Brenner potential. The incident energy was first uniformly distributed within an energy interval 20–50 eV, which was known to be the optimum energy range for chemisorption of single C36 on diamond (001) surface. More than one hundred C36 cages were impacted one after the other onto the diamond surface by randomly selecting their orientation as well as the impact position relative to the surface. The growth of films was found to be in three-dimensional island mode, where the deposited C36 acted as building blocks. The study of film morphology shows that it retains the structure of a free C36 cage, which is consistent with Low Energy Cluster Beam Deposition (LECBD) experiments. The adlayer is composed of many C36-monomers as well as the covalently bonded C36 dimers and trimers which is quite different from that of C20 fullerene-assembled film, where a big polymerlike chain was observed due to the stronger interaction between C20 cages. In addition, the chemisorption probability of C36 fullerenes is decreased with increasing coverage because the interaction between these clusters is weaker than that between the cluster and the surface. When the incident energy is increased to 40–65 eV, the chemisorption probability is found to increased and more dimers and trimers as well as polymerlike-C36 were observed on the deposited films. Furthermore, C36 film also showed high thermal stability even when the temperature was raised to 1500 K.

Characterising one-to-one conservatoire teaching : Some implications of a quantitative analysis

Despite the significant recent growth in research relating to instrumental, vocal and composition tuition in higher education, little is known about the diversity of approaches that characterise one-to-one teaching in the Conservatoire, and what counts as optimal practice for educating 21st-century musicians. Through analysis of video-recorded one-to-one lessons that draws on a ‘bottom up’ methodology for characterising pedagogical practices (Taylor, 2012; Taylor et al, 2012), this paper provides empirical evidence about the nature of one-to-one pedagogy in one Australian institution. The research aims (1) to enable a better understanding of current one-to-one conservatoire teaching; and (2) to build and improve upon existing teaching practice using authentic insights gained through systematic investigation. The authors hope the research will lead to a better understanding of the diversity and efficacy of the pedagogical practice within the specific context in which the study was conducted, and beyond, to Conservatoire pedagogy generally.