Spectral approximations to the fractional integral and derivative

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

A computationally effective alternating direction method for the space and time fractional Bloch–Torrey equation in 3-D

The space and time fractional Bloch–Torrey equation (ST-FBTE) has been used to study anomalous diffusion in the human brain. Numerical methods for solving ST-FBTE in three-dimensions are computationally demanding. In this paper, we propose a computationally effective fractional alternating direction method (FADM) to overcome this problem. We consider ST-FBTE on a finite domain where the time and space derivatives are replaced by the Caputo–Djrbashian and the sequential Riesz fractional derivatives, respectively. The stability and convergence properties of the FADM are discussed. Finally, some numerical results for ST-FBTE are given to confirm our theoretical findings.

Unnatural selection : curricular conversations in TIP 1962–2012

Our task is to consider the evolving perspectives around curriculum documented in the Theory Into Practice (TIP) corpus to date. The 50 years in question, 1962–2012, account for approximately half the history of mass institutionalized schooling. Over this time, the upper age of compulsory schooling has crept up, stretching the school curriculum's reach, purpose, and clientele. These years also span remarkable changes in the social fabric, challenging deep senses of the nature and shelf-life of knowledge, whose knowledge counts, what science can and cannot deliver, and the very purpose of education. The school curriculum is a key social site where these challenges have to be addressed in a very practical sense, through a design on the future implemented within the resources and politics of the present. The task's metaphor of ‘evolution’ may invoke a sense of gradual cumulative improvement, but equally connotes mutation, hybridization, extinction, survival of the fittest, and environmental pressures. Viewed in this way, curriculum theory and practice cannot be isolated and studied in laboratory conditions—there is nothing natural, neutral, or self-evident about what knowledge gets selected into the curriculum. Rather, the process of selection unfolds as a series of messy, politically contaminated, lived experiments; thus curriculum studies require field work in dynamic open systems. We subscribe to Raymond Williams' approach to social change, which he argues is not absolute and abrupt, one set of ideas neatly replacing the other. For Williams, newly emergent ideas have to compete against the dominant mindset and residual ideas “still active in the cultural process'” (Williams, 1977, p. 122). This means ongoing debates. For these reasons, we join Schubert (1992) in advocating “continuous reconceptualising of the flow of experience” (p. 238) by both researchers and practitioners.

Poly(2-oxazoline) hydrogels for controlled fibroblast attachment

Currently there is a lack of choice when selecting synthetic materials with the cell-instructive properties demanded by modern biomaterials. The purpose of this study was to investigate the attachment of cells onto hydrogels prepared from poly(2-oxazoline)s selectively-functionalized with cell adhesion motifs. A water-soluble macromer based on the microwave-assisted cationic ring-opening polymerization of 2-methyl-2-oxazoline and 2-(dec-9-enyl)-2-oxazoline was functionalized with the peptide CRGDSG or controls using thiol-ene photochemistry followed by facile crosslinking in the presence of a dithiol crosslinker. The growth of human fibroblasts on the hydrogel surfaces was dictated by the structure and amount of incorporated peptide. Controls without any peptide showed resistance to cellular attachment. The benignity of the crosslinking conditions was demonstrated by the incorporation of fibroblasts within the hydrogels to produce three-dimensional cell-polymer constructs.

Numerical simulation of anomalous diffusion with application to medical imaging

The first objective of this project is to develop new efficient numerical methods and supporting error and convergence analysis for solving fractional partial differential equations to study anomalous diffusion in biological tissue such as the human brain. The second objective is to develop a new efficient fractional differential-based approach for texture enhancement in image processing. The results of the thesis highlight that the fractional order analysis captured important features of nuclear magnetic resonance (NMR) relaxation and can be used to improve the quality of medical imaging.

A model of workplace safety incorporating worker interactions and simple interventions

Although there was substantial research into the occupational health and safety sector over the past forty years, this generally focused on statistical analyses of data related to costs and/or fatalities and injuries. There is a lack of mathematical modelling of the interactions between workers and the resulting safety dynamics of the workplace. There is also little work investigating the potential impact of different safety intervention programs prior to their implementation. In this article, we present a fundamental, differential equation-based model of workplace safety that treats worker safety habits similarly to an infectious disease in an epidemic model. Analytical results for the model, derived via phase plane and stability analysis, are discussed. The model is coupled with a model of a generic safety strategy aimed at minimising unsafe work habits, to produce an optimal control problem. The optimal control model is solved using the forward-backward sweep numerical scheme implemented in Matlab.

An Introduction to Constrained Control

An ubiquitous problem in control system design is that the system must operate subject to various constraints. Although the topic of constrained control has a long history in practice, there have been recent significant advances in the supporting theory. In this chapter, we give an introduction to constrained control. In particular, we describe contemporary work which shows that the constrained optimal control problem for discrete-time systems has an interesting geometric structure and a simple local solution. We also discuss issues associated with the output feedback solution to this class of problems, and the implication of these results in the closely related problem of anti-windup. As an application, we address the problem of rudder roll stabilization for ships.

Applications of Spreadsheets in Education : The Amazing Power of a Simple Tool

This e-book is devoted to the use of spreadsheets in the service of education in a broad spectrum of disciplines: science, mathematics, engineering, business, and general education. The effort is aimed at collecting the works of prominent researchers and educators that make use of spreadsheets as a means to communicate concepts with high educational value. The e-book brings some of the most recent applications of spreadsheets in education and research to the fore. To offer the reader a broad overview of the diversity of applications, carefully chosen articles from engineering (power systems and control), mathematics (calculus, differential equations, and probability), science (physics and chemistry), and education are provided. Some of these applications make use of Visual Basic for Applications (VBA), a versatile computer language that further expands the functionality of spreadsheets. The material included in this e-book should inspire readers to devise their own applications and enhance their teaching and/or learning experience.

Coffee for morning hunger pangs. An examination of coffee and caffeine on appetite, gastric emptying, and energy intake

Coffee is one of the most widely consumed beverages in the world and has a number of potential health benefits. Coffee may influence energy expenditure and energy intake, which in turn may affect body weight. However, the influence of coffee and its constituents – particularly caffeine – on appetite remains largely unexplored. The objective of this study was to examine the impact of coffee consumption (with and without caffeine) on appetite sensations, energy intake, gastric emptying, and plasma glucose between breakfast and lunch meals. In a double-blind, randomised crossover design. Participants (n = 12, 9 women; Mean ± SD age and BMI: 26.3 ± 6.3 y and 22.7 ± 2.2 kg•m−2) completed 4 trials: placebo (PLA), decaffeinated coffee (DECAF), caffeine (CAF), and caffeine with decaffeinated coffee (COF). Participants were given a standardised breakfast labelled with 13C-octanoic acid and 225 mL of treatment beverage and a capsule containing either caffeine or placebo. Two hours later, another 225 mL of the treatment beverage and capsule was administered. Four and a half hours after breakfast, participants were given access to an ad libitum meal for determination of energy intake. Between meals, participants provided exhaled breath samples for determination of gastric emptying; venous blood and appetite sensations. Energy intake was not significantly different between the trials (Means ± SD, p > 0.05; Placebo: 2118 ± 663 kJ; Decaf: 2128 ± 739 kJ; Caffeine: 2287 ± 649 kJ; Coffee: 2016 ± 750 kJ); Other than main effects of time (p < 0.05), no significant differences were detected for appetite sensations or plasma glucose between treatments (p > 0.05). Gastric emptying was not significantly different across trials (p > 0.05). No significant effects of decaffeinated coffee, caffeine or their combination were detected. However, the consumption of caffeine and/or coffee for regulation of energy balance over longer periods of time warrant further investigation.

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Texture enhancement is an important component of image processing that finds extensive application in science and engineering. The quality of medical images, quantified using the imaging texture, plays a significant role in the routine diagnosis performed by medical practitioners. Most image texture enhancement is performed using classical integral order differential mask operators. Recently, first order fractional differential operators were used to enhance images. Experimentation with these methods led to the conclusion that fractional differential operators not only maintain the low frequency contour features in the smooth areas of the image, but they also nonlinearly enhance edges and textures corresponding to high frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we apply the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other first order fractional differential operators, we find that our new algorithms provide higher signal to noise values and superior image quality.

The tensor distribution function

Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of six directions, second-order tensors (represented by three-by-three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high-angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues.