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Numerical Solutions of Fractional Differential Equations by ExtrapolationAn extrapolation algorithm is considered for solving linear fractional differential equations in this paper, which is based on the direct discretization of the fractional differential operator. Numerical results show that the approximate solutions of this numerical method has the expected asymptotic expansions.

Numerical treatment of oscillary functional differential equationsThis preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear vmethods will also be oscillatory. We conclude with some general theory

Numerical treatment of oscillatory delay and mixed functional differential equations arising in modellingThe pervading theme of this thesis is the development of insights that contribute to the understanding of whether certain classes of functional differential equation have solutions that are all oscillatory. The starting point for the work is the analysis of simple (linear autonomous) ordinary differential equations where existing results allow a full explanation of the phenomena. The Laplace transform features as a key tool in developing a theoretical background. The thesis goes on to explore the corresponding theory for delay equations, advanced equations and functional di erential equations of mixed type. The focus is on understanding the links between the characteristic roots of the underlying equation, and the presence or otherwise of oscillatory solutions. The linear methods are used as a class of numerical schemes which lead to discrete problems analogous to each of the classes of functional differential equation under consideration. The thesis goes on to discuss the insights that can be obtained for discrete problems in their own right, and then considers those new insights that can be obtained about the underlying continuous problem from analysis of the oscillatory behaviour of the analogous discrete problem. The main conclusions of the work are some semiautomated computational approaches (based upon the Principle of the Argument) which allow the prediction of oscillatory solutions to be made. Examples of the effectiveness of the approach are provided, and there is some discussion of its theoretical basis. The thesis concludes with some observations about further work and some of the limitations of existing analytical insights which restrict the reliability with which the approach developed can be applied to wider classes of problem.

Obesity and the Dysregulation of Fatty Acid Metabolism: Implications for Healthy AgingThe population of the world is aging. In 2010, an estimated 524 million people were aged 65 years or older presenting eight percent of the global population. By 2050, this number is expected to nearly triple to approximately 1.5 billion, 16 percent of the world’s population. Although people are living longer, the quality of their lives are often compromised due to illhealth. Areas covered. Of the conditions which compromise health as we age, obesity is at the forefront. Over half of the global older population were overweight or obese in 2010, significantly increasing the risk of a range of metabolic diseases. Although, it is well recognised excessive calorie intake is a fundamental driver of adipose tissue dysfunction, the relationship between obesity; intrinsic aging; and fat metabolism is less understood. In this review we discuss the intersection between obesity, aging and the factors which contribute to the dysregulation of wholebody fat metabolism. Expert Commentary. Being obese disrupts an array of physiological systems and there is significant crosstalk among these. Moreover it is imperative to acknowledge the contribution intrinsic aging makes to the dysregulation of these systems and the onset of disease.

On a degenerate nonlocal parabolic problem describing infinite dimensional replicator dynamicsWe establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u \Delta u + u \int_\Omega \nabla u^2 \] in bounded domains $\Om\sub\R^n$ which arises in game theory. We prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blowup in finite time if the initial mass is large. In particular, it is shown that in this case the blowup set coincides with $\overline{\Omega}$, i.e. the finitetime blowup is global.

On hereditary reducibility of 2monomial matrices over commutative ringsA 2monomial matrix over a commutative ring $R$ is by definition any matrix of the form $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0&tI_{nk}\end{smallmatrix}\right)$, $0<k<n$, where $t$ is a noninvertible element of $R$, $\Phi$ the compa\nion matrix to $\lambda^n1$ and $I_k$ the identity $k\times k$matrix. In this paper we introduce the notion of hereditary reducibility (for these matrices) and indicate one general condition of the introduced reducibility.

On some aspects of casual and neutral equations used in mathematical modellingThe problem that motivates the considerations here is the construction of mathematical models of natural phenomena that depend upon past states. The paper divides naturally into two parts: in the first, we expound the interconnection between ordinary differential equations, delay differential equations, neutral delaydifferential equations and integral equations (with emphasis on certain linear cases). As we show, this leads to a natural hierarchy of model complexity when such equations are used in mathematical and computational modelling, and to the possibility of reformulating problems either to facilitate their numerical solution or to provide mathematical insight, or both. Volterra integral equations include as special cases the others we consider. In the second part, we develop some practical and theoretical consequences of results given in the first part. In particular, we consider various approaches to the definition of an adjoint, we establish (notably, in the context of sensitivity analysis for neutral delaydifferential equations) roles for welldefined adjoints and ‘quasiadjoints’, and we explore relationships between sensitivity analysis, the variation of parameters formulae, the fundamental solution and adjoints.

On the behavior of the solutions for linear autonomous mixed type difference equationA class of linear autonomous mixed type difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the corresponding characteristic equation.

On the decay of the elements of inverse triangular Toeplitz matricesWe consider half–infinite triangular Toeplitz matrices with slow decay of the elements and prove under a monotonicity condition that the elements of the inverse matrix, as well as the elements of the fundamental matrix, decay to zero. We provide a quantitative description of the decay of the fundamental matrix in terms of p–norms. The results add to the classical results of Jaffard and Vecchio, and are illustrated by numerical examples.

On the Dirichlet to Neumann Problem for the 1dimensional Cubic NLS Equation on the halflineInitialboundary value problems for 1dimensional `completely integrable' equations can be solved via an extension of the inverse scattering method, which is due to Fokas and his collaborators. A crucial feature of this method is that it requires the values of more boundary data than given for a wellposed problem. In the case of cubic NLS, knowledge of the Dirichet data su ces to make the problem wellposed but the Fokas method also requires knowledge of the values of Neumann data. The study of the Dirichlet to Neumann map is thus necessary before the application of the `Fokas transform'. In this paper, we provide a rigorous study of this map for a large class of decaying Dirichlet data. We show that the Neumann data are also su ciently decaying and that, hence, the Fokas method can be applied.

On the dynamics of a nonlocal parabolic equation arising from the GiererMeinhardt systemThe purpose of the current paper is to contribute to the comprehension of the dynamics of the shadow system of an activatorinhibitor system known as a GiererMeinhardt model. Shadow systems are intended to work as an intermediate step between single equations and reactiondiffusion systems. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the GiererMeinhardt model is reduced to a single though nonlocal equation whose dynamics will be investigated. We mainly focus on the derivation of blowup results for this nonlocal equation which can be seen as instability patterns of the shadow system. In particular, a {\it diffusion driven instability (DDI)}, or {\it Turing instability}, in the neighbourhood of a constant stationary solution, which it is destabilised via diffusiondriven blowup, is obtained. The latter actually indicates the formation of some unstable patterns, whilst some stability results of globalintime solutions towards nonconstant steady states guarantee the occurrence of some stable patterns.

On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS controlWe consider a nonlocal parabolic model for a microelectromechanical system. Specifically, for a radially symmetric problem with monotonic initial data, it is shown that the solution quenches, so that touchdown occurs in the device, in a situation where there is no steady state. It is also shown that quenching occurs at a single point and a bound on the approach to touchdown is obtained. Numerical simulations illustrating the results are given.

On the study of oil paint adhesion on optically transparent glass: Conservation of reverse paintings on glassReverse painting on glass is a technique which consists of applying a cold paint layer on the reverseside of glass. The main challenge facing these artworks is the fragile adhesion of the pictorial layer – a simple movement can modify the appearance of the painting. This paper details a study into the adhesion parameters of pigments on glass and the comparison between different pigments. The relationships between the binder (linseed oil) with pigments and the glass with or without the use of an adhesive are studied. Physical analyses by surface characterisation have been carried out to better understand the influence of the pigment. The use of a sessile drop device, optical microscopy, scanning electron microscopy (SEM), a surface 3D profiler and a pencil hardness scratch tester were necessary to establish a comparison of the pictorial layer adhesion. A comparison of the effect of two adhesives; namely ox gall and gum arabic, has shown that the adhesion is not only linked to the physical parameters but that possible chemical reactions can influence the results. Finally, a treatment based on humidityextreme storage has shown the weakness of some pictorial layers.

Online conductivity calibration methods for EIT gas/oil in water flow measurementElectrical Impedance Tomography (EIT) is a fast imaging technique displaying the electrical conductivity contrast of multiphase flow. It is increasingly utilised for industrial process measurement and control. In principle, EIT has to obtain the prior information of homogenous continuous phase in terms of conductivity as a reference benchmark. This reference significantly influences the quality of subsequent multiphase flow measurement. During dynamic industrial process, the conductivity of continuous phase varies due to the effects from the changes of ambient and fluid temperature, ionic concentration, and internal energy conversion in fluid. It is not practical to stop industrial process frequently and measure the conductivity of continuous phase for taking the EIT reference. If without monitoring conductivity of continuous phase, EIT cannot present accurate and useful measurement results. To online calibrate the electrical conductivity of continuous phase and eliminate drift error of EIT measurement, two methods are discussed in this paper. Based on the linear approximation between fluid temperature and conductivity, the first method monitors fluid temperature and indirectly calibrates conductivity. In the second method, a novel conductivity cell is designed. It consists of a gravitational separation chamber with refreshing bypass and grounded shielding plate. The conductivity of continuous phase is directly sensed by the conductivity cell and fed to EIT system for online calibration. Both static and dynamic experiments were conducted to demonstrate the function and accuracy the conductivity cell.

Optimal convergence rates for semidiscrete finite element approximations of linear spacefractional partial differential equations under minimal regularity assumptionsWe consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear spacefractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear spacefractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear spacefractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.