Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

A broadcast authentication and integrity augmentation for trusted differential GPS in marine navigation

In this paper we propose an efficient authentication and integrity scheme to support DGPS corrections using the RTCM protocol, such that the identified vulnerabilities in DGPS are mitigated. The proposed scheme is based on the TESLA broadcast protocol with modifications that make it suitable for the bandwidth and processor constrained environment of marine DGPS.

Organizational ambidexterity and corporate entrepreneurship : the differential effects on venturing, innovation and renewal processes

Most corporate entrepreneurship studies have focused on either innovation, venturing or strategic renewal making comparison between the antecedents of all three aspects of corporate entrepreneurship difficult. Moreover, studies on corporate entrepreneurship hardly address organizational antecedents, while simultaneously managing and organizing CE and mainstream activities has been seen as a major challenge for incumbent firms. Our findings show that organizational ambidexterity has strong and differential effects on venturing, innovation and renewal. We find, for example, that innovation is affected by horizontal integration, while strategic renewal is significantly influenced by integration on top management team level.

Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

The differential effects of organizational antecedents on dimensions of corporate entrepreneurship

This study seeks to further delineate how organizational antecedents differentially influence the three components of corporate entrepreneurship: innovation, venturing or strategic renewal. We argue that structural differentiation may help organizations to maintain multiple and often conflicting demands of entrepreneurial and mainstream activities. Taking a social capital perspective, our study further examines two contingencies in the form of informal integration mechanisms (i.e. connectedness and TMT social integration). Our findings show structural differentiation has a positive effect on all three components of corporate entrepreneurship, yet the effect is moderated by integration mechanisms. Interunit connectedness has a positive moderation effect regarding innovation and venturing, and TMT social integration has a negative moderation effect regarding strategic renewal. This reveals that innovation is influenced by informal integration mechanisms on the organizational level, strategic renewal on top management team level, while venturing is influenced by integration mechanisms on both levels.

Differential reporting and the effect on loan evaluations : an experimental study

This study utilises a mexed design laboratory experiment to test the impact of differential reporting on one group of external financial report users-lenders. The results indicate that the judgments of bank loan officers' assessment of the ability of a borrower to repay, are not significantly affected by differential reporting (in this case, presentation on non-GAAP financial reports compared to GAAP financial reports). However, bankers request additional information from borrowers when non-GAAP financial reports are presented.

Thermal transition properties of spagehetti measured by Differential Scanning Calorimetry (DSC) and Thermal Mechanical Compression Test (TMCT)

Glass transition temperature of spaghetti sample was measured by thermal and rheological methods as a function of water content.

Simultaneous determination of three synthetic glucocorticoids by differential pulse stripping voltammetry with the aid of chemometrics

A novel voltammetric method for simultaneous determination of the glucocorticoid residues prednisone, prednisolone, and dexamethasone was developed. All three compounds were reduced at a mercury electrode in a Britton-Robinson buffer (pH 3.78), and well-defined voltammetric waves were observed. However, the voltammograms of these three compounds overlapped seriously and showed nonlinear character, and thus, it was difficult to analyze the compounds individually in their mixtures. In this work, two chemometrics methods, principal component regression (PCR) and partial least squares (PLS), were applied to resolve the overlapped voltammograms, and the calibration models were established for simultaneous determination of these compounds. Under the optimum experimental conditions, the limits of detection (LOD) were 5.6, 8.3, and 16.8 µg l-1 for prednisone, prednisolone, and dexamethasone, respectively. The proposed method was also applied for the determination of these glucocorticoid residues in the rabbit plasma and human urine samples with satisfactory results.

A differential kinetic spectrophotometric method for determination of three sulphanilamide artificial sweeteners with the aid of chemometrics

A simple and sensitive spectrophotometric method for the simultaneous determination of acesulfame-K, sodium cyclamate and saccharin sodium sweeteners in foodstuff samples has been researched and developed. This analytical method relies on the different kinetic rates of the analytes in their oxidative reaction with KMnO4 to produce the green manganate product in an alkaline solution. As the kinetic rates of acesulfame-K, sodium cyclamate and saccharin sodium were similar and their kinetic data seriously overlapped, chemometrics methods, such as partial least squares (PLS), principal component regression (PCR) and classical least squares (CLS), were applied to resolve the kinetic data. The results showed that the PLS prediction model performed somewhat better. The proposed method was then applied for the determination of the three sweeteners in foodstuff samples, and the results compared well with those obtained by the reference HPLC method.

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.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Infrastructure sustainability : differential axial shortening of concrete structures

High density development has been seen as a contribution to sustainable development. However, a number of engineering issues play a crucial role in the sustainable construction of high rise buildings. Non linear deformation of concrete has an adverse impact on high-rise buildings with complex geometries, due to differential axial shortening. These adverse effects are caused by time dependent behaviour resulting in volume change known as ‘shrinkage’, ‘creep’ and ‘elastic’ deformation. These three phenomena govern the behaviour and performance of all concrete elements, during and after construction. Reinforcement content, variable concrete modulus, volume to surface area ratio of the elements, environmental conditions, and construction quality and sequence influence on the performance of concrete elements and differential axial shortening will occur in all structural systems. Its detrimental effects escalate with increasing height and non vertical load paths resulting from geometric complexity. The magnitude of these effects has a significant impact on building envelopes, building services, secondary systems, and lifetime serviceability and performance. Analytical and test procedures available to quantify the magnitude of these effects are limited to a very few parameters and are not adequately rigorous to capture the complexity of true time dependent material response. With this in mind, a research project has been undertaken to develop an accurate numerical procedure to quantify the differential axial shortening of structural elements. The procedure has been successfully applied to quantify the differential axial shortening of a high rise building, and the important capabilities available in the procedure have been discussed. A new practical concept, based on the variation of vibration characteristic of structure during and after construction and used to quantify the axial shortening and assess the performance of structure, is presented.

A numerical method to quantify differential axial shortening in concrete buildings

Differential distortion comprising axial shortening and consequent rotation in concrete buildings is caused by the time dependent effects of “shrinkage”, “creep” and “elastic” deformation. Reinforcement content, variable concrete modulus, volume to surface area ratio of elements and environmental conditions influence these distortions and their detrimental effects escalate with increasing height and geometric complexity of structure and non vertical load paths. Differential distortion has a significant impact on building envelopes, building services, secondary systems and the life time serviceability and performance of a building. Existing methods for quantifying these effects are unable to capture the complexity of such time dependent effects. This paper develops a numerical procedure that can accurately quantify the differential axial shortening that contributes significantly to total distortion in concrete buildings by taking into consideration (i) construction sequence and (ii) time varying values of Young’s Modulus of reinforced concrete and creep and shrinkage. Finite element techniques are used with time history analysis to simulate the response to staged construction. This procedure is discussed herein and illustrated through an example.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.