The aim of this research was to investigate students' understanding of the Fundamental Theorem of Calculus (FTC). The FTC was chosen as the basis of this research because it is one of the most important topics taught in calculus, establishing the link between the concepts of differentiation and integration. Data was collected from first year undergraduate students at the Federal University of Rio de Janeiro, in Brazil. The sample comprised students from three areas: mathematics, computer sciences and engineering. A pilot study was applied in 1994 and the main study in 1995 and 1996. Questionnaire, interviews based on responses to the questionnaire and computer-task based interviews were used. The data were analysed using both quantitative and qualitative methods. The results show that some of the students' obstacles to understanding the FTC are related to difficulties with the concepts of function, continuity, derivative and integral. The definitions of these concepts were not clear in their minds and they frequently made use of images that contained only partial aspects of the definitions or were based on some particular examples. This hampered the students when they met new examples that did not fit with their pre-formed images. It was also found that definitions and theorems were so fragmented in the students' minds that there was no way they could appreciate a proof, exemplified by the proof of the FTC. Their conceptions of proof reflected the fact that they were not used to thinking of proving as fundamental to generalising a proposition, and in examining a proof, it was difficult for them to see the central ideas behind it. These results are closely associated with students' habits: they tend not to pay attention to theoretical aspects, memorising algorithmic procedures without reflecting on their applicability.

This thesis presents a novel method for active cloaking problems involving finite defects embedded in thin elastic plates. We demonstrate the approach by considering several geometries and boundary conditions, which include circular geometries with clamped boundaries, circular shaped coated inclusions and a finite cluster of rigid pins. We use results from plate theory, including problems on scattering of flexural waves from canonical geometries and platonic crystals. Such systems are governed by the fourth-order biharmonic plate equation. The method employed involves surrounding an object with a configuration of active sources whose complex amplitudes are chosen such that the main contribution to the scattered wave is cancelled in the far field. The active sources are represented by the Green's function for the biharmonic operator, describing the response to a point force. This Green's function has the beneficial property of being non-singular at the origin and the approximate cloak leads to an analytical system of linear algebraic equations for calculating the amplitudes of the active sources, required for effective cloaking. We demonstrate that the method described is applicable to arbitrarily shaped scatterers; effective cloaking is presented for a scatterer with a smooth, clamped boundary. Cloaking is also demonstrated for a circular clamped inclusion subject to flexural waves generated by a remote point source. Studying this problem enables us to find a Green's function for the cloaking problem which can be used to cloak a discretely distributed time-harmonic load. The cloaking is shown to work well at lower frequency regimes, however, at higher frequency resonant regimes, rapid changes in scattering from the inclusion are observed in narrow frequency intervals. This makes the cloak vulnerable to detection using frequency swept probe beams; time lags can arise as the sources adapt to the alterations in frequency. We extend the approach to the case of a resonant inclusion, combining active and passive techniques to tame regions of rapid variation in scattering properties. We then apply the active cloaking method approach to the coated inclusion. Finally, we modify the method for cloaking finite clusters of pins, where the geometry of the cluster causes interactions between the evanescent terms of solutions to the biharmonic equation, leading to interesting scattering patterns. Our method of active cloaking in thin elastic plates is shown to work for scattering phenomena such as localisation, wave-trapping and neutrality. We analyse band diagrams and dispersion surfaces from the infinite structure to estimate frequencies at which these phenomena occur in the finite cluster, and demonstrate effective cloaking using a finite number of sources.

Tensor C*-categories are the result of work to recast the fundamental theory of operator algebras in the setting of category theory, in order to facilitate the study of higher-dimensional algebras that are expected to play an important role in a unified model of physics. Indeed, the application of category theory to mathematical physics is itself a highly active field of research. C*-categories are the analogue of C*-algebras in this context. They are defined as norm-closed self-adjoint subcategories of the category of Hilbert spaces and bounded linear operators between them. Much of the theory of C*-algebras and their invariants generalises to C*-categories. Often, when a C*-algebra is associated to a particular structure it is not completely natural because certain choices are involved in its definition. Using C*-categories instead can avoid such choices since the construction of the relevant C*-category amounts to choosing all suitable C*-algebras at once. In this thesis we introduce and study C*-categories for which the set of objects carries topological data, extending the present body of work, which exclusively considers C*-categories with discrete object sets. We provide a construction of K-theory for topological C*-categories, which will have applications in widening the scope of the Baum-Connes conjecture, in index theory, and in geometric quantisation. As examples of such applications, we construct the C*-categories of topological groupoids, extending the familiar constructions of Renault.

No description available.

The time required to execute real-world scientific computations is a major issue. A single simulation may last hours, days, or even weeks to reach a certain level of accuracy, despite running on large-scale parallel architectures. Strict time limits may often be imposed too - 60 minutes in the case of the UK Met Office to produce a forecast. In this thesis, it is demonstrated that by raising the level of abstraction, the performance of a class of numerical methods for solving partial differential equations is improvable with minimal user intervention or, in many circumstances, with no user intervention at all. The use of high level languages to express mathematical problems enables domain-specific optimization via compilers. These automated optimizations are proven to be effective in a variety of real-world applications and computational kernels. The focus is on numerical methods based on unstructured meshes, such as the finite element method. The loop nests for unstructured mesh traversal are often irregular (i.e., they perform non-affine memory accesses, such as A[B[i]]), which makes reordering transformations for data locality essentially impossible for low level compilers. Further, the computational kernels are often characterized by complex mathematical expressions, and manual optimization is simply not conceivable. We discuss algorithmic solutions to these problems and present tools for their automation. These tools - the COFFEE compiler and the SLOPE library - are currently in use in frameworks for solving partial differential equations.

We develop and apply computational and analytical techniques to study wave-like propagation and resonant effects in periodic and quasi-periodic systems. Two themes that unify the content herein are the guidance and confinement of energy using periodic structures, and the utility of asymptotic analysis to aid computation and produce results that lend physical insight to the problems in question. In the first research chapter, we develop the method of high-frequency homogenisation (HFH) for electromagnetic waves in dielectric media, and apply this to the example of a planar array of dielectric spheres. The theory conveniently describes a range of dynamic effects, including effectively anisotropic behaviour in certain frequency regimes. In the second research chapter, we apply the HFH method to a cylindrical Bragg fibre, and use this to set up an effective eigenvalue problem in which the quasi-periodic system representing the fibre cladding is represented by a single continuous Bessel-like equation. We compare the results with those of direct numerical simulations and discuss how the theory could be developed to aid the study of photonic crystal cavities or fibres. In the remaining chapters, we consider the complex resonances of structures with angular periodicity. We demonstrate the emergence of quasi-normal modes with high Q-factors for the Helmholtz equation in such domains, and explore some of their properties using multiple scale analysis. In the final two chapters, we focus on a particular subset of these domains, and using matched asymptotic expansions show that the Q-factors for certain solutions depend exponentially on the number of inclusions arranged in a circular ring. Finally, we extend this analysis to flexural waves in thin elastic plates, and discuss the possibility of structured-ring resonators based on these solutions.

In this thesis, we explore several topics in the theory of monoidal and skew monoidal categories. In Chapter 3, we give definitions of dual pairs in monoidal categories, skew monoidal categories, closed skew monoidal categories and closed monoidal categories. In the case of monoidal and closed monoidal categories, there are multiple well-known definitions of a dual pair. We generalise these definitions to skew monoidal and closed skew monoidal categories. In Chapter 4, we introduce semidirect products of skew monoidal categories. Semidirect products of groups are a well-known and well-studied algebraic construction. Semidirect products of monoids can be defined analogously. We provide a categorification of this construction, for semidirect products of skew monoidal categories. We then discuss semidirect products of monoidal, closed skew monoidal and closed monoidal categories, in each case providing sufficient conditions for the semidirect product of two skew monoidal categories with the given structure to inherit the structure itself. In Chapter 5, we prove a coherence theorem for monoidal adjunctions between closed monoidal categories, a fragment of Grothendieck's 'six operations' formalism.

The aim of this thesis is to derive adaptive methods for discontinuous Galerkin approximations for both elliptic and parabolic interface problems. The derivation of adaptive method, is usually based on a posteriori error estimates. To this end, we present a residual-type a posteriori error estimator for interior penalty discontinuous Galerkin (dG) methods for an elliptic interface problem involving possibly curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. Respective upper and lower bounds of the error in the respective dG-energy norm with respect to the estimator are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. Moreover, a contraction property for a standard adaptive algorithm utilising these a posteriori bounds, with a bulk refinement criterion is also shown, thereby proving that the a posteriori bounds can lead to a convergent adaptive algorithm subject to some mesh restrictions. This work is also concerned with the derivation of a new L1∞(L2)-norm a posteriori error bound for the fully discrete adaptive approximation for non-linear interface parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the interior penalty discontinuous Galerkin finite element method. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto [48], enabling us to use the a posteriori error estimators derived for elliptic interface models and to obtain optimal order in both L1∞(L2) and L1∞(L2) + L2(H¹) norms. The effectiveness of all the error estimators and the proposed algorithms is confirmed through a series of numerical experiments.

Exponential smoothing is one of the most popular forecasting methods in practice. It has been used and researched for more than half a century. It started as an ad-hoc forecasting method and developed to a family of state-space models. Still all exponential smoothing methods are based on time series decomposition and the usage of such components as "level", "trend", "seasonality" and "error". It is assumed that these components may vary from one time series to another and take different forms depending on data characteristics. This makes their definition arbitrary and in fact there is no single way of identifying these components. At the same time the introduction of different types of exponential smoothing components implies that a model selection procedure is needed. This means that a researcher needs to select an appropriate type of model out of 30 different types either manually or automatically for every time series analysed. However several recent studies show that an underlying statistical model may have a form completely different than the one assumed by specific exponential smoothing models. These modelling questions motivate our research. We propose a model without strictly defined "level", "trend" and "seasonality". The model greatly simplifies the selection procedure, distinguishing only between seasonal and non-seasonal time series. Although we call it "Complex Exponential Smoothing" (CES), due to the use of complex-valued functions, its usage simplifies the forecasting procedure. In this thesis we first discuss the main properties of CES and propose an underlying statistical model. We then extend it in order to take seasonality into account and conduct experiments on real data to compare its performance with several well-known univariate forecasting models. We proceed to discuss the parameters estimation for exponential smoothing and propose a "Trace Forecast Likelihood" function that allows estimating CES components more efficiently. Finally we show that Trace Forecast Likelihood has desirable statistical properties, is connected to shrinkage and is generally advisable to use with any univariate model.

This thesis, entitled Spectral Theory Using Linear Systems and Sampling from the Spectrum of Hill’s Equation is submitted by Caroline Brett, Master of Science for the degree of Doctor of Philosophy, September 2015. It uses linear systems to solve various problems connected with Hill’s equation, −f + qf = λf for q ∈ C2, real-valued and π-periodic. Introducing a new operator, Rx constructed from a linear system, (−A, B, C) allows us to solve Hill’s equation and the inverse spectral problem. We use Rx to construct a function, T(x, y) that satisfies a Gelfand–Levitan integral equation and then derive a PDE for T(x, y). Solving this PDE recovers q. Extending Hill’s work in [28], we show that there exist Hilbert–Schmidt operators, Rp and Rc analogous to Rx, such that the roots of their Carleman determinants are elements of the periodic spectrum of Hill’s equation. The latter half concerns sampling from entire functions in Paley–Wiener space. From the periodic spectrum of Hill’s equation we derive a sampling sequence, (tn)n∈Z. Whittaker, Kotel’nikov and Shannon give a sampling result for (n)n∈Z where samples occur at a constant rate. Samples taken from the periodic spectrum do not occur at a constant rate, nevertheless we provide analogous results for this case. From (tn)n∈Z we also construct Riesz bases for L2[0, π] and L2[−π, π], the Fourier transform space of PW(π). In L2[0, π] we construct the dual Riesz basis using linear systems. Furthermore, we show that the determinant of the Gram matrix associated with the Riesz basis is a Lipschitz continuous function of (tn)n∈Z. Finally, we look at an integral, Ia associated with Ramanujan and use it to create a basis for PW π2. We conclude with an evaluation of various determinants associated with Ia.