Gromov’s compactness theorem for metric spaces, a compactness theorem for the space of compact metric spaces equipped with the Gromov-Hausdorff distance, is a theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give a compactness theorem for the space of distance measure spaces equipped with the generalised Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple (X, d,µ), where (X, d) forms a distance space (a generalisation of a metric space where, we allow the distance between two points to be infinity) and µ is a finite Borel measure. Using this result we prove that the Deligne-Mumford compactification is the completion of the moduli space of Riemann surfaces under the generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics. While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and topological classification of “almost all" leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations, as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation, we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.

❆❜str❛❝t ❚❤❡ ❞✐♣❧♦♠❛ t❤❡s✐s ❞❡❛❧s ✇✐t❤ ♣♦ss✐❜❧❡ ♣❤✐❧♦s♦♣❤✐❝❛❧ ❛♥❛❧②s❡s ♦❢ ●ö✲ ❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠s ❛♥❞ t❤❡✐r ✐♥t❡r♣r❡t❛t✐♦♥s ✐♥ ❞✐✛❡r❡♥t ❜r❛♥❝❤❡s♦❢♣❤✐❧♦s♦♣❤②✭♣❤❡♥♦♠❡♥♦❧♦❣②✱❛♥❛❧②t✐❝❛❧♣❤✐❧♦s♦♣❤②♦❢♠✐♥❞✱ ❑❛♥t✬s ♣❤✐❧♦s♦♣❤②✮✳ P❛rt ♦❢ t❤❡ t❤❡s✐s ✐s ❞❡❞✐❝❛t❡❞ t♦ t❤❡ ❛tt✐t✉❞❡s t♦ ♠❛t❤❡♠❛t✐❝❛❧ ❞✐s❝✐♣❧✐♥❡s ❛♥❞ t❤❡✐r ❢✉♥❞❛♠❡♥t❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ❝❛✉s❡❞ ❜② r❡✈♦❧✉t✐♦♥❛r② ❞✐s❝♦✈❡r✐❡s s✉❝❤ ❛s ◆♦♥✲❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡t✲ r✐❡s ❛♥❞ ✐♥❝♦♠♣❧❡t♥❡ss t❤❡♦r❡♠s✳ ❚❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ s❡❝♦♥❞ ●ö❞❡❧✬s ✐♥❝♦♠♣❧❡t❡♥❡ss t❤❡♦r❡♠✱ ●❡♥t③❡♥✬s ❝♦♥s✐st❡♥❝② ♣r♦♦❢ ♦❢ P❡❛♥♦ ❛r✐t❤♠❡t✐❝ ❛♥❞ ❍✐❧❜❡rt✬s ♣r♦❣r❛♠♠❡ ✐s ❛❧s♦ ❞✐s❝✉ss❡❞✳

The aim of this thesis is to show that the use of heavy-tailed distributions in finance is theoretically unfounded and may cause significant misunderstandings and fallacies in model interpretation. The main reason seems to be a wrong understanding of the concept of the distributional tail. Also in models based on real data it seems more reasonable to concentrate on the central part of the distribution not tails. Powered by TCPDF (www.tcpdf.org)

It is well known that there exist domains Ω in Cn,n ≥ 2, such that all holomorphic functions in Ω continue analytically beyond the boundary. We wish to study this remarkable phenomenon. The first chapter seeks to motivate this theme by offering some well-known extension results on domains in Cn having many symmetries. One important result, in this regard, is Hartogs’ theorem on the extension of functions holomorphic in a certain neighbourhood of (D x {0} U (∂D x D), D being the open unit disc in C. To understand the nature of analytic continuation in greater detail, in Chapter 2, we make rigorous the notions of ‘extensions’ and ‘envelopes of holomorphy’ of a domain. For this, we use methods similar to those used in univariate complex analysis to construct the natural domains of definitions of functions like the logarithm. Further, to comprehend the geometry of these abstractly-defined extensions, we again try to deal with some explicit domains in Cn; but this time we allow our domains to have fewer symmetries. The subject of Chapter 3 is a folk result generalizing Hartogs’ theorem to the extension of functions holomorphic in a neighbourhood of S U (∂D x D), where S is the graph of a D-valued function Φ, continuous in D and holomorphic in D. This problem can be posed in higher dimensions and we give its proof in this generality. In Chapter 4, we study Chirka and Rosay’s proof of Chirka’s generalization (in C2) of the above-mentioned result. Here, Φ is merely a continuous function from D to itself. Chapter 5 — a departure from our theme of Hartogs-Chirka type of configurations — is a summary of the key ideas behind a ‘non-standard’ proof of the so-called Hartogs phenomenon (i.e., holomorphic functions in any connected neighbourhood of the boundary of a domain Ω Cn , n ≥ 2, extend to the whole of Ω). This proof, given by Merker and Porten, uses tools from Morse theory to tame the boundary geometry of Ω, hence making it possible to use analytic discs to achieve analytic continuation locally. We return to Chirka’s extension theorem, but this time in higher dimensions, in Chapter 6. We see one possible generalization (by Bharali) of this result involving Φ where is a subclass of C (D; Dn), n ≥ 2. Finally, in Chapter 7, we consider Hartogs-Chirka type configurations involving graphs of multifunctions given by “Weierstrass pseudopolynomials”. We will consider pseudopolynomials with coefficients belonging to two different subclasses of C(D; C), and show that functions holomorphic around these new configurations extend holomorphically to D2 .

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It is well known that mathematics students have to be able to understand and prove theorems. From our experience we know that engineering students should also be able to do the same, since a good theoretical knowledge of mathematics is essential for solving practical problems and constructing models. Proving theorems gives students a much better understanding of the subject, and helps them to develop mathematical thinking. The proof of a theorem consists of a logical chain of steps. Students should understand the need and the legitimacy of every step. Moreover, they have to comprehend the reasoning behind the order of the chain’s steps. For our research students were provided with proofs whose steps were either written in a random order or had missing parts. Students were asked to solve the \"puzzle\" – find the correct logical chain or complete the proof. These \"puzzles\" were meant to discourage students from simply memorizing the proof of a theorem. By using our examples students were encouraged to think independently and came to improve their understanding of the subject.

We construct and investigate random geometric structures that are based on a homogeneous Poisson point process. We investigate the random Vietoris-Rips complex constructed as the clique complex of the well known gilbert graph as an infinite random simplicial complex and prove that every realizable finite sub-complex will occur infinitely many times almost sure as isolated complex and also in the case of percolations connected to the unique giant component. Similar results are derived for the Cech complex. We derive limit theorems for the f-vector of the Vietoris-Rips complex on the unit cube centered at the origin and provide a central limit theorem and a Poisson limit theorem based on the model parameters. Finally we investigate random polytopes that are given as convex hulls of a Poisson point process in a smooth convex body. We establish a central limit theorem for certain linear combinations of intrinsic volumes. A multivariate limit theorem involving the sequence of intrinsic volumes and the number of i-dimensional faces is derived. We derive the asymptotic normality of the oracle estimator of minimal variance for estimation of the volume of a convex body.

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The topic of this dissertation lies at the interface between the areas of Harmonic Analysis, Partial Differential Equations, and Geometric Measure Theory, with an emphasis on the study of singular integral operators associated with second and higher order elliptic boundary value problems in non-smooth domains. The overall aim of this work is to further the development of a systematic treatment of second and higher order elliptic boundary value problems using singular integral operators. This is relevant to the theoretical and numerical treatment of boundary value problems arising in the modeling of physical phenomena such as elasticity, incompressible viscous fluid flow, electromagnetism, anisotropic plate bending, etc., in domains which may exhibit singularities at all boundary locations and all scales. Since physical domains may exhibit asperities and irregularities of a very intricate nature, we wish to develop tools and carry out such an analysis in a very general class of non-smooth domains, which is in the nature of best possible from the geometric measure theoretic point of view. The dissertation will be focused on three main, interconnected, themes: A. A systematic study of the poly-Cauchy operator in uniformly rectifiable domains in $\mathbb{C}$; B. Solvability results for the Neumann problem for the bi-Laplacian in infinite sectors in ${\mathbb{R}}^2$; C. Connections between spectral properties of layer potentials associated with second-order elliptic systems and the underlying tensor of coefficients. Theme A is based on papers [16, 17, 18] and this work is concerned with the investigation of polyanalytic functions and boundary value problems associated with (integer) powers of the Cauchy-Riemann operator in uniformly rectifiable domains in the complex plane. The goal here is to devise a higher-order analogue of the existing theory for the classical Cauchy operator in which the salient role of the Cauchy-Riemann operator $\overline{\partial}$ is now played by $\overline{\partial}^m$ for some arbitrary fixed integer $m\in{\mathbb{N}}$. This analysis includes integral representation formulas, higher-order Fatou theorems, Calderón-Zygmund theory for the poly-Cauchy operators, radiation conditions, and higher-order Hardy spaces. Theme B is based on papers [3, 19] and this regards the Neumann problem for the bi-Laplacian with $L^p$ data in infinite sectors in the plane using Mellin transform techniques, for $p\in(1,\infty)$. We reduce the problem of finding the solvability range of the integrability exponent $p$ for the $L^{p}$ biharmonic Neumann problem to solving an equation involving quadratic polynomials and trigonometric functions employing the Mellin transform technique. Additionally, we provide the range of the integrability exponent for the existence of a solution to the $L^{p}$ biharmonic Neumann problem in two-dimensional infinite sectors. The difficulty we are overcoming has to do with the fact that the Mellin symbol involves hypergeometric functions. Finally regarding theme C, based on the ongoing work in [2], the emphasis is the investigation of coefficient tensors associated with second-order elliptic operators in two dimensional infinite sectors and properties of the corresponding singular integral operators, employing Mellin transform. Concretely, we explore the relationship between distinguished coefficient tensors and $L^{p}$ spectral and Hardy kernel properties of the associated singular integral operators. / Mathematics

This thesis is an account of our study of two branches of dynamical systems theory, namely the mean and pointwise ergodic theory. In our work on mean ergodic theorems, we investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces which includes Fr echet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems. Next we turn to the topic of pointwise ergodic theorems. We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of -compact locally compact Hausdor groups acting measure-preservingly on - nite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on di erent Banach function spaces, and how the properties of these function spaces in- uence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages. Our investigations of these two parts of ergodic theory are uni ed by the techniques used - locally convex vector spaces, harmonic analysis, measure theory - and by the strong interaction of the nal results, which are obtained in greater generality than hitherto achieved. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2014