A research report submitted to the Faculty of Science, in partial fulfilment of the requirements for the degree of Master of Science, University of the Witwatersrand. Johannesburg, 2016. / The challenges facing humanity are presenting at an unprecedented scale and complexity. Climate change, biodiversity loss, land degradation, ocean depletion, poverty, inequality, and failing health care and educational systems are among the problems that have come to be defined as "wicked" because of their spread and interconnectedness. At the same time, there has been an increasing realisation that multi-, inter- and transdisciplinary (MIT) research is critical to addressing these complex global challenges. This research identified some of the barriers and enablers to MIT research at the University of the Witwatersrand (Wits), based on interviews with researchers and research managers at the University who have been involved in conducting or facilitating MIT research. [Abbreviated Abstract. Open document to view full version] / LG2017

Nesta tese, obtemos estados quânticos que satisfazem a equação de Schrödinger, para Hamiltonianos quadráticos de forma geral e, ao mesmo tempo, permitem de maneira natural obter a correspondência com a descrição clássica. Usamos o método de integrais de movimento para construir operadores de criação e aniquilação, que satisfazem a álgebra de Weyl-Heisenberg. Dessa forma, construímos os estados de número generalizados (ENG) de maneira análoga ao que é feito para os estados de Fock. Obtemos diferentes famílias de estados coerentes (EC), através de uma superposição dos ENG, que chamamos de estados coerentes generalizados (ECG). Esses estados são rotulados pela constante complexa z escrita em termos do valor esperado inicial da coordenada e do momento. Escrevemos os ECG em função do desvio padrão inicial na coordenada, $\\sigma_q$, de modo a minimizar a relação de incerteza de Heisenberg no instante de tempo inicial. Obtemos, de forma pioneira, os ECG para partícula livre e discutimos em detalhes suas propriedades, tal como a relação de completeza, a minimização das relações de incerteza e a evolução da correspondente densidade de probabilidade. Mostramos que o valor esperado da coordenada e do momento segue ao longo da trajetória clássica no espaço de fase. Mostramos que, quando o comprimento de onda da partícula livre é muito menor que $\\sigma_q$, os EC se comportam como estados semiclássicos. Além da partícula livre, construímos pela primeira vez, os ECG para o oscilador invertido e discutimos em detalhes suas propriedades. Mostramos que os ECG de sistemas diferentes podem ser relacionados, impondo condições sobre os parâmetros do Hamiltoniano. Por fim, consideramos Hamiltonianos dependentes do tempo, em particular, construímos os ECG, de forma exata, para um oscilador harmônico cuja frequência varia explicitamente no tempo. Mostramos ainda modelos úteis para obter solução exata de sistemas dependentes do tempo, fazendo analogia com a equação de spin ou equação de Schrödinger unidimensional independente do tempo. Além disso, desenvolvemos um método próprio, que fixa a solução e em seguida determinamos a forma da frequência. / In this thesis we obtain quantum states that satisfy the Schrödinger equation for quadratic Hamiltonians in the general form and at the same time allow, naturally, to obtain the correspondence with the classical description. For this, we use the method of integrals of motion to construct creation and annihilation operators, which satisfy the algebra of Weyl-Heisenberg. Thus, we obtain the generalized number states (GNS) in the same way that is done for the Fock states. We obtain diferent families of coherent states (CS) that we call generalized CS (GCS), by a superposition of GNS. These states are labeled by a complex constant z which is written in terms of the initial expected values of the coordinate and momentum. We write the GCS in terms of the initial standard deviation of the coordinate, $\\sigma_q$, which provides the minimization of Heisenberg uncertainty relation at the initial instant time. In particular, we obtain for the first time the GCS for the free particle and discuss in detail their properties, such as the completeness relation, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We show that the expected values of coordinated and momentum propagate along the classical trajectory in phas espace. When the Compton wavelength is much smaller than $\\sigma_q$, the CS can be considered a semiclassical state. In addition to the free particle, we obtain for the first time the GCS for the inverted oscillator and discuss in detail their properties. We show that the GCS of diferent systems can be related by imposing conditions on the parameters of the Hamiltonian. Finally, we consider the time-dependent Hamiltonian, especially to obtain the GCS for a harmonic oscillator whose frequency varies explicitly in time. We also show useful models to obtain exact solution for time-dependent systems, by analogy with the spin equation or one-dimensionaltime-independent Schrödinger equation, as well as a method which consists first to find the solution and then determine the shape of the frequency.

A quantização geométrica e um método desenvolvido para prover uma construção geométrica que relacione a mecânica clássica com a quântica. O primeiro passo consiste em apresentar uma forma simplética, \'omega\'!, sobre uma variedade simplética, M, como a forma curvatura da conexão abla de um brado linear, L, sobre M. As funções sobre M operam como seções de L. Mas o espaço de todas as seções é grande demais. Queremos considerar seções constantes em certa direção, com respeito a derivada covariante dada por abla, e para isso precisamos o conceito de polarizações, essas seções são chamadas de seções polarizadas. Para obter uma estrutura de espaco de Hilbert nestas seções, precisamos de certos objetos chamados de meias densidades. Além disso, também temos um empareamento sesquilinear entre seções de polarizações diferentes. Neste trabalho, primeiramente consideraremos o empareamento para seções polarizadas adaptadas a polarizações reais não transversais, como método para obter aplicações integrais entre estes espaços de Hilbert que em combinação com a convolução do par grupóide M x \' M BARRA\', pode definir um produto integral de funções definidas na variedade simplética. Este produto, no caso do plano euclidiano e do plano de Bieliavsky, coincide com produto de Weyl integral e o produto de Bieliavsky, respectivamente. Jáa no caso do plano hiperbólico, este tipo de polarizações reais não são transversais nem são não transversais, dessa forma, escolhemos o empareamento entre uma polarização real e uma polarização holomorfa do par grupóide, as quais são transversais, para obter um produto integral no plano hiperbólico, que no caso do plano euclidiano e o produto de Weyl / The geometric quantization is a method developed to provide a geometrical construction relating classical to quantum mechanics. The first step consists of realizing the symplectic form, \'omega\', on a symplectic manifold, M, as the curvature form of a line bundle, L, over M. The functions on M then operate as sections of L. However, the space of all sections of L is too large. One wants to consider sections which are constant in certain directions (polarized sections) and for that one needs to introduce the concept of a polarization. To get a Hilbert space structure on the polarized sections, one needs to consider objects known as half densities. In this work, first we consider a sesquilinear pairing between objects associated to certain different polarizations, which are nontransverse real polarizations, to obtain integral applications between their associated Hilbert spaces, and to use the convolution of the pair groupoid M x \' M BARRA\' to obtain an integral product of functions on M. In the euclidian plane case, we recover the integral Weyl product and, in the Bieliavsky plane case, we obtain the Bieliavsky product. On the other hand, for the hyperbolic plane, such real polarizations are neither transverse nor nontransverse, so we use the pairing between a real polarization and a holomorphic polarization, which are transverse polarizations on the pair groupoid, to obtain an integral product of functions on the hyperbolic plane. This same procedure, in the euclidian plane case, also produces the integral Weyl product

Dans le cas générique Yu. S. Ilyashenko a donné une solution pour le problème tangentielle du centre et le probème de la monodromie. Néanmoins, on ne connaît pas la solution pour tous les cas non-génériques. Dans cette thèse on étudie une famille des équations hamiltoniennes non-génériques dont l'hamiltonien est un produit de polynômes réels irréductibles de dégre supérieur ou égal à 1. On étudie cette famille dans le but d'avoir un modèle d'équation hamiltonienne qui nous permette de comprendre d'autres cas non-génériques. Cette famille ne satisfait pas necessairement les conditions de généricité de transversalité à l'infini et n'a pas nécessairement tous les points singuliers aux niveaux distincts. Nous considerons quelques conditions géomètriques sur les hamiltoniens qu'on appelle bon partage du plan proyective réel et bonne multiplicité à l'infini. Ces conditions nous servent pour calculer l'orbite par monodromie des cycles évanescents. On résout le problème de la monodromie pour deux sous-familles dans cette famille d'hamiltoniennes. Une d'elles satisfait que tous les points critiques de type centre sont à des niveux critiques distincts, et l'autre satisfait que l'hamiltonien est invariant par la réflexion par rapport à l'axe des y. En utilisant la solution du problème de la monodromie on résout aussi le problème tangentiel du centre pour ces familles. / In the generic case Yu. S. Ilyashenko gave a solution of the tangential center problem and the monodromy problem. However, a solution for all non-generic cases is not known. In this thesis we study a family of non-generic Hamiltonians, whose Hamiltonian is a product of real polynomials of degree equal or bigger than 1. We study this family with the idea that a good understanding of this Hamiltonian model could help us to understand other non-generic cases later. In this family the genericity assumption of transversality at infinity fails and the coincidence of the critical values for different critical points is allowed. We consider some geometric conditions on the Hamiltonians of this family that we call good divide of the real projective plane and good multiplicity at infinity. These conditions help us to compute the orbit under monodromy of vanishing cycles. We give a solution of the monodromy problem of two sub-families in this family. One of them satisfying that all the center critical points are at different critical levels, and the other satisfying that the Hamiltonian is invariant under the reflection with respect to the y-axis. Using the solution of the monodromy problem we also provide a solution of the tangential center problem for those families.

The unsteady nature of coastal hydrodynamics is associated with complex boundary layer dynamics and hence engineering predictions of shear stresses and sediment transport are difficult. This thesis explores some of the complex hydrodynamic problems and boundary layer behaviour in the coastal zone and seeks to provide new and improved modelling approaches. The latest experimental results are used to inform the model development process. New laboratory experiments carried out as part of this thesis illustrate the value of convolution integral calculations for both pressure and skin friction forces on particles on the bed. The experiments also highlight the importance of the phase differences between free stream velocity and boundary layer shear stresses. The use of a “bed” shear stress as a model input is found to be problematic whenever there is a large vertical gradient in the boundary layer shear stress. New experimental and modelling work has helped to improve our understanding of sheet flow boundary layer dynamics. This thesis builds on some of these new discoveries to propose a new simplified model framework for sheet flow sediment transport prediction using convolution integrals. This time domain technique has the advantage of simplicity while incorporating the most important physical processes from more detailed models. The new model framework could be incorporated into any depth averaged coastal hydrodynamic modelling software package. Boundary layer analysis techniques presented in the thesis provide an improved understanding of the effective roughness of mobile beds and can be used to calculate instantaneous shear stress profiles throughout the mobile bed boundary layer. New solutions for swash zone hydrodynamics are presented which illustrate the limitations of the previous benchmark analytical model for swash hydrodynamics. It is shown that real swash necessarily involves a much larger influx of mass and momentum than the analytical solution which was previously used by many in the swash sediment transport research community. Models for swash boundary layer development are also presented.

Plasmas seeded with solid particulates of nanometer to micron sizes (complex plasma systems) are a ubiquitous feature of intergalactic, interstellar and planetary environments but also of plasma processing applications or even fusion devices. Their novel aspects compared with ideal multi-component plasmas stem from (i) the large number of elementary charges residing on the grain surface, (ii) the variability of the charge over mass ratio of the dust component, (iii) the inherent openness and dissipative nature of such systems.   Their statistical description presents a major challenge; On one hand by treating dust grains as point particles new phase space variables must be introduced augmenting the classical Hamiltonian phase space, while the microphysics of interaction between the plasma and the grains will introduce additional coupling between the kinetic equations of each species, apart from the usual fine-grained electromagnetic field coupling. On the other hand complex plasma systems do not always exist in a gaseous state but can also condensate, i.e. form liquid, solid or crystalline states.   In this thesis we study gaseous partially ionized complex plasma systems from the perspective of the Klimontovich technique of second quantization in phase space. Initially, in regimes typical of dust dynamics. Starting from the Klimontovich equations for the exact phase space densities, theory deliverables such as the permittivity, the spectral densities of fluctuations and the collision integrals are implemented either for concrete predictions related to low frequency electrostatic waves or for diagnostic purposes related to the enhancement of the ion density and electrostatic potential fluctuation spectra due to the presence of dust grains. Particular emphasis is put to the comparison of the self-consistent kinetic model with multi-component kinetic models (treating dust as an additional massive charged species) as well as to the importance of the nature of the plasma particle source. Finally, a new kinetic model of complex plasmas (for both constant and fluctuating sources) is formulated. It is valid in regimes typical of ion dynamics, where plasma discreteness can no longer be neglected, and, in contrast to earlier models, does not require relatively large dust densities to be valid. / QC 20120316

DE LA TESI DOCTORAL:Aquesta memòria estudia bàsicament el comportament asimptòtic de la densitat de diferents famílies de vectors aleatoris. Al començament es dóna una introducció on es comenten diversos treballs anteriors que tracten sobre estudis asimptòtics de densitats, es pot observar el gran lligam que hi ha entre les estimacions de Varadhan i l'anomenat desenvolupament de Taylor de la densitat. Les estimacions són un primer pas cap a un estudi més extens del comportament asimptòtic.Un cop feta l'introducció general (Capítol 1), el Capítol 2 de la memòria està dedicat a l'estudi de les anomenades estimacions de Varadhan. Al tercer Capítol realitzarem un estudi més acurat i exhaustiu del comportament asimptòtic de la densitat. Al Capítol 4, sota les mateixes condicions que s'utilitzen per demostrar l'existència i regularitat d'una densitat "pe(y)", nosaltres trobarem el desenvolupament asimptòtic amb d = 1, on ara els coeficients "c-1" dependran de les derivades del procés solució de l'equació estocàstica pertorbada avaluades en e >> 0; a més a més, aquestes derivades satisfarán equacions d'evolució que seran descrites. Finalment, al Capítol 5, estudiarem el comportament densitat que al Capítol 4, però per a tot "y" pertanyent a R.Els Capítols 2, 3, 4 i 5 contenen una introducció on s'explica la metodologia que nosaltres hem seguit en aquell capítol, donant les idees més importants. Les Seccions d'aquests Capítols constaran quasi sempre de tres parts. Una primera, anomenada Objectiu, està dedicada a explicar el propòsit de la Secció. Una segona, dita Preliminars, on es donaran els prerequisits necessaris, quan s'escaigui, per poder portar a terme la demostració dels Objectius. A l'última es provaran els resultats.

The study of the electromagnetic phenomena along the last two centuries has brought about outstanding contributions for the human progress. The electromagnetism represents still now, at the beginning of the third millenium, a very important research area. The radiation pattern of particular types of antennas -for example, fractal or microstrip-, the analysis of the effect of the cellular communications on human beings or the detection of buried mines represent specific examples of the wide variety of problems of great interest nowadays. The study of such a variety of problems relies on the application of the Maxwell equations, which rule all the electromagnetic behaviour. Since the analytical solution can only be obtained for very particular cases of canonical forms, to tackle the analysis of an arbitrary problem, one makes use of the numerical methods. The discretization of electromagnetic integral equations by the Method of Moments -MoM- excels as a powerful and reliable tool for analysing bodies composed of locally homogeneous regions -penetrable or perfectly conducting- immerse in a wide and nearly uniform medium -typically the ground or the free-space-. These integral methods result from the surface equivalence theorem, which allows in general two different formulations, the Electric Field Integral Equation (EFIE) and the Magnetic Field Integral Equation (MFIE). For the case of penetrable bodies, the Poggio, Miller, Chang, Harrington and Wu (PMCHW) formulation, that results from the subtraction of the EFIE and MFIE at both sides of the surfaces, can also be employed.The Method of Moments is based on the full expansion of the physical magnitudes, field and current, over the interface surfaces between the regions. In consequence, the solution of the problem is obtained through the inversion of a full-matrix, which, for electrically large problems, requires excessive memory resources and computation time. That is why the MoM is widely considered a brute-force method. The expansion of the magnitudes is carried out through the discretization of the surface; that is, patches spreading over the interface. The first half of this dissertation Thesis tackles the development of the MoM applied to problems with bodies with symmetry of revolution -BoR-. Since in this case the physical magnitudes present an azimuthal periodicity, they can be expressed as a Fourier series. The orthogonality between the different modes enables to obtain separately each azimuthal mode of the solution. It is thus only required to spread the patches along the generating arc of the bodies for each mode, which is very advantageous because the electromagnetic analysis can be carried out indeed for dimensionally large problems. A well-known PeC-EFIE BoR formulation is developed. Accordingly, PeC-MFIE and PMCHW formulations are developed from scratch. Furthermore, it is commented in detail and corrected to some extent the numerical error associated to the fastest-varying part of the PeC-MFIE BoR operator. The BoR-codes are particularly useful in modelling the electromagnetic behaviour of buried mines, which very often show revolution symmetry. The most outstanding contribution of this dissertation Thesis is the study of the appropriate conditions to develop correctly the 3D operators so as to yield accurate results for any structure. Since the discretization implies a break on the continuity properties of the physical magnitudes -field and current- the valid 3D-operators must ensure the physical electromagnetic requirements in the discretized surface. In mathematical terms, these requirements set the rank -field- and domain- -current- spaces, which essentially require the enforcement of the continuity across the edges of either the tangential or the normal component of the expanded magnitudes.For the case of an arbitrary perfectly conducting -PeC- body, it is recommended in this work the use of the divergence-conforming and of the curl-conforming functions respectively in the development of the PeC-EFIE and the PeC-MFIE operators. Low-order sets over triangular facets -RWG and unxRWG- are chosen to develop the PeC-operators. Furthermore, it is reasoned theoretically the inherent misbehaviour of the PeC-MFIE in case the current expansion relies on a divergence-conforming set. A heuristic correction is provided. The better behaviour of PeC-EFIE(RWG) and PeC-MFIE(unxRWG) is confirmed with examples. In view of the results, it is reasoned the suitability of PeC-EFIE(RWG) for the analysis of physical polyhedrons, which makes PeC-MFIE(unxRWG) excel as a more appropriate operator for curved bodies. A procedure for improving the performance of PeC-EFIE(RWG) for coarsely meshed spheres is given.For the case of arbitrary penetrable bodies, the same low-order sets are used to expand the operators EFIE, MFIE and PMCHW. It is shown their compatibility with the combination of the right PeC-operators. In the dielectric case, in addition to the required continuity of the magnitudes across the edges at each region, the fields at both sides of the surface must satisfy the interface continuity, which is ignored in the conducting case -the fields are null inside the conductor-. The impossibility of meeting both continuity requirements at the same time justifies the apparition of inherent and different errors in the dual EFIE-MFIE and in PMCHW. It is thoroughly reasoned and confirmed with examples the suitability of PMCHW for problems with only penetrable regions. It is also shown and discussed in detail the robustness of EFIE-MFIE since its behaviour is appropriate for electrically not too small structures with perfectly conducting or penetrable regions. The analysis of composite structures -very useful to model microstrip antennas- can be considered as a group of disjoint bodies with null distances of separation. For this type of problems, it is recommended in this work the use of EFIE-MFIE since, unlike PMCHW, they can ensure the continuous transition to zero of a distance of separation increasingly small. Finally, efficient methods -IE-MEI and MLFMM- relying on the previous 3D-operators. The development of the PeC 3D IE-MEI cannot maintain the advantages present in the 2D case since the harmonic metrons are not valid in the 3D general case. A new set of metrons that ensures little discontinuity of the current across the edges is presented. It is confirmed with examples how these metrons, so-called quasi-continuous, reduce the number of required coefficients per row for a certain current error. Some examples of penetrable spheres with moderate electrical dimensions analysed under a MLFMM implementation are shown and commented.

In this thesis a novel approximate/analytical approach based on the concepts of stochastic averaging and of statistical linearization is developed for the response determination of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems subject to evolutionary stochastic excitation. The significant advantage of the approach relates to the fact that it is readily applicable for excitations possessing even non-separable evolutionary power spectra (EPS) circumventing ad hoc pre-filtering and pre-processing excitation treatments associated with existing alternative schemes of linearization. Further, the approach can be used, in a rather straightforward manner, in conjunction with recently developed design spectrum based analyses for obtaining peak response estimates without resorting to numerical integration of the nonlinear equations of motion. Furthermore, a novel approximate/analytical Wiener path integral based solution (PIS) is developed and a numerical PIS approach is extended to determine the response and first-passage probability density functions (PDFs) of nonlinear/hysteretic systems subject to evolutionary stochastic excitation. Applications include the versatile Preisach hysteretic model, recently applied in modeling systems equipped with smart material (shape memory alloys) devices used for seismic hazard risk mitigation. The approach is also applied to determine the capsizing probability of a ship, whose rolling dynamics is captured by a softening Duffing oscillator. Finally, novel harmonic wavelets based joint time-frequency response analysis and identification approaches are developed capable of determining the time-varying frequency content of non-stationary complex stochastic phenomena encountered in engineering applications. Specifically, a harmonic wavelets based statistical linearization approach is developed to determine the EPS of the response of nonlinear/hysteretic systems subject to stochastic excitation. In a similar context, an identification approach for nonlinear time-variant systems based on the localization properties of the harmonic wavelet transform is also developed. It can be construed as a generalization of the well established reverse multiple-input/single-output (MISO) spectral identification approach to account for non-stationary inputs and time-varying system parameters. Several linear and nonlinear time-variant systems are used to demonstrate the reliability of the approach.

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev completion of its fundamental group G and the cohomology groups of X with coefficients in rank one local systems. In this thesis, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. The relative completion S_p is a prosolvable group that generalizes the classical Malcev completion; when p is the trivial representation, S_p is the Malcev completion of G. The group S_p is tightly controlled by the cohomology groups H^1(X,L_{p^k}) with coefficients in the irreducible local systems L_{p^k} associated to the representation p.The pronilpotent Lie algebra u_p of the prounipotent radical U_p of S_p has been described by Hain. If p is the trivial representation, then u_p is the holonomy Lie algebra, which is well-known to be quadratically presented. In contrast, we show that when X is the complement of the braid arrangement in complex two-space, there are infinitely many representations p from G into (C^*)^2 for which u_p is not quadratically presented.We show that if Y is a subtorus of the character torus T containing the trivial character, then S_p is combinatorially determined for general p in Y. We do not know whether S_p is always combinatorially determined. If S_p is combinatorially determined for all characters p of G, then the characteristic varieties of the arrangement X are combinatorially determined.When Y is an irreducible subvariety of T^N, we examine the behavior of S_p as p varies in Y. We define an affine group scheme S_Y over Y such that if Y = {p}, then S_Y is the relative Malcev completion S_p. For each p in Y, there is a canonical homomorphism of affine group schemes from S_p into the affine group scheme which is the restriction of S_Y to p. This is often an isomorphism. For example, if there exists p in Y whose image is Zariski dense in G_m^N, then this homomorphism is an isomorphism for general p in Y. / Dissertation