### Browsing by Subject "Mathematics and Applied Mathematics"

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- ItemOpen Access1+1+2 covariant approach to gravitational lensing in f(R) gravity(2009) Nzioki, Anne Marie; Dunsby, PeterIn this thesis, we develop the 1 + 1 + 2 formalism, a technique originally devised for General Relativity, to treat spherically symmetric spacetimes in for fourth order theories of gravity. Using this formalism, we derive equations for a static and spherically symmetric spacetime for general f(R) gravity. We apply these master eqautions to derive some exact solutions, which are used to gain insight on Birkhoff's theorem in this framework. Additionally, we derive a covariant form of the lensing angle for a specific spherically symmetric solution in f(R) = Rn gravity.
- ItemOpen AccessA disformally coupled quintessence mimicking the ΛCDM background(2022) Dusoye, Avishek; Dunsby, Peter Klaus; de la Cruz-Dombriz, Alvaro; Dunsby, P K S; Nunes, N JAlthough the currently-accepted Concordance model of the Universe has been very successful observationally, it cannot resolve two main issues. Firstly, it cannot untangle the unknown nature of the cosmological constant in the Einstein Field Equations, which is responsible for the accelerated cosmological expansion. Secondly, it cannot explain the σ8 tension, which occurs because the constraints upon galactic clustering by the Cosmic Microwave Background Planck experiments diverge from the large-scale measurement by the Dark Energy Survey. As an alternative to the cosmological constant, this thesis will be using a scalar field, namely the quintessence. Our studied cosmological model assumes that the quintessence is coupled with a generic fluid. It also assumes a theory of gravity with two geometries. The gravitational geometry describes the curvature of space-time while the physical geometry describes the propagation of matter fields. The conformal transformation, which relates the gravitational metric and the physical metric, is extended here to a disformal transformation. In this thesis, the disformally coupled quintessence model mimics the expansion history of the Concordance model, in order to reproduce its observational success and yet have additional degrees of freedom to attempt to address those two issues. Using this approach, the quintessential potential is not specified. The dynamical system for our model is analysed using phase portraits for various studied scenarios. We investigate the expansion history of the DCQ model, where the quintessence couples disformally with dark matter (Scenario I). Our investigation confirms that the quintessential mass influences the disformal characteristics of the dynamical system. Furthermore, the evolution of the density perturbations for the disformally coupled dark matter is reviewed. A disformal effect due to the quintessential mass is seen in the growth rate of the cosmological structures on large scales. The disformal parameter renders no appreciable effect on the evolution of total matter perturbation. A Bayesian analysis of the relevant parameters for the perturbative model (i.e., conformal parameter and quintessential mass) is then carried out using the Redshift Space Distortion data to constrain the best-fit parameters, which might elucidate the σ8 tension. The best fit set of parameters indicates that the data prefers the model to behave conformally.
- ItemOpen AccessA review of the Hubble tension(2022) Houliston, Rebecca; Larena, Julie; Weltman, AmandaThe Hubble constant H0 is the rate of expansion of the universe today. The discrepancy between the early universe H0 value, inferred using ΛCDM from Planck observations of the CMB, and the late universe H0 value, obtained using luminosity and distance measurements from a Type Ia Supernova (SNIa) distance ladder, has now reached 4.2σ. Despite improvements in precision, this tension has increased. This thesis studies these two measurements, as well as various other H0 determinations, which are independent of both the CMB and the SNIa distance ladder and corroborate the Hubble tension (with the caveat that many have large uncertainties), with a particular focus on lensing and gravitational waves. Some of the very many solutions proposed to resolve the Hubble tension are also explored, with an emphasis on late universe solutions and Early Dark Energy. The improvement in precision, the growing discrepancy, and the supporting independent measurements of the Planck and SNIa distance ladder H0 values are strong evidence that a significant tension between the early and late universe exists. This indicates that some modification to or expansion of ΛCDM is required. A great deal of models and solutions have been proposed to do so, however none have managed to fully resolve the tension yet.
- ItemOpen AccessA study of circuit Complexity for Coherent States(2022) Tladi, Mpho; Haque, Shajid; Murugan, Jeffrey; Weltman, AmandaComputational complexity is a popular quantity in quantum information theory. It has made huge strides in recent years in the study of black hole dynamics. A brief definition of complexity is the measure of how difficult it is to implement a task. For a quantum system, complexity evaluates the difficulty of preparing a quantum state from a given reference state by unitary transformations. However, in the dual gravity theory complexity has a geometric meaning. In some black hole context, Leonard Susskind and collaborators proposed two holographic conjectures. The Complexity=Volume (CV) states that complexity of the boundary field theory is dual to the volume of a co dimension one maximal surface that extends to the boundary of the Ads space. Complexity=Action (CA) posits that complexity of the boundary is the same as the action evaluated as an action on patch in the bulk defined as the Wheeler De Witt patch. In recent years, these two conjectures have initiated an extensive study of complexity. This thesis is also motivated by these conjectures and will investigate complexity in the field theory side of the story. Specifically, we will explore the complexity for coherent states. We will start with a review of different methods of computing complexity. Finally, we then investigate the complexity for coherent states by using the methods of circuit complexity and operator complexity
- ItemOpen AccessA study on complexity(2023) Rapotu, Dimakatso; Haque, Shajid; Murugan JeffreyThis thesis explores quantum complexity for various quantum systems. Quantum complexity is a well defined quantity in quantum information theory that measures the difficulty of constructing a quantum state from a given reference state and so far, various methods within high energy physics communities have been proposed for computing complexity. In this thesis, we will first review the computations of the different methods used for computing complexity, such as the circuit complexity that uses the wave function, Fubini-Study complexity, and finally the recently proposed Krylov complexity for closed quantum systems. We then extend our investigation and review the complexity for some open quantum systems that have already been explored in literature and finally, we will make some progress by also extending the investigation towards computing the complexity of a new open quantum system, namely the non-gaussian random matrix model.
- ItemOpen AccessActive Inference in Multi-Objective Dynamic Environments(2022) Hodson, Rowan; Shock, Jonathan; Smith, Ryan; Solms, MarkArtificial Intelligence holds the promise of not only creating intelligent entities, but also unlocking the mysteries of our brains, and the nature of the subjective consciousness that accompanies them. Many paradigms of artificial intelligence are attempting to push the boundaries of the field, in order to catch a glimpse of the secrets behind general intelligence and the nature of the human mind. A less explored, yet promising paradigm is that of Active Inference - a theory which details a first-principled explanation of how agents use action and perception to successfully operate within an external environment. Much work has been done to explore the framework's viability in modelling scenarios both related to neural process theory and more classical agent-based machine learning. However, due to the relative recency of the theory, there are still many areas of comparison and evaluation to explore. This dissertation aims to investigate Active Inference's algorithmic capacity to solve more complex decision-based environments. Specifically, with varying degrees of complexity, I make use of a dynamic environment with a multi-objective reward function to investigate the Active Inference agent's ability to learn and plan while balancing exploration and exploitation, and compare this to other Bayesian Machine Learning algorithms. In doing so, I investigate some novel approaches and additions to Active Inference's algorithmic structure which include a dynamic preference distribution, a two-tiered hierarchical approach to the state space (using model-free Reinforcement Learning to solve the lower level), and the introduction of the Propagated Parameter Belief Search algorithm - a modification to Active Inference which allows the agent to perform more complex counterfactual reasoning.
- ItemOpen AccessAlgebraic exponentiation and internal homology in general categories(2010) Gray, James Richard Andrew; Janelidze, GWe study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories.
- ItemOpen AccessAlternative theories of gravity and their application to cosmology(2008) Leach, Jannie A; Bruyns, PeterIn this thesis we study extended theories of gravity in the context of cosmology. The first part is dedicated to the application of the theory of dynamical systems, which allow us to investigate the global dynamics of some cosmological models resulting from scalar-tensor and higher-order theories of gravity. We use the dynamical systems approach with non–compact expansion normalised variables to study the isotropisation of Bianchi type I models in Rn–gravity. We find that these type of models can isotropise faster or slower than their general relativity counterparts. We extend this analysis to the full class of orthogonal spatially homogeneous Bianchi models to study the effect of spatial curvature on the isotropisation of these models. A compact state space is constructed by dividing the state space into different sectors, that allows us to also investigate static solutions and bouncing or recollapsing behaviours which is not possible when using non-compact expansion normalised variables. We find no Einstein static solutions, but there do exist cosmologies with bounce behaviours. We also find that all isotropic points are flat Friedmann like. We discuss the advantages and disadvantages of compactifying the state space, and illustrate this using two examples. We next study the phase-space of Friedmann models derived from scalar-tensor gravity where the non-minimal coupling is F(φ) = ξφ2 and the self-interaction potential is V (φ) = λφn. Transient almost-Friedmann phases evolving towards accelerated expansion and unstable inflationary phases evolving towards stable ones are found. In the last part of this work, we set out a framework to analyse tensor anisotropies in the cosmic microwave background of scalar-tensor cosmologies. As an example, we consider one of the exact solutions found for the class of scalar-tensor theories considered above.
- ItemOpen AccessAlternatives to the Black-Scholes model(2001) Durrell, Fernando; Ouwehand, PeterIn this paper, I consider alternative models to the one posited by Black and Scholes. I consider discontinuous security price movements, non-constant volatility, and models very different from the Black-Scholes model. I found that most of the model prices for the close to at-the-money options are very different from the market prices. In general, the models did poorly in producing similar prices as the market.
- ItemOpen AccessAn analysis of frictional effects in non-stationary contact problems for metal forming simulations(2021) Colville, Kevin; Laurie, Henri; Ronda; JacekThe finite element method (FEM) is widely used for the simulation of metal forming processes and has been successfully used in contact problems which arise in processes such as deep-drawing, punching, extrusion and rolling. All these processes involve friction between the contact surfaces: the sheet-metal workpiece and the toolpieces. The model of friction is thus an important part of any simulation of metal forming processes. Most FEM codes use a friction model that assumes that the contact surface is a plane. Attempts to address this problem have focused on the convective description of deformation, which has the advantage of being naturally extended to numerical methods like the FEM at the expense of additional computation and numerical complexity. The convective description is used in this work, which focuses on the numerical implementation of the objective measure. The effects of the rotation of the material contact point is taken into account by including objective time derivatives of the slipping (tangential) direction function. The objective rate of the direction function includes the surface spin induced by the rigid motion of a contact point sliding over the tool surface, and the material spin occurring during the elastic-plastic deformation of the blank. This is introduced by adapting the incremental relations of the friction slip. This thesis presents the results of numerical experiment to determine the influence that the rotation and convection of contact points has on the frictional stresses and slipping energy. Four different friction models are implemented within the finite element program ABAQUS and applied to simulations of standardmetal forming benchmark processes: the square-cup and s-rail deep drawing benchmarks of the Numisheet conferences, for which several experimental and numerical results are available to compare with the solution of a finite element simulation. The results for each metal-forming simulation are calculated for different friction models, and are compared and a choice made as to which is the “best” friction model for the process. Further, the reverse problem of determining the values of friction parameters by comparison of simulation and experimental results is performed for these benchmark problems. As there is yet no ideal friction model for all processes that are modelled, finding the most appropriate friction model by numerical means is proposed to improve the quality of a simulation.
- ItemOpen AccessAnalysis of convertible bonds(2004) Thompson, Kevin; Ouwehand, PeterIncludes bibliographical references ( leaves 90-92).
- ItemOpen AccessAn analysis of spatial percolation structures using a network approach(2006) Fadul, Mohammed Altaj Mohammed; Witten, G; Perrier, EIn this thesis we analyse several spatial structures, built from percolation models, by means of an approach used so far in the field of network science. In the first chapter we summarize the major network concepts and characterizations that have been obtained as regards the statistical properties of several data sets or theoretical models, We also give a brief introduction to percolation theory and its applications, adding details in two particular cases where mathematical results are available. In the second chapter we then study one particular application of percolation theory to the modelling of distribution and species abundance at different seales. We mainly focus on the way percolation theory was used to compare two diffcrcnt spatial patterns, particularly the random and the aggrergated distribution.
- ItemOpen AccessAnalytical approximations of surface fields induced on convex scatters by exteriorly incident scalar fields(1989) Nel, W J F; Du Plessis, N MThe boundary value problems for the Helmholtz equation give rise to boundary integral equations for the unknown surface field or its normal derivative. These integral equations involve the Helmholtz surface potentials in the form of weakly singular surface integrals. This thesis is based on a method of parameterisation of the surface integrals which removes the weak singularities provided that the surface satisfies certain convexity conditions. Firstly this method of parameterisation is applied to investigate the properties of the Helmholtz surface potentials on convex surface elements, and some new proofs are given. The theory is then applied to the boundary integral equations which arise when a scalar field is incident on a bounded scatterer. The surface integrals in these integral equations are Helmholtz potentials and can be regularised by suitable parameterisation. It is assumed that the unknoWn density function is an analytical function on the boundary of the scatterer, and can therefore be expanded as a Taylor series at any point of the surface. If this expansion is substituted into the regularised integral equation and if the operations of integration and summation are formally interchanged, then the end result is a partial differential equation of infinite order involving only the field coordinates and having analytical coefficients. However, if the Taylor expansions are truncated then partial differential equations of finite orders result. The view is taken that analytical solutions of such differential equations of finite orders can serve as _approximations for the surface field or its normal derivative provided that suitable initial conditions are imposed to ensure uniqueness. On the other hand the general solution of such a differential equation can serve as a local approximation at any point on the surface. Some basic properties of the differential equations and their solutions, called analytical approximations, are discussed and the theory is then applied to the problem of acoustic scattering from a sound hard sphere.
- ItemOpen AccessThe annihilation graphs of commutator posets and lattices(2015) Mehdinezhad, Elham; Janelidze, GeorgeWe propose a new, widely generalized context for the study of the zero-divisor/ annihilating-ideal graphs, where the vertices of graphs are not elements/ideals of a commutative ring, but elements of an abstract ordered set (imitating the lattice of ideals), equipped with a binary operation (imitating products of ideals). The intermediate level of congruences of any algebraic structure admitting a "good" theory of commutators is also considered.
- ItemOpen AccessApplication of extreme value theory to the calculation of value-at-risk(2001) Seymour, Anthony; Polakow, DanielThe main aim of the study was to test the applicability of published EVT-based VaR calculation methods to the South African market. Two methods were tested on a hypothetical portolio of South African stocks, using the standard backtesting technique.
- ItemOpen AccessThe application of general linear modelling methods to estimate trends in abundance of the hake and rock lobster stocks off South Africa(1999) Glazer, Jean Patricia; Butterworth, Doug SThe two species of Cape hake, Merluccius capensis (shallow-water hake) and M paradoxus (deepwater hake), fomi the mainstay of the bottom trawl industry off South Africa and constitute the country's most valuable fishery. It is therefore important that the status of this resource be assessed regularly to ensure that exploitation is at a sustainable level. The two Cape hake species are morphologically similar and no distinction is made between them in commercial catch statistics. Consequently, for assessment purposes, the Cape hakes are treated as a single species. It is assumed that two stocks of Cape hake exist, one off the West Coast and another off the South Coast of South Africa. Central to the assessments of these stocks are the catch per unit effort (CPUE) data because it assumed that CPUE is proportional to abundance. The nominal CPUE (hake catch divided by actual time trawled) for both the West and South Coast stocks has shown a steady growth over the period 1978 - 1994, increasing at a rate of 3.8% per annum on the West Coast and 4.2% per annum on the South Coast. The bulk of this thesis is concerned with determining whether these increases in CPUE are the result of an increasing biomass, or are rather, in part, the result of improved vessel efficiency due to technological advancement or of changes in fishing strategy. The existing CPUE time series had previously been standardised by means of applying power factors which were crudely estimated in the early 1970s and which are likely inappropriate for the current fishing fleets. These CPUE series have therefore been re-standardised by applying the internationally accepted approach of General Linear Modelling (GLM).
- ItemOpen AccessApplications of the gauge theory/gravity correspondence(2010) Prinsloo, Andrea Helen
- ItemOpen AccessApproaches to ensembles of universes(2003) Kirchner, Ulrich; Ellis, GFRThis thesis consists of three parts - each of them focusing on different aspects relating to ensembles of universes and causally disconnected regions. In the first part I investigate possible measures over the space of FLRW models. In the second part I examine the behaviour of such transition regions for spherically symmetric space-times. In the last part of this thesis I discuss philosophical, physical, and probabilistic issues related to the concept of a multiverse - an ensemble of universes. The difference between ensembles of really existing universes and ensembles of possible universes is emphasized.
- ItemOpen AccessApproaches to reflective hulls of subcategories(1994) Vajner, Václav; Bargenda, H W; Brümmer, G C LReflective subcategories originated as a formal mathematical concept in the 1960's. Perhaps the first abstract definition of reflectivity can be attributed to P. Freyd who, in [Freyd 1960] and [Freyd 1964], gave a definition in terms of reflection arrows. Already in [Isbelll964] the general definition of a reflective subcategory was applied to some concrete situations, and used to formulate one of the first problems concerning reflectivity, namely, whether the intersection of (full, isomorphism-closed) reflective subcategories of the category of uniform spaces is again reflective. This problem, together with analogous questions posed in other contexts (e.g., by H. Herrlich for the category of topological spaces), led to the formulation of the reflective hull problem for subcategories in general, namely, whether a given subcategory is contained in a smallest reflective supercategory. Much of the research concerned with reflectivity and the reflective hull problem considers sufficient conditions for a category such that the reflectivity of (certain) subcategories can be described and the existence of reflective hulls can be guaranteed. These conditions are usually given in terms of (co)completeness and (co)wellpoweredness (see, for example, [Tholen 1987), [Kelly 1987]). A primary objective of this thesis is to provide sufficient and necessary conditions, formulated in subcategory-related terms, for the reflectivity of a given subcategory, and for the characterisation of the existence of reflective hulls. Our approach to finding appropriate descriptions of reflective hulls is essentially a constructive one, in the sense that we attempt to "generate" the reflective hull of a given subcategory (and hence give a concrete description of the hull) by means of certain closure processes applied to the given subcategory. We should also emphasise that our philosophy is not a conservative one in that, apart from applications of our constructions to particular situations, we make as few global assumptions as possible in our considerations. Intuitively, there are several ways in which reflectivity can be viewed as a mathematical concept; the results in this thesis emphasise these points of view. First, reflectivity may be viewed as a completeness property, i.e., as a kind of limit procedure; we study the correspondence between reflective hulls and closures of subcategories under certain types of limits. Reflectivity may also be considered as a cocompleteness property; appropriately we also consider the closure of a given subcategory under certain kinds of colimits and its relation to a possible reflective hull. Both of these constructions are generalisations of natural descriptions of reflection arrows in the special case of partially-ordered classes. Finally, reflectivity can be considered as a (subcategory-related) factorisation property; in this context we consider closures of a subcategory in terms of factorisations relative to the given subcategory, and, related to this, closures under special kinds of colimits relative to the given subcategory. In this thesis we also obtain results concerning the relation between reflectivity and weaker concepts; in particular results concerning intersections of reflective subcategories, and reflective hulls of almost reflective subcategories, are given, and applied to concrete situations, for example, the following problem posed in Rosicky and Tholen 1988: Is the category of complete Boolean algebras an intersection of reflective subcategories of the category of frames?
- ItemOpen AccessApproximations to the Lévy LIBOR Model(2014) Al-Hassan, Hassana; Becker, Ronald; Mataramvura, SureIn this thesis, we study the LIBOR Market Model and the Lévy-LIBOR. We first look at the construction of LIBOR Market Model (LMM) and address the major problems associated with specifically the drift component of LMM. Due to the complexity of the drift for LMM, the Monte Carlo method seems to be the ideal tool to use. However, the Monte Carlo method is time consuming and therefore an expensive tool to use. To improve on the process we look beyond the dynamics of the lognormal distribution, where Brownian motion (the only Lévy process with continuous paths), is the driving process and apply other Lévy processes with jumps as the driving process in the dynamics of LIBOR. The resulting process is called Lévy LIBOR Model constructed in the framework of Eberlein and Özkan (2005). The Lévy LIBOR model is a very flexible and a general process to use but has a complicated drift part in the terminal measure. The complicated drift term has random terms in the drift part as a result of change of measure. We employ Picard approximation and cumulant expansions in the resulting drift component to make the processes tractable in the framework of Papapantoleon and Skovmand (2010).