Includes bibliographical references. / The gauge theory/gravity correspondence encompasses a variety of di_erent specific dualities. We investigate various topics in the context of Super–Yang- Mills/type IIB string theory and superconformal Chern-Simons-matter/type IIA string theory dualities. We carry out a rather extensive study of the type IIA AdS3_S3_S3_S1 Green- Schwarz superstring, up to quadratic order in fermions. We discuss issues related to fixing its _-symmetry and show the one-loop finiteness of two-point functions of bosonic fields. We then perform a Hamiltonian analysis and compare SU(2) string states with predictions from the conjectured Bethe equations. Furthermore, we show that, at least at tree-level, the two-body S-matrix is reflectionless. We then concern ourselves with extending Mikhailov’s construction of giant gravitons from holomorphic functions to include meromorphic functions, which lead to giants with non-trivial topologies in AdS5_S5. We explore what topological configurations giants, whose dynamics preserve a certain amount of supersymmetry, assume. We are particularly interested in solutions created by a localised modification of a set of intersecting spherical giant gravitons, as this seems the most tractable limit. We finally explore some aspects of holographic particle-vortex duality, in particular its realisation in the ABJM model and a possible relation to Maxwell duality in AdS4. We formulate a symmetric version of the transformation that acts as a self-duality, show how to embed it as an abelian duality in the (2+1)-dimensional, N = 6 super–Chern-Simons-matter theory that is the ABJM model, and speculate on a possible non-abelian extension.

This thesis deals with the theoretical and numerical analysis of coupled problems in thermoelasticity. Of particular interest are models that support propagation of thermal energy as waves, rather than the usual mechanism by diffusion. The thesis consists of two parts. The first deals with the non-classical, linear thermoelastic model first proposed and developed by Green and Naghdi in the years between 1991 and 1995, as a possible alternative that potentially removes the shortcomings of the standard Fourier based model. The non-classical theory incorporates three models: the classical model based on Fourier's law of heat conduction, resulting in a hyperbolic-parabolic coupled system; a non-classical theory of a fully-hyperbolic extension; and a combination of the two. An efficient staggered time-stepping algorithm is proposed based on operator-splitting and the time-discontinuous Galerkin finite element method for the non-classical, linear thermoelastic model. The coupled problem is split into two contractive sub-problems, namely, the mechanical phase and thermal phase, on the basis of an entropy controlling mechanism. In the mechanical phase temperature is allowed to vary so as to ensure the entropy remains constant, while the thermal phase is a purely non-classical heat conduction problem in a fixed configuration. Each sub-problem is discretized using the time-discontinuous Galerkin finite element method, resulting in stable time-stepping sub-algorithms. A global stable algorithm is obtained by combining the algorithms for the sub-problems by way of a product method. A number of numerical examples are presented to demonstrate the performance and capability of the method. The second part of this work concerns the formulation of a thermodynamically consistent generalized model of nonlinear thermoelasticity, whose linearization about a natural reference configuration includes the theory of Green and Naghdi. The generalized model is based on the fundamental laws of continuum mechanics and thermodynamics, and is realized through two basic assumptions: The first is the inclusion into the state space of a vector field, which is known as the thermal displacement, and is a time primitive of the absolute temperature. The second is that the heat flux vector is additively split into two parts, which are referred to as the energetic and dissipative components of the heat flux vector. The application of the Coleman-Noll procedure leads to find constitutive relations for the stress, entropy, and energetic component of the heat flux as derivatives of the free energy function. Furthermore, a Clausius-Duhem-type inequality is assumed on a constitutive relation for the dissipative component of the heat flux vector to ensure thermodynamic consistency. A Lyapunov function is obtained for the generalized problem with finite strains; this serves as the basis for the stability analysis of the numerical methods designed for generalized thermoelasticity at finite strains. Due to the lack of convexity of the elastic potential in the finite strain case, a direct extension of the time-discontinuous formulation from the linear to the finite strain case does not guarantee stability. For this reason, various numerical formulations both in monolithic and staggered approaches with fully or partially time-discontinuity assumptions are presented in the framework of the space-time methods. The stability of each of the numerical algorithms is thoroughly analysed. The capability of the newly formulated generalized model of thermoelasticity in predicting various expected features of non-Fourier response is illustrated by a number of numerical examples. These also serve to demonstrate the performance of the space-time Galerkin method in capturing fine solution features.

Bibliography: p. 153-160.

abstract: Dengue is a mosquito-borne arboviral disease that causes significant public health burden in many trophical and sub-tropical parts of the world (where dengue is endemic). This dissertation is based on using mathematical modeling approaches, coupled with rigorous analysis and computation, to study the transmission dynamics and control of dengue disease. In Chapter 2, a new deterministic model was designed and used to assess the impact of local fluctuation of temperature and mosquito vertical (transvasorial) transmission on the population abundance of dengue mosquitoes and disease in a population. The model, which takes the form of a deterministic system of nonlinear differential equations, was parametrized using data from the Chiang Mai province of Thailand. The disease-free equilibrium of the model was shown to be globally-asymptotically stable when a certain epidemiological quantity is less than unity. Vertical transmission was shown to only have marginal impact on the disease dynamics, and its effect is temperature-dependent. Dengue burden in the province is maximized when the mean monthly temperature lie in the range [26-28] C. A new deterministic model was designed in Chapter 3 to assess the impact of the release of Wolbachia-infected mosquitoes on curtailing the mosquito population and dengue disease in a population. The model, which stratifies the mosquito population in terms of sex and Wolbachia-infection status, was rigorously analysed to characterize the bifurcation property of the model as well as the asymptotic stability of the various disease-free equilibria. Simulations, using Wolbachia-based mosquito control from Queensland, Australia, showed that the frequent release of mosquitoes infected with the bacterium can lead to the effective control of the local wild mosquito population, and that such effective control increases with increasing number of Wolbachia-infected mosquitoes released (up to 90% reduction in the wild mosquito population, from their baseline values, can be achieved). It was also shown that the well-known feature of cytoplasmic incompatibility has very little effect on the effectiveness of the Wolbachia-based mosquito control. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2020

This thesis focuses on Fast Radio Bursts (FRBs) and presents computational techniques that can be used to understand these enigmatic events and the Universe around them. Chapter 1 provides a theoretical overview of FRBs; providing a foundation for the chapters that follow. Chapter 2 details current understandings by providing a review of FRB properties and progenitor theories. In Chapter 3, we implement non-parametric techniques to measure the elusive baryonic halo of the Milky Way. We show that even with a limited data set, FRBs and an appropriate set of statistical tools can provide reasonable constraints on the dispersion measure of the Milky Way halo. Further, we expect that a modest increase in data (from fewer than 100 FRB detections to over 1000) will significantly tighten constraints, demonstrating that the technique we present may offer a valuable complement to other analyses in the near future. In Chapter 4, we study the fine time-frequency structure of the most famous FRB: FRB 121102. Here, we use autocorrelation functions to maximise the structure of 11 pulses detected with the MeerKAT radio telescope. The study is motivated by the low time-resolution of MeerKAT data, which presents a challenge to more traditional techniques. The burst profiles that are unveiled offer unique insight into the local environment of the FRB, including a possible deviation from the expected cold plasma dispersion relationship. The pulse features and their possible physical mechanisms are critically discussed in a bid to uncover the nature and origin of these transients.

The dynamics of many biological processes rely on an interplay between spatial transport and chemical reactions. In particular, spatial dynamics can play a critical role in the successful functioning of cellular signaling processes, where as basic a prop- erty as cell shape can significantly influence the behavior of signaling pathways. The inside of cells is a complex spatial environment, filled with organelles, filaments and proteins. We investigate the question of how cell signaling pathways function robustly in the presence of such spatial heterogeneity for the most basic of chemical signals. Due to the noisy environment of a cell, particle-based stochastic reaction-diffusion models are a widely used approach for studying such cellular processes, explicitly modeling the diffusion of, and reactions between, individual molecules. However, the computational expense of such methods can greatly limit the size of chemical systems that can be studied. To overcome this challenge, we rigorously derive coarse-grained deterministic partial integro-differential equation models that provide a mean field ap- proximation to the particle-based stochastic reaction-diffusion model. Relationships between the mean field models and standard reaction-diffusion partial differential equation models are further investigated for general biochemical reaction systems. Comparisons between these models are illustrated through mathematical analysis and numerical examples.

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We consider the heat equation forced by a space-time white noise and with periodic boundary conditions in one dimension. The equation is discretized in space using four different methods; spectral collocation, spectral truncation, finite differences, and finite elements. For each of these methods we derive a space-time white noise approximation and a formula for the covariance structure of the solution to the discretized equation. The convergence rates are analyzed for each of the methods as the spatial discretization becomes arbitrarily fine and this is confirmed numerically. Dirichlet and Neumann boundary conditions are also considered. We then derive covariance structure formulas for the two dimensional stochastic heat equation using each of the different methods. In two dimensions the solution does not have a finite variance and the formulas for the covariance structure using different methods does not agree in the limit. This means we must analyze the convergence in a different way than the one dimensional problem. To understand this difference in the solution as the spatial dimension increases, we find the Sobolev space in which the approximate solution converges to the solution in one and two dimensions. This result is then generalized to n dimensions. This gives a precise statement about the regularity of the solution as the spatial dimension increases. Finally, we consider a generalization of the stochastic heat equation where the forcing term is the spatial derivative of a space-time white noise. For this equation we derive formulas for the covariance structure of the discretized equation using the spectral truncation and finite difference method. Numerical simulation results are presented and some qualitative comparisons between these two methods are made. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester, 2015. / April 8, 2015. / Space-Time White Noise, Stochastic Heat Equation / Includes bibliographical references. / Xiaoming Wang, Professor Co-Directing Dissertation; Brian Ewald, Professor Co-Directing Dissertation; Laura Reina, University Representative; Philip L. Bowers, Committee Member; Bettye Anne Case, Committee Member; Giray Okten, Committee Member.

The sensitivity analysis of options is as important as pricing in option theory since it is used for hedging strategies, hence for risk management purposes. This dissertation presents new sensitivities for options when the underlying follows an exponential Lévy process, specifically Variance Gamma and Normal Inverse Gaussian processes. The calculation of these sensitivities is based on a finite dimensional Malliavin calculus and the centered finite difference method via Monte-Carlo simulations. We give explicit formulas that are used directly in Monte-Carlo simulations. By using simulations, we show that a localized version of the Malliavin estimator outperforms others including the centered finite difference estimator for the call and digital options under Variance Gamma and Normal Inverse Gaussian processes driven option pricing models. In order to compare the performance of these methods we use an inverse Fourier transform method to calculate the exact values of the sensitivities of European call and digital options written on S&P 500 index. Our results show that a variation of localized Malliavin calculus approach gives a robust estimator while the convergence of centered finite difference method in Monte-Carlo simulations varies with different Greeks and new sensitivities that we introduce. We also discuss an approximation method for the Variance Gamma process. We introduce new random number generators for the path wise simulations of the approximating process. We improve convergence results for a type of sensitivity by using a mixed Malliavin calculus on the increments of the approximating process. / A Dissertation Submitted to the Department of Mathematics in Partial FulfiLlment of the Requirements for the Degree of Doctor of Philosophy. / Summer Semester, 2010. / April 12, 2010. / Centered Finite Difference, Monte-Carlo simulations, FFT, Malliavin calculus, Inverse Fourier Transform method, Normal Inverse Gaussian process, Approximation of Lévy processes, Variance Gamma process, Greeks / Includes bibliographical references. / Craig A. Nolder, Professor Directing Thesis; Fred Huffer, University Representative; Bettye Anne Case, Committee Member; David Kopriva, Committee Member; Giray Okten, Committee Member; Jack Quine, Committee Member.

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