In this thesis, we study the exact eigenvalue distribution of product of independent rectangular complex Gaussian matrices and also that of product of independent truncated Haar unitary matrices and inverses of truncated Haar unitary matrices. The eigenvalues of these random matrices form determinantal point processes on the complex plane. We also study the limiting expected empirical distribution of appropriately scaled eigenvalues of those matrices as the size of matrices go to infinity. We give the first example of a random matrix whose eigenvalues form a non-rotation invariant determinantal point process on the plane. The second theme of this thesis is infinite products of random matrices. We study the asymptotic behaviour of singular values and absolute values of eigenvalues of product of i .i .d matrices of fixed size, as the number of matrices in the product in-creases to infinity. In the special case of isotropic random matrices, We derive the asymptotic joint probability density of the singular values and also that of the absolute values of eigenvalues of product of right isotropic random matrices and show them to be equal. As a corollary of these results, we show probability that all the eigenvalues of product of certain i .i .d real random matrices of fixed size converges to one, as the number of matrices in the product increases to infinity.

Mild traumatic brain injury (mTBI) is a common and often inconspicuous wound that is frequently associated with chronic low-grade symptoms and cognitive dysfunction. Previous evidence suggests that daily blue wavelength light therapy may be effective at reducing fatigue and improving sleep in patients recovering from mTBI. However, the effects of light therapy on recovering brain structure remain unexplored. In this study, we analyzed white matter diffusion properties, including generalized fractional anisotropy, and the quantity of water diffusion in isotropic (i.e., isotropic diffusion) and anisotropic fashion (i.e., quantitative anisotropy, QA) for fibers crossing 11 brain areas known to be significantly affected following mTBI. Specifically, we investigated how 6 weeks of daily morning blue light exposure therapy (compared to an amber-light placebo condition) impacted changes in white matter diffusion in individuals with mTBI. We observed a significant impact of the blue light treatment (relative to the placebo) on the amount of water diffusion (QA) for multiple brain areas, including the corpus callosum, anterior corona radiata, and thalamus. Moreover, many of these changes were associated with improvements in sleep latency and delayed memory. These findings suggest that blue wavelength light exposure may serve as one of the potential non-pharmacological treatments for facilitating structural and functional recovery following mTBI; they also support the use of QA as a reliable neuro-biomarker for mTBI therapies.

This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S- curvature. We show that the class of all such subsets can be naturally divided into two subclasses: The first subclass consists of those sets S for which the following holds: If two Riemannian manifolds have positive S- curvature then their connected sum also admits a Riemannian metric of positive S- curvature. The other subclass consists of those sets for which the normalized Ricci flow on a closed Riemannian manifold with positive S-curvature converges to a metric of constant positive sectional curvature. In the second part of the thesis, we study the behavior of Ricci flow for a manifold having positive S - curvature, where S is in the first subclass. More specifically, we study the Ricci flow for a special class of metrics on Sp+1 x S1 , p ≥ 4, which have positive isotropic curvature.

This study investigates the through-thickness behaviour of carbon/epoxy laminates. A through-thickness compression test regime was conducted utilising three specimen designs, which are waisted, hollow cylindrical and cubic specimens. An assessment and comparison of each specimen is given regarding their advantages and disadvantages in characterising the through-thickness response of [+45/-45/90/0]s quasi-isotropic AS4/8552 carbon/epoxy laminates. A finite element (FE) study of the three specimens is presented which results in specimen geometries that provided a macroscopically uniform stress response throughout the gauge length whilst also minimising other features such as stress concentrations. Further to the final geometries being presented, the method of manufacture for the laminate and machining processes for each of the specimens is given. A mesoscopic FE study is presented relating to the free-edge effects induced by through-thickness loading in quasi-isotropic laminates. The results presented show that free-edge effects will be present in the test specimens and will have a larger overall impact on the hollow cylindrical specimen. The free-edge effects also increase the stress concentrations present in the corners of the waisted and cubic specimens. Characteristic stress strain curves are presented for each specimen with strain data taken from post yield strain gauges attached to the specimens. The extracted initial Young's modulus Ez and Poisson's ratios vzx and vzy show a small variation between specimens. The strength values for the three specimens vary greatly with the waisted specimen being the strongest and cylindrical specimen the weakest, indicating that the chosen specimen geometry dominates failure. The experimental data will be used for test case 12 in the Second World Wide Failure Exercise (WWFE-II). A study is presented to predict the effective elastic properties of Z-pinned laminates. The materials under consideration are UD and [0/90]s cross-ply AS4/3501-6 carbon/epoxy laminates. Estimates on the effective properties are provided by two FE approaches and two analytical bounding approaches; namely Voigt and Reuss bounds and Walpole's bounding theory. The two FE approaches are based on extreme assumptions about the in-plane fibre volume fraction in the presence of Z-pins and provide a tight range of values in which the real result should lie. Furthermore, whilst the bounding methods are simple and in the case of Young's moduli produce very wide bounds the selection of the suitable bound result can lead to a good estimate in comparison with the FE data. Typically the best bounding method result for each elastic property is within 10% of the FE predictions.

The deep mining industry and civil engineering need to perform rock stability analyses during excavation projects. These analyses are closely related with displacements in tunnel contours. The ground reaction curve is a powerful tool to characterize these displacements that is widely used in the New Austrian Tunneling Method. However, the analytical solutions that exist are only applicable under isotropic stress conditions for deep tunnels. This study aims to investigate when it is possible using the analytical methods to determine the ground reaction curves with enough accuracy in the case of shallow tunnels under anisotropic in-situ stress conditions. The method begins with a literature study. After that, with the help of a 2D model, a comparison between the analytical and the numerical solutions for ground reaction curves at different depths and at different initial in-situ stress ratios was carried out. The results show that both crown and floor displacements deviate more from the analytical solution than the wall displacement. The crown and floor can even move upwards under high initial in-situ stress ratios for shallow tunnels. Because of that, the analytical solution of the ground reaction curve at shallow depths under anisotropic stress conditions should not be used. In the case of isotropic stress field conditions for the analysis in this study, the results given by the analytical solution agree with the numerical ones at depths higher than 14 times the radius of the tunnel. On the other hand, the difference between numerical and analytical solutions becomes higher while increasing the initial in-situ stress ratio, even for very deep tunnels. Furthermore, an empirical equation to obtain the displacements of the ground surface, tunnel wall and tunnel crown has been obtained after a multiple linear regression analysis.

The extensive industrialization and human expansion has caused environmental protection wildlife conservation to become paramount concerns of the 21st century. The ecosystems of oceans and seas have particularly been affected due to activities like oil spills and increased fishing. This has led to a growing interest in monitoring of the oceans and marine animals to detect signs of distress in aquatic species. However, acquisition of data from oceans to land has been a challenging and expensive task. The concept of Internet of Sea provides a solution to this data transfer between the ocean nodes, like animal tags or deployed floating transceivers, and our land Internet, and can potentially eliminate the need of expensive monitoring ships or underwater cables. The Internet of Sea is system that comprises of sensor nodes in the form of detachable marine animal tags as the data acquisition platforms and distributed floating transceivers as the intermedium nodes which then transfer the data to the base-stations located on lands. The data acquired by animal tags are first to be stored in the tag, and once the tag comes to the sea surface, the data is transferred to the nearby floating transceivers. The floating transceivers have multi- hopping capability so the data can be passed to the land base-stations through a small number of transceivers. Due to the specific geometric shapes and size constraints of the tag and floating transceivers, as well as the harsh ocean environment, novel integrated antennas are required for this type of system. In this thesis, we propose several antenna designs suitable for Internet of Sea applications. The first design is a quasi-isotropic Antenna in Package (AiP), operating in the Bluetooth band, which has been designed for semi-real-time monitoring. Secondly, a large frequency-ratio dual- band microstrip antenna array, working at Extended Global System for Mobile communications (E-GSM900), Long Range (LoRa), and Bluetooth bands, has been designed for large-area wireless communication. Lastly, a circularly polarized microstrip antenna array has also been designed for Global Positioning System (GPS). Throughout the work, the measured results are consistent with the design strategies and simulation results.

Laboratory experiments on components of complex systems such as gas turbines require many conditions to be met. Requirements to be met in order to simulate real world conditions include inlet flow conditions such as velocity profile, Reynold's number, and temperature. The methodology to be introduced designs a velocity profile generating screen to match real world conditions through the use of perforated plates. The velocity profile generating screen is an array of jets arranged in a manner to produce sections of different solidities, a ratio of area that obstructs fluid flow compared to that of the total area. In an effort to better understand the interaction between perforated plate sections of different solidities, a collection of experimental data sets is presented to characterize the plates. This includes identification of fluid flow regions with characterization of the flow dynamics, though the analysis of velocity and turbulence decay. The aim of this characterization is to determine how the perforated plate's solidity affects the velocity development downstream and the location at which the velocity profile being produced can be considered complete.

Cosmology is an attempt to mathematically describe the behaviour of the universe, the most commonly used models are the Friedmann-Lemaître-Robertson-Walker solutions. These models seem to be accurate for an old universe, which is homogeneous with low anisotropy. However for an earlier universe these models might not be that accurate or even correct. The almost non-existent anisotropy observed today might have played a bigger role in the earlier universe. For this reason we will study another model known as Bianchi Type I, where the universe is not necessarily isotropic. We utilize a 1+3-covariant formalism to obtain the equations that determine the behaviour of the universe and then use a tetrad formalism to complement the 1+3-covariant equations. Using these equations we examine the geometry of space-time and its dynamical properties. Finally we briefly discuss the different singularities possible and examine some special cases of geodesic movement.

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