Master of Architecture / This thesis explores the transformative potential of designing spaces that prioritize healing, connection, and mindfulness, especially for those with post-traumatic stress disorder (PTSD). It uses intentional design choices like natural light, wood, water, and granite to create a relaxing and grounding environment. Geometric Islamic patterns are incorporated to enhance visual appeal and provide a sense of order. By prioritizing healing, connection, and mindfulness, we can create spaces that alleviate PTSD symptoms and empower individuals on their journey towards healing and well-being.

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This thesis describes an analytical approach to the optical design of four-mirror anastigmatic optical systems. In all cases investigated here the object is at infinity. In the introduction the field of reflecting, or "catoptric", optical system design is discussed and given some historical context. The concept of the "simplest possible reflecting anastigmat" is raised in connection with Plate Diagram analysis. It is shown that four-plate systems are in general the simplest possible anastigmats, and that four-plate systems comprised of four spherical mirrors are the last family of "simplest possible reflecting anastigmats" for which the complete solution set remains unknown. In chapter 2 third-order aberration coefficients in wavefront measure are derived in a form that is particularly suitable for Plate Diagram analysis. These coefficients are subsequently used to describe the Plate Diagram, and to detail the application of the Plate Diagram to the survey of all possible solutions for four-spherical-mirror anastigmats. The Plate Diagram technique is also generalized to investigate its use as an optical design tool. In the example given a generalized Plate Diagram approach is used to determine solutions for four-mirror anastigmats with a prescribed first-order layout and a minimum number of conicoids. In chapter 3 results are presented for the survey of four-spherical-mirror anastigmats in which all elements are required to be smaller than the primary mirror. Two novel families of four-spherical-mirror anastigmats are presented and these are shown to be the only examples of four-spherical-mirror systems that exist under the given constraints. Chapter 4 gives an example of the application of Plate Diagram analysis to the design of an anastigmatic system with a useful first-order layout and a minimum number of conicoid mirrors. It is shown that systems with useful first-order layouts and only one conicoid mirror can be obtained using this method. In chapter 5 results are presented of the survey of all remaining four-spherical-mirror anastigmatic systems: that is systems in which elements are allowed to exceed the diameter of the entrance pupil, which includes systems with concave and convex primary mirrors. A wide variety of solutions are presented and classified according to both the underlying geometry of the solutions and the first-order layouts. Of these systems only one has been reported in previously published literature. The results presented in this thesis complete the set of "four-plate" reflecting anastigmats, and it can now be said that all possible solutions for four-spherical-mirror anastigmatic systems have been determined.

A subaperture configuration to test aspheric optical components or systems is developed. The method developed has potential for testing rotationally symmetric aspheres without the use of null lenses. The aperture is divided into annular subaperture regions and an interferometer is refocused for each region to reduce the fringe density. Essential mathematical treatments involving annular subapertures, such as annular Zernike polynomials, are provided in detail. Numerical and experimental validations of the algorithm are described. Tolerance analyses on subaperture measurements are given. Computer programs were written to link the subaperture Zernike coefficients and to determine the full-aperture Zernike coefficients.

Wave radiation is an important process in many geophysical flows. In particular, it is by wave radiation that flows may adjust to a state for which the dynamics is slow. Such a state is described as “balanced”, meaning there is an approximate balance between the Coriolis force and horizontal pressure gradients, and between buoyancy and vertical pressure gradients. In this thesis, wave radiation processes relevant to these enormously complex flows are studied through the use of some highly simplified models, and a parallel aim is to develop accurate numerical techniques for doing so. This thesis is divided into three main parts. 1. We consider accurate numerical boundary conditions for various equations which support wave radiation to infinity. Particular attention is given to discretely non-reflecting boundary conditions, which are derived directly from a discretised scheme. Such a boundary condition is studied in the case of the 1-d Klein-Gordon equation. The limitations concerning the practical implementation of this scheme are explored and some possible improvements are suggested. A stability analysis is developed which yields a simple stability criterion that is useful when tuning the boundary condition. The practical use of higher-order boundary conditions for the 2-d shallow water equations is also explored; the accuracy of such a method is assessed when combined with a particular interior scheme, and an analysis based on matrix pseudospectra reveals something of the stability of such a method. 2. Large-scale atmospheric and oceanic flows are examples of systems with a wide timescale separation, determined by a small parameter. In addition they both undergo constant random forcing. The five component Lorenz-Krishnamurthy system is a system with a timescale separation controlled by a small parameter, and we employ it as a model of the forced ocean by further adding a random forcing of the slow variables, and introduce wave radiation to infinity by the addition of a dispersive PDE. The dynamics are reduced by deriving balance relations, and numerical experiments are used to assess the effects of energy radiation by fast waves. 3. We study quasimodes, which demonstrate the existence of associated Landau poles of a system. In this thesis, we consider a simple model of wave radiation that exhibits quasimodes, that allows us to derive some explicit analytical results, as opposed to physically realistic geophysical fluid systems for which such results are often unavailable, necessitating recourse to numerical techniques. The growth rates obtained for this system, which is an extension of one considered by Lamb, are confirmed using numerical experiments.

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In the first part of this thesis, we demonstrate theory and computations for finite-energy line defect solutions in an improvement of Ericksen-Leslie liquid crystal theory. Planar director fields are considered in two and three space dimensions, and we demonstrate straight as well as loop disclination solutions. The possibility of static balance of forces in the presence of a disclination and in the absence of ow and body forces is discussed. The work exploits an implicit conceptual connection between the Weingarten-Volterra characterization of possible jumps in certain potential fields and the Stokes-Helmholtz resolution of vector fields. The theoretical basis of our work is compared and contrasted with the theory of Volterra disclinations in elasticity. Physical reasoning precluding a gauge-invariant structure for the model is also presented. In part II of the thesis, the time-harmonic Schrodinger equation with periodic potential is considered. We derive the asymptotic form of the scattering wave function in the periodic space and investigate the possibility of its application as a DtN non-reflecting boundary condition. Moreover, we study the perfectly matched layer method for this problem and show that it is a reliable method, which converges rapidly to the exact solution, as the thickness of the absorbing layer increases. Moreover, we use the tight-binding method to numerically solve the Schrodinger equation for Graphene sheets, symmetry-adapted Carbon nanotubes and DNA molecules to demonstrate their electronic behavior in the presence of local defects. The results for Y-junction Carbon nanotubes depict very interesting properties and confirms the predictions for their application as new transistors.

This thesis describes an analytical approach to the optical design of four-mirror anastigmatic optical systems. In all cases investigated here the object is at infinity. In the introduction the field of reflecting, or "catoptric", optical system design is discussed and given some historical context. The concept of the "simplest possible reflecting anastigmat" is raised in connection with Plate Diagram analysis. It is shown that four-plate systems are in general the simplest possible anastigmats, and that four-plate systems comprised of four spherical mirrors are the last family of "simplest possible reflecting anastigmats" for which the complete solution set remains unknown. In chapter 2 third-order aberration coefficients in wavefront measure are derived in a form that is particularly suitable for Plate Diagram analysis. These coefficients are subsequently used to describe the Plate Diagram, and to detail the application of the Plate Diagram to the survey of all possible solutions for four-spherical-mirror anastigmats. The Plate Diagram technique is also generalized to investigate its use as an optical design tool. In the example given a generalized Plate Diagram approach is used to determine solutions for four-mirror anastigmats with a prescribed first-order layout and a minimum number of conicoids. In chapter 3 results are presented for the survey of four-spherical-mirror anastigmats in which all elements are required to be smaller than the primary mirror. Two novel families of four-spherical-mirror anastigmats are presented and these are shown to be the only examples of four-spherical-mirror systems that exist under the given constraints. Chapter 4 gives an example of the application of Plate Diagram analysis to the design of an anastigmatic system with a useful first-order layout and a minimum number of conicoid mirrors. It is shown that systems with useful first-order layouts and only one conicoid mirror can be obtained using this method. In chapter 5 results are presented of the survey of all remaining four-spherical-mirror anastigmatic systems: that is systems in which elements are allowed to exceed the diameter of the entrance pupil, which includes systems with concave and convex primary mirrors. A wide variety of solutions are presented and classified according to both the underlying geometry of the solutions and the first-order layouts. Of these systems only one has been reported in previously published literature. The results presented in this thesis complete the set of "four-plate" reflecting anastigmats, and it can now be said that all possible solutions for four-spherical-mirror anastigmatic systems have been determined.

Conventional wireless techniques are becoming inadequate for beyond fifth-generation (5G) networks due to latency and bandwidth considerations. To increase the wireless network throughput and improve wireless communication systems’ error performance, we propose physical layer network coding (PNC) in an Intelligent Reflecting Surface (IRS)-assisted environment. We consider an IRS-aided butterfly network, where we propose an algorithm for obtaining the optimal IRS phases. Also, analytic expressions for the bit error rate (BER) are derived. The numerical results demonstrate that the scheme proposed in this thesis significantly enhances the BER performance. The proposed scheme is compared to traditional network coding without IRS. For instance, at a target BER of 10−3, 28 dB and 0.75 dB signal to noise ratio (SNR) gains are achieved at the relay and destination node of the 32-element IRS-assisted butterfly network model.

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