In this thesis we introduce the spherical, central, angular, halfspace and regression symmetry of random vectors and their measures. Firstly we deal with their mutual relations and equivalent expressions. We also study the uniqueness of the center of individual symmetries and other interesting properties. Then we define the halfspace, projection, spatial and regression multidimensional median and show their properties. Finally we look at the relationships between these medians and symmetric distributions. 1

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18794号 / 理博第4052号 / 新制||理||1583(附属図書館) / 31745 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川合 光, 准教授 福間 將文, 教授 田中 貴浩 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2014. / Symmetries and conservation laws of partial di erential equations (pdes) have been instrumental in giving new approaches for reducing pdes. In this dissertation, we study the symmetries and conservation laws of the two-dimensional Schr odingertype equation and the Benney-Luke equation, we use these quantities in the Double Reduction method which is used as a way to reduce the equations into a workable pdes or even an ordinary di erential equations. The symmetries, conservation laws and multipliers will be determined though di erent approaches. Some of the reductions of the Schr odinger equation produced some famous di erential equations that have been dealt with in detail in many texts.

Friction Stir Welding (FSW) is a solid-state joining process that is of special interest in joining aluminum and other alloys that are traditionally difficult to fusion weld. The energy required for this joining process is transmitted to the work-pieces through a rotating FSW tool. Modeling attempts, aimed at perfecting the process, rely on assumptions of the contact conditions present between the work-pieces and the FSW tool. Various studies have attempted to define these contact conditions. Both theoretical and experimental studies indicate the contact conditions between the work-piece and weld tool are unknown and may vary during the FSW process. To provide insight into the contact conditions, the objective of this study is to characterize the FSW nugget in terms of swept volume as indicated by the cross-sectional area and symmetry of the FSW nugget over a range of processing conditions.

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Mirror symmetry has been a significant area of research for geometry and physics for over two decades. Berglund and Hubsch proposed that for a certain family of singularities W, the so called "transposed" singularity WT should be the mirror partner of W. cite{BH} The techniques for constructing the orbifold LG models to test this conjecture were developed by FJR in cite{FJR} with a cohomological field theory generalized from the study of r-spin curves. The duality of LG A- and B-models became more elaborate when Krawitz cite{Krawitz} generalized the Intriligator-Vafa orbifold B-model to include contributions from more than one sector.This thesis presents a program written in Maple for explicitly computing bases for both LG A- and B-model rings, as well as the correlators for A-models to the extent of current knowledge. Included is a list of observations and conjectures drawn from computations done in the program.

Symmetry plays a crucial role in exploring the laws of nature. By exploiting some of the underlying analogies between the mathematical formalism of quantum mechanics and that of electrodynamics, in this dissertation we show that optics can provide a fertile ground for studying, observing, and utilizing some of the peculiar symmetries that are currently out of reach in other areas of physics. In particular, in this work, we investigate two important classes of symmetries, parity-time symmetry (PT) and supersymmetry (SUSY), within the context of classical optics. The presence of PT symmetry can lead to entirely real spectra in non-Hermitian systems. In optics, PT-symmetric structures involving balanced regions of gain and loss exhibit intriguing properties which are otherwise unattainable in traditional Hermitian systems. We show that selective PT symmetry breaking offers a new method for achieving single mode operation in laser cavities. Other interesting phenomena also arise in connection with PT periodic structures. Along these lines, we introduce a new class of optical lattices, the so called mesh lattices. Such arrays provide an ideal platform for observing a range of PT-related phenomena. We show that defect sates and solitons exist in such periodic environments exhibiting unusual behavior. We also investigate the scattering properties of PT-symmetric particles and we show that such structures can deflect light in a controllable manner. In the second part of this dissertation, we introduce the concept of supersymmetric optics. In this regard, we show that any optical structure can be paired with a superpartner with similar guided wave and scattering properties. As a result, the guided mode spectra of these optical waveguide systems can be judiciously engineered so as to realize new families of mode filters and mode division multiplexers and demultiplexers. We also present the first experimental demonstration of light dynamics in SUSY ladders of photonic lattices. In addition a new type of transformation optics based on supersymmetry is also explored. Finally, using the SUSY formalism in non-Hermitian settings, we identify more general families of complex optical potentials with real spectra.

Graphene nanoribbons, graphene based optical sensors, and grating based plasmonics are explored experimentally. Graphene nanoribbons exhibit highly insulating states that may allow for graphene based digital applications. We investigate the sensitivity of these states to local charged impurities in ultra high vacuum. We look into the possibility of isolating two-dimensional films of H-BN and BSCCO, and test for any interesting phenomena. We also assess graphene*s applicability for optical sensing by implementing a new style of spectral detector. Utilizing surface plasmon excitations nearby a graphene field-effect transistor we are able to produce a detector with wavelength sensitivity and selectivity in the visible range. Finally, we study another plasmonic phenomenon, and observe the resonant enhancement of diffraction into a symmetry-prohibited order in silver gratings.

We discuss the role of symmetries in the dissociative recombinations (DR) of three polyatomic ions, namely the linear HCO+ (formyl) ion and the two highly symmetric H+3 and H3O+ (hydronium) molecular ions. Regarding the HCO+ ion, we apply a quantum mechanical treatment using the Multi-channel Quantum Defect Theory (MQDT) formalism to describe the ion-electron scattering process. Our study takes into account the Renner-Teller effect in order to model the non Born-Oppenheimer vibronic coupling in linear polyatomic ions. The coupling has shown to represent the main mechanism responsible for electronic capturing in highly excited Rydberg states associated with excited vibrational levels of the ionic core. We consider all internal degrees of freedom of HCO+ and obtain the dissociative cross section as a function of the incident electron kinetic energy. We have also improved the theoretical approach by including the large permanent dipole moment of HCO+ using a generalization of the MQDT formalism. To our knowledge, this is the rst time the permanent dipole moment of an ion is included in a DR study. The obtained results are in good agreement with experimental data. We also study the DR of H+3 and H3O+ symmetric ions using a simpli ed theoretical treatment, which focuses on the key ingredient of the DR process, the electron capture in the rst excited degenerate vibrational normal mode of the ions through non Born-Oppenheimer Jahn-Teller coupling. For both ions the obtained cross sections are in very good agreement with the available experimental data. Moreover, in the case of H+3 , the results reproduce previous calculations from two independent theoretical studies. Finally, we investigate the role of symmetries in few body ultra-cold collisions by considering both three and four identical atoms systems. We derive allowed rearrangements of different fragments of the system, satisfying the complete symmetry of the molecular Hamiltonian. For that purpose we establish a correspondence between constants of motion of the system in di erent large-distance con gurations and irreducible representations of the total symmetry group. Selection rules (forbidden transitions) and allowed states, which depend on the fermionic or bosonic nature of the atoms, can be derived from these results.

This work investigates the modal properties of general cyclically symmetric systems and the multi-stage systems with cyclically symmetric stages. The work generalizes the modal properties of engineering applications, such as planetary gears, centrifugal pendulum vibration absorber (CPVA) systems, multi-stage planetary gears, etc., and provides methods to improve the computational efficiency to numerically solve the system modes when cyclically symmetric structures exist. Modal properties of cyclically symmetric systems with vibrating central components as three-dimensional rigid bodies are studied without any assumptions on the system matrix symmetries: asymmetric inertia matrix, damping, gyroscopic, and circulatory terms can be present. In the equation of motion of such a cyclically symmetric system, the matrix operators are proved to have properties related to the cyclic symmetry. These symmetry-related properties are used to prove the modal properties of general cyclically symmetric systems. Only three types of modes can exist: substructure modes, translational-tilting modes, and rotational-axial modes. Each mode type is characterized by specific central component modal deflections and substructure phase relations. Instead of solving the full eigenvalue problem,all vibration modes and natural frequencies can be obtained by solving smaller eigenvalue problems associated with each mode type. This computational advantage is dramatic for systems with many substructures or many degrees of freedom per substructure. Group theory is applied to further generalize the modal properties of cyclically symmetric systems when both rigid-body and compliant central components exist, such as planetary gears with an elastic continuum ring gear. The group theory for symmetry groups is introduced, and the group-theory-based modal analysis does not rely on any knowledge of the properties of system matrices in system equations of motion. The three types of modes (substructure modes, translational-tilting modes, and rotational-axial modes) are characterized by specific rigid-body central component modal defections, substructure phase relations, and nodal diameter components of compliant central components. The general formulation of reduced eigenvalue problems for each mode type is obtained through group-theory-based method, and it applies to discrete, continuous, or hybrid discrete-continuous cyclically symmetric systems. The group-theory-based modal analysis also applies to systems with other symmetry types. The group-theory-based modal analysis is generalized to analyze the multi-stage systems that are composed of symmetric stages coupled through the motions of rigid-body central components. The proposed group-theory-based modal analysis applies to multi-stage systems with cyclically symmetric stages, such as multi-stage planetary gears and CPVA systems with multiple groups of absorbers. The method also applies to multi-stage systems with component stages that have different types of symmetry. For a multi-stage system with symmetric stages, a unitary transformation matrix can be built through an algorithmic and computationally inexpensive procedure. The obtained unitary transformation matrix provides the foundation to analyze the modal properties based on the principles of group-theory-based modal analysis. For general multi-stage systems with symmetric component stages, the vibration modes are classified into two general types, single-stage substructure modes and overall modes, according to the non-zero modal deflections in each component stage. Reduced eigenvalue problems for each mode type are formulated to reduce the computational cost for eigensolutions. Finite element models of multi-stage bladed disk assemblies consist of multiple cyclically symmetric bladed disks that are coupled through the boundary nodes at the inter-stage interface. To improve the computational efficiency of calculating the full system modes, a numerical method is proposed by combination of the multi-stage cyclic symmetry reduction method and the subspace iteration method. Compared to the multi-stage cyclic symmetry reduction method, the proposed method improves the accuracy of obtained eigensolutions through an iterative process that is derived from the subspace iteration method. Based on the cyclic symmetry in each component stage of bladed disk, the proposed iterative method that can be performed using single stage sector models only, instead of using matrix operators for the full multi-stage bladed disks. Parallel computations can be performed in the proposed iterative method, and the computational speed for eigensolutions can be increased significantly. / Doctor of Philosophy / Cyclically symmetric structures exist in many engineering applications such as bladed disks, circular plates, planetary gears, centrifugal pendulum vibration absorbers (CPVA), etc. During steady operation, these cyclically symmetric systems are subjected to traveling wave dynamic loading. Component vibrations result in undesirable effects, including high cycle fatigue (HCF) failure, noise, performance reduction, etc. Knowledge of the modal properties of cyclically symmetric systems is helpful to analyze the system forced response and understand experimental modal testing. In this work, single stage cyclically symmetric systems are proved to have highly structured modes. The single stage systems considered in this work can have both rigid bodies and elastic continua as components. Group theory is used to study the modal properties, including the system mode types and the characteristics of different mode types. All the vibration modes of single stage cyclically symmetric systems can be solved from reduced eigenvalue problems. The methodology also applies to systems with other types of symmetry. For multi-stage systems with cyclically symmetric substructures, such as multi-stage planetary gears, a group-theory-based method is developed to analyze the modal properties. For industrial applications, such as multi-stage bladed disk assemblies, this work develops an iterative method to facilitate the calculations of system modes. The modal properties and methods for solving system modes apply to mechanical systems, including CPVA systems, the single/multi-stage planetary gears in power transmission systems, bladed disk assemblies in turbines, circular plates, elastic rings, etc.