Inequalities for discrete and continuous supremum operators Rastislav O , lhava In this thesis we study continuous and discrete supremum operators. In the first part we investigate general properties of Hardy-type operators involving suprema. The boundedness of supremum operators is used for characterization of interpo- lation spaces between two Marcinkiewicz spaces. In the second part we provide equivalent conditions for boundedness of supremum operators in the situation when the domain space in one of the classical Lorentz spaces Λp w1 or Γp w1 and the target space Λq w2 or Γq w2 . In the case p ≤ q we use inserting technique obtaining continuous conditions. In the setting of coefficients p > q we provide only partial results obtaining discrete conditions using discretization method. In the third part we deal with a three-weight inequality for an iterated discrete Hardy-type operator. We find its characterization which enables us to reduce the problematic case when the domain space is a weighted ℓp with p ∈ (0, 1) into the one with p = 1. This leads to a continuous analogue of investigated discrete inequality. The work consists of author's published and unpublished results along with material appearing in the literature.

A statistical investigation of the relationship between Lorentz force and Bursty Bulk Flow event (BBF) spatial location in the magnetotail is undertaken. Data is obtained in situ by the ESA Cluster II mission during the period July to October 2004. Firstly, a short introduction to BBFs and the Cluster mission is presented. Secondly, the curlometer method for determining Current densities in the Inner Central Plasma Sheet and its approximations are discussed. The curlometer method uses magnetic field density data from the Fluxgate Magnetometer (FGM) instrument and plasma velocities are obtained by the Hot Ion Analyzer (HIA) instrument. The satellite separation at the time of the measurement in the year 2004 was on the order of 1000 km. Results of the investigation are inconclusive. A few possible sources of error and reference material are mentioned.

Plasmonic metasurfaces enable functionalities that extend beyond the possibilities of classical optical materials and as a result, have gained significant research interest over the years. This thesis aims towards introducing plasmonic metamaterials and metasurfaces, a two-dimensional subset of metamaterials. The thesis also provides insights into the nonlinear optical responses from subwavelength metallic nanostructures manifesting as extraordinary physical phenomena like the second harmonic generation (SHG). The hydrodynamic Drude model is a theory that characterizes electron conduction in a hydrodynamic way to predict optical responses of metals. The thesis discusses the various contributions to the second-order optical nonlinearities from the terms in the hydrodynamic model: Coulomb, convection, and the Lorentz magnetic force. The significance of these terms, specifically the Lorentz magnetic term, is validated in contrast with existing research. The details of the work carried out to achieve a significant contribution to SHG from the Lorentz magnetic term are provided. A dominant Lorentz magnetic force for SHG was achieved through engineering T-shaped aperture arrays milled into a thin gold film. The dimensions of these structures were tuned for fundamental wavelength resonance. The structures exhibit both magnetic and electric field enhancements at the plasmonic resonance. Furthermore, a revised theoretical model is developed to accurately predict both linear and nonlinear optical responses of metamaterials. The model is based on the hydrodynamic Drude model and nonlinear scattering theory. Results from the finite difference time domain simulations performed on the metasurface are presented. It is observed that the T-shaped structure provides 65% greater nonlinear generation from the Lorentz magnetic term than the sum of the other two hydrodynamic terms. The influence of incident beam polarization on SHG conversion efficiency was also investigated. It was discovered that even though the contributions of hydrodynamic (Coulomb and convection) terms are maximum at 0◦ and 90◦, the metasurface shows maximum SHG intensity at 45◦ which indicates a dominant Lorentz magnetic term. Experimental validation was performed using the fabricated metasurface and a good agreement between the experiment and theoretical calculations was observed. Another aspect of the magnetic Lorentz force contribution, Bethe’s aperture theory was evaluated for a circular aperture at off-normal incident light. It is shown that the Lorentz force dominates the SHG by an order of magnitude at angled incidence where the generation is maximized. The angular dependence was observed to match the magnetic and electric dipole interaction effects as predicted from Bethe’s theory. The revised theory developed in this thesis predicts the linear and nonlinear optical responses of metamaterials including their angular dependency. The analysis and numerical calculations for a circular aperture agree well with past experiments. To conclude, the thesis provides an outlook on future developments in the field of nonlinear plasmonic research with regards to the development of highly efficient nonlinear metasurfaces through optimization of the Lorentz contributions. An insight into the recent developments in nanofabrication capabilities, design methodologies, nano-characterization techniques, modern electromagnetic simulations is discussed as avenues for future research in nanophotonic and nanoplasmonic device design and development. / Graduate

Motivated by A. Z. Petrov's classification of four-dimensional Lorentzian metrics, we provide an algebraic classification of the isometry-isotropy pairs of four-dimensional pseudo-Riemannian metrics admitting local slices with five-dimensional isometries contained in the Lorentz algebra. A purely Lie algebraic approach is applied with emphasis on the use of Lie theoretic invariants to distinguish invariant algebra-subalgebra pairs. This method yields an algorithm for identifying isometry-isotropy pairs subject to the aforementioned constraints.

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This thesis addresses the issue of the time variability of the fine structure constant, alpha. Recent claims of a varying alpha are set against the established standards of quantum electrodynamical theory and experiments. A study of the feasibility of extracting data on the time dependence of alpha using particles in Penning traps is compared to the results obtained by existing methods, including those using astrophysical data and those obtained in atomic clock experiments. Suggestions are made on the nature of trapped particles and the trapping fields.

Over the years numerous computational fluid dynamics models have been developed in order to study the fluid flow in gas and induction stirred ladles. These models are used to gain insight in the industrial processes used in ladle treatment of steel. A unified model of an induction stirred Ladle in two and three dimensions is presented. Induction stirring of molten steel is a coupled multi-physics phenomena involving electromagnetic and fluid flow. Models presented in this thesis gives a more accurate description of the real stirring conditions and flow pattern, by taking into account the multi-physics behavior of the induction stirring process in an induction stirred ladle. This thesis presents a formulation of coupled electromagnetic and fluid flow equations. The coupled electromagnetic and fluid flow equations are solved using the finite element method in two and three-dimensions. The simulation model is used to predict values of steel velocities and magnetic flux density. The simulation model is also used to predict the effect of increased current density on flow velocity. Magnetic flux density values obtained from the model are verified against experimental values. / QC 20120615

This Thesis is devoted to the study of topological phases of matter in optical platforms, focusing on non-Hermitian systems with gain and losses involving nonreciprocal elements, and fractional quantum Hall liquids where strong interactions play a central role.In the first part we investigated nonlinear Taiji micro-ring resonators in passive and active silicon photonics setups. Such resonators establish a unidirectional coupling between the two whispering-gallery modes circulating in their perimeter. We started by demonstrating that a single nonlinear Taiji resonator coupled to a bus waveguide breaks Lorentz reciprocity. When a saturable gain is added to a single Taiji resonator, a sufficiently strong unidirectional coupling rules out the possibility of lasing in one of the whispering-gallery modes with independence of the type of optical nonlinearity and gain saturation displayed by the material. This can be regarded as a dynamical time-reversal symmetry breaking. This effect is further enhanced by an optical Kerr nonlinearity. We showed that both ring and Taiji resonators can work as optical isolators over a broad frequency band in realistic operating conditions. Our proposal relies on the presence of a strong pump in a single direction: as a consequence four-wave mixing can only couple the pump with small intensity signals propagating in the same direction. The resulting nonreciprocal devices circumvent the restrictions imposed by dynamic reciprocity. We then studied two-dimensional arrays of ring and Taiji resonators realizing quantum spin-Hall topological insulator lasers. The strong unidirectional coupling present in Taiji resonator lattices promotes lasing with a well-defined chirality while considerably improving the slope efficiency and reducing the lasing threshold. Finally, we demonstrated that lasing in a single helical mode can be obtained in quantum spin-Hall lasers of Taiji resonators by exploiting the optical nonlinearity of the material. In the second part of this Thesis we dived into more speculative waters and explored fractional quantum Hall liquids of cold atoms and photons. We proposed strategies to experimentally access the fractional charge and anyonic statistics of the quasihole excitations arising in the bulk of such systems. Heavy impurities introduced inside a fractional quantum Hall droplet will bind quasiholes, forming composite objects that we label as anyonic molecules. Restricting ourselves to molecules formed by one quasihole and a single impurity, we find that the bound quasihole gives a finite contribution to the impurity mass, that we are able to ascertain by considering the first-order correction to the Born-Oppenheimer approximation. The effective charge and statistical parameter of the molecule are given by the sum of those of the impurity and the quasihole, respectively. While the mass and charge of such objects can be directly assessed by imaging the cyclotron orbit described by a single molecule, the anyonic statistics manifest as a rigid shift of the interference fringes in the differential scattering cross section describing a collision between two molecules.

A representation is a group homomorphism whose image is a group of invertible matrices. Representations and their associated matrices are analyzed through well-established techniques from linear algebra. We characterize representations by a unique decomposition into irreducible representations just as we characterize the decomposition of matrices into their eigenspaces. Through the study of these representations, we uncover mathematical relationships that underlie groups with physical applications. Due to physical symmetries, we study how the irreducible representations of groups that embody the actions of even the most basic rotations are utilized in the computation of irreducible representations groups that reflect more complicated mechanics, like the Poincar\'e Group. Further, we utilize the representations of the abstract braid group to gain key insights into understanding the behavior of anyonic systems in quantum mechanics. Finally, we explore the behavior of Fibonacci anyons for ways to understand to illustrate the underlying braid relations.