Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions

We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we further describe.Comment: v1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added connections with Masuoka's families KA and KB, description of K3 in KD(n), lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for KD(6), misc improvements, accepted for publication in Journal of Algebra and Its Application

Numerical Solution of Static and Dynamic Problems of Nanobeams and Nanoplates

Recently nanotechnology has become a challenging area of research. Accordingly, a new class of materials with revolutionary properties and devices with enhanced functionality has been developed by various researchers. Structural elements such as beams, membranes and plates in micro or nanoscale have a vast range of applications. Conducting experiments at nanoscale size is quite difficult and so development of appropriate mathematical models plays an important role. Among various size dependent theories, nonlocal elasticity theory pioneered by Eringen is being increasingly used for reliable and better analysis of nanostructures. Finding solutions for governing partial differential equations are the key factor in static and dynamic analyses of nanostructures. It is sometimes difficult to find exact or closed-form solutions for these differential equations. As such, few approximate methods have been developed by other researchers. But, the existing methods may not handle all sets of boundary conditions and sometimes those are problem dependent. Accordingly, computationally efficient numerical methods have been developed here for better understanding of static and dynamic behaviors of nanostructures. Also these numerical methods can handle all classical boundary conditions of the static and dynamic problems of nanobeams and nanoplates with ease. It may be noted that application of numerical methods converts bending problem to system of equations while buckling and vibration problems to generalized eigen value problems. The present thesis first investigates bending of nanobeams and nanoplates. Next buckling and vibration of the above nanostructural members are studied by solving the corresponding partial differential equations. In the above regard, various beam and plate theories are considered for the analysis and corresponding results are reported after the convergence study and validation in special cases wherever possible. Finally, few complicating effects are also considered in some of the problems. As regards, structural members (nanobeams and nanoplates) with variable material properties are frequently used in engineering applications to satisfy various requirements. For efficient design of nanostructures, sometimes non-uniform material properties of the nano-components should also be studied. As such, we have considered here non-uniform material properties of nanobeams and nanoplates and investigated the deflection in static problems and vibration characteristics in vibration problems. Similarly, other complicating effects such as surrounding medium and temperature are important in the nanotechnology applications too. Accordingly, the effect of these complicating effects on the nanobeams and nanoplates have also been investigated in detail. It is worth mentioning that Rayleigh-Ritz and differential quadrature methods have been used to solve the above said problems. In the Rayleigh-Ritz method, simple and boundary characteristic orthogonal polynomials have been used as shape functions. Use of boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method has some advantages over other shape functions. This is because of the fact that some of the entries of stiffness, mass and buckling matrices become either one or zero. On the other hand, Differential Quadrature (DQ) method is also a computationally efficient method which can be used to solve higher order partial differential equations that may handle all sets of classical boundary conditions. Accordingly DQ method has also been used in solving the problems of nanobeams with complicating effects. In view of the above, systematic study of static and dynamic problems of nanobeams and nanoplates are done after reviewing the existing ones. Various new results of the above problems are reported in term of Figures and Tables. The new results obtained through the above mathematical models may serve as bench mark and those may certainly be used by design engineers and practitioners to validate their experimental work for better design of the related nanostructures

Slip table dynamic behavior

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1995.Includes bibliographical references (leaves 224-227).by Elizabeth Therese Smith.M.S

Polarisation properties of exciton-polaritons in semiconductor microcavities

Interactions of exciton-polaritons in semiconductor microcavities and the resulting polarisation dynamics are investigated theoretically. Within the coboson framework of polariton-polariton scattering, it is shown that the matrix element of direct Coulomb scattering is proportional to the transferred momentum, q, cubed in the limit of small q. In the same limit, the magnitude of superexchange/exchange interactions can be considered constant. These results are applied to the elastic circle geometry, where a system of equations describing the steady-state pseudospin components is derived. It is shown, that for this geometry, polariton-polariton scattering can account for the generation of circularly/linearly polarised final states from linearly/circularly polarised initial states, depolarisation and the generation of spin currents. In the low density regime polaritons are good bosons and the dynamics of polariton Bose-Einstein condensation (BEC) are investigated. A stochastic model is derived, resulting in a Langevin type equation describing the time dynamics of the condensate spinor order parameter. The build up in condensate polarisation degree is shown to evidence macroscopic ground state population, while the stochastic choice of polarisation vector evidences the symmetry breaking nature of the phase transition. The decrease of polarisation degree above threshold is demonstrated to be a consequence of polariton-polariton interactions, a result which is complemented by recent experimental work. The stochastic model is extended to include Josephson coupling of spatially separate condensates. The coupling results in polarisation and phase correlations between the condensates, explaining the polarisation locking and spatial coherence seen experimentally. Finally, the effect of polarisation pinning by local effective fields is examined

Active control of particulate contamination for a Leo spacecraft environment

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1993.Includes bibliographical references (p. 95-97).by John Charles Conger.M.S