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Tsoi, Hung Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 88-90). / Abstracts in English and Chinese. / Preface --- p.7 / Chapter 1 --- Kahler Manifold --- p.10 / Chapter 1.1 --- Hermitian Manifold --- p.12 / Chapter 1.2 --- Kahler Manifold --- p.13 / Chapter 1.2.1 --- "Positive (l,l)-form" --- p.15 / Chapter 2 --- Vector Bundle --- p.16 / Chapter 2.1 --- Holomorphic Vector Bundle and Connection --- p.17 / Chapter 2.2 --- Hermitian Connection and Chern Connection --- p.18 / Chapter 2.2.1 --- Existence of Chern connection on a holomorphic vector bundle --- p.19 / Chapter 2.3 --- Curvature --- p.21 / Chapter 2.4 --- Positivity of Vector Bundles --- p.23 / Chapter 2.5 --- Chern Classes and Holomorphic Line Bundle --- p.24 / Chapter 2.5.1 --- Chern class in axiomatic approach --- p.25 / Chapter 2.5.2 --- Chern class in algebraic topology --- p.26 / Chapter 2.5.3 --- Chern class in terms of curvature --- p.27 / Chapter 2.5.4 --- In the case of hermitian line bundle --- p.28 / Chapter 3 --- Analytic Technique on Kahler Manifold --- p.30 / Chapter 3.1 --- Dolbeault Cohomology --- p.30 / Chapter 3.2 --- Commutator Relations on Kahler Manifold --- p.31 / Chapter 3.2.1 --- Commutator relation on a line bundle --- p.32 / Chapter 3.3 --- Hodge Theory --- p.33 / Chapter 3.4 --- Bochner Technique --- p.35 / Chapter 3.4.1 --- Bochner-Kodaira-Nakano identity --- p.36 / Chapter 4 --- Kodaira Vanishing Theorem and L2 estimate of d --- p.38 / Chapter 4.1 --- Kodaira Vanishing Theorem --- p.39 / Chapter 4.2 --- Extension of Kodaira Vanishing Theorem by L2 Method --- p.44 / Chapter 4.2.1 --- Plurisubharmonic functions and weakly pseudoconvex Kahler manifold --- p.47 / Chapter 5 --- Multiplier Ideal Sheaf --- p.55 / Chapter 5.1 --- Algebraic Properties of Multiplier Ideal Sheaf --- p.56 / Chapter 5.2 --- Some Calculations of Multiplier Ideal Sheaf --- p.59 / Chapter 6 --- Nadel Vanishing Theorem --- p.62 / Chapter 6.1 --- Nadel Vanishing Theorem by L2 Estimate of d --- p.62 / Chapter 6.2 --- The Original Setting of Nadel --- p.64 / Chapter 6.2.1 --- S-bounded and S-null sequence --- p.65 / Chapter 6.2.2 --- Multiplier ideal sheaf by Nadel --- p.67 / Chapter 6.3 --- Nadel Vanishing Theorem by Computation of Cech Cohomology --- p.69 / Chapter 6.3.1 --- L2 estimate of d --- p.69 / Chapter 6.3.2 --- Koszul cochain --- p.70 / Chapter 6.3.3 --- The cohomology vanishing theorem --- p.73 / Chapter 7 --- Kawamata-Viehweg Vanishing Theorem --- p.77 / Chapter 7.1 --- Numerically Effective Line Bundle --- p.77 / Chapter 7.2 --- Kawamata-Viehweg Vanishing Theorem --- p.85 / Bibliography --- p.88

Recent results of string theory have shown that while the traditional cycles studied in Calabi-Yau 4-manifolds preserve half the spacetime supersymmetry, the more general class of Cayley cycles are novel in that they preserve only one quarter of it. Moreover, Cayley cycles play a crucial role in understanding mirror symmetry on Calabi-Yau 4-manifolds and Spin manifolds. Nonetheless, only very few nontrivial examples of Cayley cycles are known. In particular, it would be very useful to know interesting examples of Cayley cycles on the complex 4-torus. This thesis will develop key techniques for finding and constructing lattice periodic Cayley manifolds in Euclidean 8-space. These manifolds will project down to the complex 4-torus, yielding nontrivial Cayley cycles.

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

<p>The aim of this work has been to develop new knowledge about macromodelling of microsystems. Doing that, we have followed two different approaches for generating macromodels, namely model order reduction and lumped modelling. The latter is a rather mature method that has been widely recognized and used for a relatively long period of time. Model order reduction, on the other hand, is a relatively new area still in rapid development. Due to this, the parts considering reduced order modelling is strongly biased towards methodology and concepts, whereas parts on lumped modelling are biased towards systems and devices.</p><p>In the first part of this thesis, we focus on model order reduction. We introduce some approaches for reducing model order for linear systems, and we give an example related to squeeze-film damping. We then move on to investigate model order reduction of nonlinear systems, where we present and use the concept of invariant manifolds. While the concept of invariant manifolds is general, we utilize it for reducing models. An obvious advantage of using invariant manifold theory is that it offers a conceptually clear understanding of effects and behaviour of nonlinear system.</p><p>We exemplify and investigate the accuracy of one method for identifying invariant manifolds. The example is based on an industrialized dual-axis accelerometer.</p><p>A new geometrical interpretation of external forcing, relating to invariant manifolds, is presented. We show how this can be utilized to deal with external forcing in a manner consistent with the invariance property of the manifold. The interpretation also aids in reducing errors for reduce models.</p><p>We extend the asymptotic approach in a manner that makes it possible to create design-parameter sensitive models. We investigate an industrialized dual-axis accelerometer by means of the method and demonstrate capabilities of the method. We also discuss how manifolds for nonlinear dissipative systems can be found.</p><p>Focusing on lumped modelling, we analyse a microresonator. We also discuss the two analogies that can be used to build electrical equivalents of mechanical systems. It is shown how the f → V analogy, linking velocity to voltage, is the natural choice. General properties of lumped modelling are investigated using models with varying degrees of freedom.</p><p>Finally, we analyse an electromagnetic system, intended for levitating objects, and we demonstrate the scaling effects of the system. Furthermore, we prove the intrinsic stability of the system, although the floating disc will be slightly tilted. This is the first analysis done assessing the stability criterions of such a systems. The knowledge arising from the analysis gives strong indications on how such a system can be utilized, designed, and improved.</p>

Thesis (Ph. D. in Mathematics)--Vanderbilt University, May 2007. / Title from title screen. Includes bibliographical references.

The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.

For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.

We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.

On a compact closed manifold with edges live pseudodifferential operators which are block matrices of operators with additional edge conditions like boundary conditions in boundary value problems. They include Green, trace and potential operators along the edges, act in a kind of Sobolev spaces and form an algebra with a wealthy symbolic structure. We consider complexes of Fréchet spaces whose differentials are given by operators in this algebra. Since the algebra in question is a microlocalization of the Lie algebra of typical vector fields on a manifold with edges, such complexes are of great geometric interest. In particular, the de Rham and Dolbeault complexes on manifolds with edges fit into this framework. To each complex there correspond two sequences of symbols, one of the two controls the interior ellipticity while the other sequence controls the ellipticity at the edges. The elliptic complexes prove to be Fredholm, i.e., have a finite-dimensional cohomology. Using specific tools in the algebra of pseudodifferential operators we develop a Hodge theory for elliptic complexes and outline a few applications thereof.