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With time going by, urban morphological structures of Beijing and Shanghai have dramatic changes during last decades. These changes often ignored by citizen, but have big influence for human daily life. And the changes of urban morphologies should be easily recognized by citizen. There are many previous comparative studies between these two Chinese cities, and these studies focus on types of areas, such as environment, traffic, city planning and cultures etc.. There are also many comparative studies about using space syntax theory and geometrical statistics to study urban morphologies. However, there are not direct comparison urban morphological study between Beijing and Shanghai, which from multiple perspectives. In order to gain a better understanding of urban morphologies, this thesis take street networks of two Chinese cites as a research object, based on space syntax theory, as well the combination of traditional geometrical statistics, comparative analysis methods to systematic quantitative analyze and comparative study the different street networks of urban space in Beijing and Shanghai. This project work analyzes hierarchy of axial lines, which automatically generated from street networks, to do a morphological comparison from topological perspective. And it analyzes frequency distribution of axial lines’ included angles and length of axial lines to study urban morphologies from geometrical perspective. Results in the project seem to empirical study that, the well-connected streets are minority part, which all most distributed in the sample cities’ ring structures and center areas. Street networks constitute an obvious regular grid pattern of Beijing and a curves pattern of Shanghai. Based on the hierarchical levels of street networks, research samples have same hierarchical levels but without the same number of street lines. The included angles of axial lines have an exceptionally sharply peaked bimodal distribution for both cities and number of most connected street’s length do not increase so much from ring1 to ring6 for Beijing, but they have much change for Shanghai.

Jiang, Lei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 68-75). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Background and Motivation --- p.5 / Chapter 2.1 --- Foreshortening --- p.5 / Chapter 2.1.1 --- Vanishing Point --- p.6 / Chapter 2.1.2 --- Metric Rectification --- p.8 / Chapter 2.2 --- Content Aware Image Resizing --- p.11 / Chapter 2.3 --- Texture Deformation --- p.15 / Chapter 2.3.1 --- Shape from texture --- p.16 / Chapter 2.3.2 --- Shape from lattice --- p.18 / Chapter 3 --- Resizing on Facade --- p.21 / Chapter 3.1 --- Introduction --- p.21 / Chapter 3.2 --- Related Work --- p.23 / Chapter 3.3 --- Algorithm --- p.24 / Chapter 3.3.1 --- Facade Detection --- p.25 / Chapter 3.3.2 --- Facade Resizing --- p.32 / Chapter 3.4 --- Results --- p.34 / Chapter 4 --- Cell Texture Editing --- p.42 / Chapter 4.1 --- Introduction --- p.42 / Chapter 4.2 --- Related Work --- p.44 / Chapter 4.3 --- Our Approach --- p.46 / Chapter 4.3.1 --- Cell Detection --- p.47 / Chapter 4.3.2 --- Local Affine Estimation --- p.49 / Chapter 4.3.3 --- Affine Transformation Field --- p.52 / Chapter 4.4 --- Photo Editing Applications --- p.55 / Chapter 4.5 --- Discussion --- p.58 / Chapter 5 --- Conclusion --- p.65 / Bibliography --- p.67

In the context of moment maps and diffeomorphisms of Kähler manifolds, Donaldson introduced a fully nonlinear Monge-Ampère type equation. Among the conjectures he made about this equation is that the existence of solutions is equivalent to a positivity condition on the initial data. Weinkove later affirmed Donaldson's conjecture using a gradient flow for the equation in the space of Kähler potentials of the initial data. The topic of this thesis is the case when the initial data is merely semipositive and the domain is a closed Kähler surface. Regularity techniques for degenerate Monge-Ampère equations, specifically those coming from pluripotential theory, are used to prove the existence of a bounded, unique, weak solution. With the aid of a Nakai criterion, due to Lamari and Buchdahl, it is shown that this solution is smooth away from some curves of negative self-intersection.

In this thesis, results will be presented that pertain to the global regularity of solutions to boundary value problems having the general form \begin{align} F\left[D^2u-A(\,\cdot\,,u,Du)\right] &= B(\,\cdot\,,u,Du),\quad\text{in}\ \Omega^-,\notag\\ T_u(\Omega^-) &= \Omega^+, \end{align} where $A$, $B$, $T_u$ are all prescribed; and $\Omega^-$ along with $\Omega^+$ are bounded in $\mathbb{R}^n$, smooth and satisfying notions of c-convexity and c^*-convexity relative to one another (see [MTW05] for definitions). In particular, the case where $F$ is a quotient of symmetric functions of the eigenvalues of its argument matrix will be investigated. Ultimately, analogies to the global regularity result presented in [TW06] for the Optimal Transportation Problem to this new fully-nonlinear elliptic boundary value problem will be presented and proven. It will also be shown that the A3w condition (first presented in [MTW05]) is also necessary for global regularity in the case of (1). The core part of this research lies in proving various a priori estimates so that a method of continuity argument can be applied to get the existence of globally smooth solutions. The a priori estimates vary from those presented in [TW06], due to the structure of F, introducing some complications that are not present in the Optimal Transportation case.¶ In the final chapter of this thesis, the A3 condition will be reformulated and analysed on round spheres. The example cost-functions subsequently analysed have already been studied in the Euclidean case within [MTW05] and [TW06]. In this research, a stereographic projection is utilised to reformulate the A3 condition on round spheres for a general class of cost-functions, which are general functions of the geodesic distance as defined relative to the underlying round sphere. With this general expression, the A3 condition can be readily verified for a large class of cost-functions that depend on the metrics of round spheres, which is tantamount (combined with some geometric assumptions on the source and target domains) to the classical regularity for solutions of the Optimal Transportation Problem on round spheres.

We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.

We study a number of local and global classification problems in generalized complex geometry. Generalized complex geometry is a relatively new type of geometry which has applications to string theory and mirror symmetry. Symplectic and complex geometry are special cases. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a complex point arises from a holomorphic Poisson structure. In the proof we use a smoothed Newton’s method along the lines of Nash, Moser and Conn. In the second topic, we consider whether a given regular Poisson structure and transverse complex structure come from a generalized complex structure. We give cohomological criteria, and we find some counterexamples and some unexpected examples, including a compact, regular generalized complex manifold for which nearby symplectic leaves are not symplectomorphic. In the third topic, we consider generalized complex structures with nondegenerate type change; we describe a generalized Calabi-Yau structure induced on the type change locus, and prove a local normal form theorem near this locus. Finally, in the fourth topic, we give a classification of generalized complex principal bundles satisfying a certain transversality condition; in this case, there is a generalized flat connection, and the classification involves a monodromy map to the Courant automorphism group.

In 1976, Uhlenbeck used transversality theory to show that for certain families of elliptic operators, the property of having only simple eigenvalues is generic. As one application, she proved that on a closed Riemannian manifold, the eigenvalues of the Laplace-Beltrami operator Δg are all simple for a residual set of Cr metrics. In 2012, Enciso and Peralta-Salas established an analogue of Uhlenbeck's theorem for differential forms, showing that on a closed 3-manifold, there exists a residual set of Cr metrics such that the nonzero eigenvalues of the Hodge Laplacian Δg(k) on k-forms are all simple for 0 ≤ k ≤ 3. In this dissertation, we continue to address the question of whether Uhlenbeck's theorem can be extended to differential forms. In particular, we prove that for a residual set of Cr metrics, the nonzero eigenvalues of the Hodge Laplacian Δg(2) acting on coexact 2-forms on a closed 5-manifold have multiplicity 2. To prove our main result, we structure our argument around a study of the Beltrami operator *gd, which is related to the Hodge Laplacian by Δg(2) = -(*gd)2 when the operators are restricted to coexact 2-forms on a 5-manifold. We use techniques from perturbation theory to show that the Beltrami operator has only simple eigenvalues for a residual set of metrics. We further establish even eigenvalue multiplicities for the Hodge Laplacian acting on coexact k-forms in the more general setting n = 4 ℓ + 1 and k = 2 ℓ for ℓ ϵ N.

Title: Ricci flow and geometric analysis on manifolds Author: Jakub Eliáš Ústav: Matematický ústav UK Supervisor: doc. RNDr. Petr Somberg Ph.D., Matematický ústav UK Abstract: This thesis discusses basis aspects of the Ricci flow on manifolds with a view towards the ambient space construction. We start with the back- ground review of the Riemannian geometry and parabolic partial differential equations, and the Ricci flow problem on manifolds is established. Then we aim towards the formulation of the Ricci flow problem on ambient spaces and provide several basic examples. There are two main parts: the first consists of general theory needed to formulate our problem and strategy, while the second part consists of particular calculations associated with the Ricci flow problem. Keywords: Ricci flow, Ambient space, Ambient metric, Poincaré-Einstein metric. 1

In a recent study of the geometric analysis of various masterpieces of many periods dating from early Egyptian to contemporary times, the author noted with particular interest the structure of the paintings of the German Renaissance masters. It seemed that the Germans used a simpler geometric plan in their compositions than did the Italian Renaissance painters. The writer was inspired to make further investigation to determine if such a theory were true.