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Wu, Lijiang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 102-105). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.6 / Chapter 1 --- Background knowledge --- p.9 / Chapter 1.1 --- Comparison theorems --- p.9 / Chapter 1.2 --- Bochner techniques --- p.13 / Chapter 1.3 --- Eigenvalue estimates for Laplacian operator --- p.14 / Chapter 1.4 --- Spectral theory for Schrodinger operator on Rieman- nian manifolds --- p.16 / Chapter 2 --- Vanishing theorems --- p.20 / Chapter 2.1 --- Liouville type theorem for Lp subharmonic functions --- p.20 / Chapter 2.2 --- Generalized type of vanishing theorem --- p.21 / Chapter 3 --- Finite dimensionality results --- p.28 / Chapter 3.1 --- Three types of integral inequalities --- p.28 / Chapter 3.2 --- Weak Harnack inequality --- p.34 / Chapter 3.3 --- Li's abstract finite dimensionality theorem --- p.37 / Chapter 3.4 --- Applications of the finite dimensionality theorem --- p.42 / Chapter 4 --- Ends of Riemannian manifolds --- p.48 / Chapter 4.1 --- Green's function --- p.48 / Chapter 4.2 --- Ends and harmonic functions --- p.53 / Chapter 4.3 --- Some topological applications --- p.72 / Chapter 5 --- Splitting theorems --- p.79 / Chapter 5.1 --- Splitting theorems for manifolds with non-negative Ricci curvature --- p.79 / Chapter 5.2 --- Splitting theorems for manifolds of Ricci curvature with a negative lower bound --- p.83 / Chapter 5.3 --- Manifolds with the maximal possible eigenvalue --- p.93 / Bibliography --- p.102

In this thesis we present a general framework of geometric partial differential equations from the viewpoint of geometric energy functional. The proposed geometric functional involves the Gaussian curvature, the mean curvature and the squared norms of their gradients. The geometric partial differential equations are given as the Euler-Lagrangian Equations of the geometric energy functionals by using the calculus of variation method. As a special example, we focus on Gaussian curvature related geometric energy functionals and the corresponding partial differential equations. We present three numerical methods to solve the resulting geometric partial differential equations: the direct discretization method, the finite element method and the level set method. We test these numerical schemes with a large class of geometric models. Potential applications of our proposed geometric partial differential equations include mesh optimization, surface smoothing, surface blending, surface restoration and physical simulation. Finally, we point out some possible directions of future work including singular analysis of the derived geometric partial differential equations and numerical error estimates of our numerical schemes. / Yan, Yinhui. / "September 2008." / Adviser: Kwong Chung Piney. / Source: Dissertation Abstracts International, Volume: 73-01, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 121-134). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.

<p>In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere, and subsets thereof. In particular, we are interested in Min-Oo's conjecture that there exist no metrics on the upper hemisphere having scalar curvature greater or equal to that of the standard spherical metric, while satisfying certain natural geometric boundary conditions.</p><p>While the conjecture as originally stated has recently been disproved, there are still many interesting modications to consider. For instance, it has been shown that Min-Oo's rigidity conjecture holds on sufficiently small geodesic balls contained in the upper hemisphere, for metrics sufficiently close to the spherical metric. We show that this local rigidity phenomena can be extended to a larger class of domains in the hemisphere, in particular finding that it holds on larger geodesic balls, and on certain domains other than geodesic balls (which necessarily have more complicated boundary geometry). We discuss a possible method for finding the largest possible domain on which the local rigidity theorem is true, and give a Morse-theoretic interpretation of the problem.</p><p>Another interesting open question is whether or not such a rigidity statement holds for metrics that are not close to the spherical metric. We find that a scalar curvature rigidity theorem can be proved for metrics on sufficiently small geodesic balls in the hemisphere, provided certain additional geometric constraints are satisfied.</p> / Dissertation

By applying geometric analysis to some experimental maze situations the present study attempted to determine if a continuity in the responding of experimental Ss existed. This continuity in responding might suggest the presence of alternative explanations for the behavior of these Ss in some maze problems. The study made use of a modified version of the Tolman, Ritchie, and Kalish (1946a) experiment using six runways during training rather than one. The results of the study show that three of the six groups obtained the identical angle of choice, angle between the runway trained on and the runway chosen during the experimental trial, indicating the possibility of an underlying behavioral factor determining this continuity in responding.

In part I of this dissertation, certain rich geometric properties hidden behind quadratic 0-1 programming are investigated. Especially, we derive new lower bounding methods and variable fixation techniques for quadratic 0-1 optimization problems by investigating geometric features of the ellipse contour of a (perturbed) convex quadratic function. These findings further lead to some new optimality conditions for quadratic 0-1 programming. Integrating these novel solution schemes into a proposed solution algorithm of a branch-and-bound type, we obtain promising preliminary computational results. / In part II of this dissertation, we present new results of the duality gap between the binary quadratic optimization problem and its Lagrangian dual. We first derive a necessary and sufficient condition for the zero duality gap and discuss its relationship with the polynomial solvability of the problem. We then characterize the zeroness of duality gap by the distance, delta, between the binary set and certain affine space C. Finally, we discuss a computational procedure of the distance delta. These results provide new insights into the duality gap and polynomial solvability of binary quadratic optimization problems. / The unconstraint quadratic binary problem (UBQP), as a classical combinatorial problem, finds wide applications in broad field and human activities including engineering, science, finance, etc. The NP-hardness of the combinatorial problems makes a great challenge to solve the ( UBQP). The main purpose of this research is to develop high performance solution method for solving (UBQP) via the geometric properties of the objective ellipse contour and the optimal solution. This research makes several contributions to advance the state-of-the-art of geometric approach of (UBQP). These contributions include both theoretical and numerical aspects as stated below. / Liu, Chunli. / Adviser: Duan Li. / Source: Dissertation Abstracts International, Volume: 70-06, Section: B, page: 3764. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 140-153). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

In the setting of Riemannian manifolds with nonnegative Ricci curvature, we provide geometric inequalities as consequences of the Monotonicity Formulas holding along the flow of the level sets of the p-capacitary potential. The work is divided into three parts. (1) In the first part, we describe the asymptotic behaviour of the p-capactitary potential in a natural class of Riemannian manifolds. (2) The second part is devoted to the proof of our Monotonicity-Rigidity Theorems. (3) In the last part, we apply the Monotonicity Theorems to obtain geometric inequalities, focusing on the Extended Minkowski Inequality.

This thesis introduces a new lacunarity analysis add-in for ArcGIS.

<p>We explore a model of dark matter called wave dark matter (also known as scalar field dark matter and boson stars) which has recently been motivated by a new geometric perspective by Bray. Wave dark matter describes dark matter as a scalar field which satisfies the Einstein-Klein-Gordon equations. These equations rely on a fundamental constant Upsilon (also known as the ``mass term'' of the Klein-Gordon equation). Specifically, in this dissertation, we study spherically symmetric wave dark matter and compare these results with observations of dwarf spheroidal galaxies as a first attempt to compare the implications of the theory of wave dark matter with actual observations of dark matter. This includes finding a first estimate of the fundamental constant Upsilon.</p><p>In the introductory Chapter 1, we present some preliminary background material to define and motivate the study of wave dark matter and describe some of the properties of dwarf spheroidal galaxies.</p><p>In Chapter 2, we present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-areal coordinates and its properties. In particular, we show how the metric component functions chosen are extremely compatible with notions in Newtonian mechanics. We also show the monotonicity of the Hawking mass in these coordinates. Finally, we discuss how these coordinates and the metric can be used to solve the spherically symmetric Einstein-Klein-Gordon equations.</p><p>In Chapter 3, we explore spherically symmetric solutions to the Einstein-Klein-Gordon equations, the defining equations of wave dark matter, where the scalar field is of the form f(t,r) = exp(i omega t) F(r) for some constant omega in R and complex-valued function F(r). We show that the corresponding metric is static if and only if F(r) = h(r)exp(i a) for some constant a in R and real-valued function h(r). We describe the behavior of the resulting solutions, which are called spherically symmetric static states of wave dark matter. We also describe how, in the low field limit, the parameters defining these static states are related and show that these relationships imply important properties of the static states.</p><p>In Chapter 4, we compare the wave dark matter model to observations to obtain a working value of Upsilon. Specifically, we compare the mass profiles of spherically symmetric static states of wave dark matter to the Burkert mass profiles that have been shown by Salucci et al. to predict well the velocity dispersion profiles of the eight classical dwarf spheroidal galaxies. We show that a reasonable working value for the fundamental constant in the wave dark matter model is Upsilon = 50 yr^(-1). We also show that under precise assumptions the value of Upsilon can be bounded above by 1000 yr^(-1).</p><p>In order to study non-static solutions of the spherically symmetric Einstein-Klein-Gordon equations, we need to be able to evolve these equations through time numerically. Chapter 5 is concerned with presenting the numerical scheme we will use to solve the spherically symmetric Einstein-Klein-Gordon equations in our future work. We will discuss how to appropriately implement the boundary conditions into the scheme as well as some artificial dissipation. We will also discuss the accuracy and stability of the scheme. Finally, we will present some examples that show the scheme in action.</p><p>In Chapter 6, we summarize our results. Finally, Appendix A contains a derivation of the Einstein-Klein-Gordon equations from its corresponding action.</p> / Dissertation

Thesis (Ph. D.)--University of Alberta, 2010. / Title from pdf file main screen (viewed on July 13, 2010). A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics, Department of Mathematical and Statistical Sciences, University of Alberta. Includes bibliographical references.