The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

Recent work has demonstrated the impressive efficacy of computing representations in hyperbolic space rather than in Euclidean space. This is especially true for multi-relational data and for data containing latent hierarchical structures. In this work, we seek to understand why this is the case. We reflect on the intrinsic properties of hyperbolic geometry and then zero in on one of these as a possible explanation for the performance improvements --- projection. To validate this hypothesis, we propose our projected cone model, $\mathcal{PC}$. This model is designed to capture the effects of projection while not exhibiting other distinguishing properties of hyperbolic geometry. We define the $\mathcal{PC}$ model and determine all of the properties we need in order to conduct machine learning experiments with it. The model is defined as the stereographic projection of a cone into a unit disk. This is analogous to the construction of the Beltrami-Poincaré model of hyperbolic geometry by stereographic projection of one sheet of a two-sheet hyperboloid into the unit disk. We determine the mapping formulae between the cone and the unit disk, its Riemannian metric, and the distance formula between two points in the $\mathcal{PC}$ model. We investigate the learning capacity of our model. Finally, we generalize our model to higher dimensions so that we can perform representation learning in higher dimensions with our $\mathcal{PC}$ model. Because generalizing models into higher dimensions can be difficult, we also introduce a baseline model for comparison. This is a product space model, $\mathcal{PCP}$. It is built up from our rigourously developed, two-dimensional version of the $\mathcal{PC}$ model. We run experiments and compare our results with those obtained by others using the Beltrami-Poincaré model. We find that our model performs almost as well as their Beltrami-Poincaré model, far outperforming representation learning in Euclidean space. We thus conclude that projection indeed is key in explaining the success which hyperbolic geometry brings to representation learning.

In this paper we use the Sobolev steepest descent method introduced by John W. Neuberger to solve the hyperbolic Monge-Ampère equation. First, we use the discrete Sobolev steepest descent method to find numerical solutions; we use several initial guesses, and explore the effect of some imposed boundary conditions on the solutions. Next, we prove convergence of the continuous Sobolev steepest descent to show local existence of solutions to the hyperbolic Monge-Ampère equation. Finally, we prove some results on the Sobolev gradients that mainly arise from general nonlinear differential equations.

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Yiu, Fa Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-58). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Elementary properties and notations of Hyperbolic space --- p.9 / Chapter 3 --- Poisson kernel and Conformal densities --- p.16 / Chapter 3.1 --- Poisson kernel --- p.17 / Chapter 3.2 --- Conformal densities --- p.19 / Chapter 4 --- Patterson construction and decomposition --- p.27 / Chapter 4.1 --- Patterson construction --- p.27 / Chapter 4.2 --- Patterson decomposition --- p.33 / Chapter 5 --- Bonahon surfaces and Grided surfaces --- p.39 / Chapter 5.1 --- Bonahon surfaces --- p.40 / Chapter 5.2 --- Grided surfaces --- p.46 / Chapter 6 --- Margulis number of Hyperbolic Manifolds --- p.51 / Margulis Number for Hypcrbolic 3-manifolds --- p.5 / Chapter 6.1 --- Gcomertrically finite groups --- p.51 / Chapter 6.2 --- Margulis number of Closed Hyperbolic Manifolds --- p.53 / Bibliography --- p.55

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

In this thesis, a novel fiber structure with advantages of high coupling efficiency, long working distance and better alignment tolerance has been presented. In this structure, the front-end of the singe-mode fiber (SMF) was spliced a graded-index fiber (GIF) with 50£gm core diameter. A hyperbolic-shaped lens was fabricated in the front-end of GIF. This novel fiber structure can increase the coupling efficiency of SMF effectively by using the wavefront-transfer characteristic of hyperbolic-shaped lens and the focusing characteristic of GIF. According to the simulation results, the optimized length of the GIF was 1160£gm. This novel fiber structure can reach to the coupling efficiency of 77% and working distance of 16£gm when the output power of laser diode was operated at 10mW and the radius curvature of lensed fiber was 12.74£gm. The lateral and longitudinal alignment tolerances of this fiber were 0.8£gm and 1.3£gm, respectively. In comparison with the conventional SMF lens, this novel fiber structure has longer working distance and better fiber alignment tolerance. Therefore, this structure can increase the package yield and reduce the fabrication cost for the application of laser module package.

Thesis (Ph. D.)--Rutgers University, 2009. / "Graduate Program in Mathematics." Includes bibliographical references (p. 51-52).

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This thesis develops two families of numerical methods, based upon rational approximations having distinct real poles, for solving first- and second-order parabolic/ hyperbolic partial differential equations. These methods are thirdand fourth-order accurate in space and time, and do not require the use of complex arithmetic. In these methods first- and second-order spatial derivatives are approximated by finite-difference approximations which produce systems of ordinary differential equations expressible in vector-matrix forms. Solutions of these systems satisfy recurrence relations which lead to the development of parallel algorithms suitable for computer architectures consisting of three or four processors. Finally, the methods are tested on advection, advection-diffusion and wave equations with constant coefficients.