Thesis (Ph.D.)--University of Melbourne, Dept. of Mathematics and Statistics, 2006. / Typescript. Includes bibliographical references (leaves 87-90).

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

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The structure of Euclidean, spherical, and hyperbolic geometries are compared, considering specifically postulates, curvature of the plane, and visual models. Implications for distance, the sum of the angles of triangles, and circumference to diameter ratios are investigated. / text

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The primary aim of this thesis is to construct explicit examples of four dimensional hyperbolic link complements. Using the theory of Kirby diagrams and Kirby calculus we set up a general framework that one can use to attack such a problem. We use this framework to construct explicit examples in a smooth standard S⁴ and a smooth standard S² x S². We then characterise which homeomorphism types of smooth simply connected closed 4-manifolds can admit a hyperbolic link complement, along the way giving constructions of explicit examples.

<p>In this thesis we investigate and compare two recently developed models of the option valuation according to the Swedish market. The first model is the Stochastic Volatility model with jumps in the stock price and the volatility (SVJJ) and the second is the Hyperbolic model. First of all we make brief introduction about the valuation of derivatives and considered models. Then we introduce methods for the estimation of parameters for each model. To solve this problem for the SVJJ model we use the Empirical Characteristic Function Estimation and for the Hyperbolic we use the Maximum Likelihood Method. Before explicit calculations (with estimated parameters) we describe the derivation of the pricing formula which is based on characteristic functions and densities. In conclusion we made numerical valuations of the call option prices for the OMXS30 index on the Swedish Stock Exchange. The main idea of this thesis is to compare 2 different models using numerical methods and the real data sets. To achieve this goal we firstly, compare the empirical characteristic function obtained from the market and the analytical ones for estimated parameters in case of both models. Secondly, we make a comparison of calculated call option prices and produce the summary.</p>

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.