http://hdl.handle.net/1957/15739 2013-05-25T13:03:06Z http://hdl.handle.net/1957/38663 Continued fractions and the divisor at infinity on a hyperelliptic curve : examples and order bounds Daowsud, Katthaleeya We use the theory of continued fractions over function fields in the setting of hyperelliptic curves of equation y²=f(x), with deg(f)=2g+2. By introducing a new sequence of polynomials defined in terms of the partial quotients of the continued fraction expansion of y, we are able to bound the sum of the degrees of consecutive partial quotients. This allows us both (1) to improve the known naive upper bound for the order N of the divisor at infinity on a hyperelliptic curve; and, (2) to apply a naive method to search for hyperelliptic curves of given genus g and order N. In particular, we present new families defined over ℚ with N=11 and 2 ≤ g ≤ 10. Graduation date: 2013 2013-04-25T00:00:00Z http://hdl.handle.net/1957/38592 Some results in probability from the functional analytic viewpoint Gelbaum, Zachary A. This dissertation presents some results from various areas of probability theory, the unifying theme being the use of functional analytic intuition and techniques. We first give a result regarding the existence of certain stochastic integral representations for Banach space valued Gaussian random variables. Next we give a spectral geometric construction of Gaussian random fields over various manifolds that generalize classical fractional Brownian motion. Lastly we present a result describing the limiting distribution for the largest eigenvalue of a product of two random matrices from the β-Laguerre ensemble. Graduation date: 2013 2013-04-17T00:00:00Z http://hdl.handle.net/1957/38186 Representations of fractional Brownian motion Wichitsongkram, Noppadon Integral representations provide a useful framework of study and simulation of fractional Browian motion, which has been used in modeling of many natural situations. In this thesis we extend an integral representation of fractional Brownian motion that is supported on a bounded interval of ℝ to integral representation that is supported on bounded subset of ℝ[superscript d]. These in turn can be used to give new series representations of fractional Brownian motion. Graduation date: 2013 2013-03-13T00:00:00Z http://hdl.handle.net/1957/37360 Gaussian random fields related to Levy's Brownian motion : representations and expansions; Gaussian random fields related to Lévy's Brownian motion : representations and expansions Rode, Erica S. This dissertation examines properties and representations of several isotropic Gaussian random fields in the unit ball in d-dimensional Euclidean space. First we consider Lévy's Brownian motion. We use an integral representation for the covariance function to find a new expansion for Lévy's Brownian motion as an infinite linear combination of independent standard Gaussian random variables and orthogonal polynomials. Next we introduce a new family of isotropic Gaussian random fields, called the p-processes, of which Lévy's Brownian motion is a special case. Except for Lévy's Brownian motion the p-processes are not locally stationary. All p-processes also have a representation as an infinite linear combination of independent standard Gaussian random variables. We use these expansions of the random fields to simulate Lévy's Brownian motion and the p-processes along a ray from the origin using the Cholesky factorization of the covariance matrix. Graduation date: 2013 2013-02-25T00:00:00Z