http://hdl.handle.net/1957/13737 Sat, 25 May 2013 04:02:41 GMT 2013-05-25T04:02:41Z 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 Thu, 25 Apr 2013 00:00:00 GMT http://hdl.handle.net/1957/38663 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 Wed, 17 Apr 2013 00:00:00 GMT http://hdl.handle.net/1957/38592 2013-04-17T00:00:00Z http://hdl.handle.net/1957/38475 Convergence Analysis of Yee Schemes for Maxwell's Equations in Debye and Lorentz Dispersive Media Bokil, V. A.; Gibson, N. L. We present discrete energy decay results for the Yee scheme applied to Maxwell's equations in Debye and Lorentz dispersive media. These estimates provide stability conditions for the Yee scheme in the corresponding media. In particular, we show that the stability conditions are the same as those for the Yee scheme in a nondispersive dielectric. However, energy decay for the Maxwell-Debye and Maxwell-Lorentz models indicate that the Yee schemes are dissipative. The energy decay results are then used to prove the convergence of the Yee schemes for the dispersive models. We also show that the Yee schemes preserve the Gauss divergence laws on its discrete mesh. Numerical simulations are provided to justify the theoretical results. Mon, 06 May 2013 00:00:00 GMT http://hdl.handle.net/1957/38475 2013-05-06T00:00:00Z http://hdl.handle.net/1957/38394 Time dependent wavemaker problem for linear waves Cooper, Julia M. The classical two-dimensional wavemaker problem is formulated for linear waves. Two conformal mappings are applied to the mathematical formulation to transform the wavemaker problem into a unit disk. It is then shown that this technique cannot produce in practice a numerical representation of the fluid motion throughout time for any position in the wavemaker channel. An analytic solution to the classical wavemaker problem is developed and then solved numerically. This solution contibutes information about the fluid motion for all time and for any position in the wavemaker channel. In addition to this problem, a related initial value problem is solved theoretically and then numerically. This results in the surface wave and fluid motion being described for all time and all positions in a two dimensional, semi-infinite channel. Graduation date: 1990 Thu, 18 May 1989 00:00:00 GMT http://hdl.handle.net/1957/38394 1989-05-18T00:00:00Z