Ph.D. Theses (Mathematics)http://hdl.handle.net/1957/157392014-04-16T10:35:47Z2014-04-16T10:35:47ZThe local conservation laws of the nonlinear Schrödinger equationBarrett, Johnnerhttp://hdl.handle.net/1957/447492013-12-31T18:13:03Z2013-11-08T00:00:00ZThe local conservation laws of the nonlinear Schrödinger equation
Barrett, Johnner
The nonlinear Schrödinger equation is a well-known partial differential equation that provides a successful model in nonlinear optic theory, as well as other applications. In this dissertation, following a survey of mathematical literature, the geometric theory of differential equations is applied to the nonlinear Schrödinger equation. The main result of this dissertation is that the known list of local conservation laws for the nonlinear Schrödinger equation is complete. A theorem is proved and used to produce a sequence of local conservation law characteristics of the nonlinear Schrödinger equation. The list of local conservation laws as given by Faddeev and Takhtajan and a theorem of Olver, which provides a one-to-one correspondence between equivalence classes of conservation laws and equivalence classes of their characteristics, are then used to prove the main result.
Graduation date: 2014; Access restricted to the OSU Community, at author's request, from Dec. 12, 2013 - June 12, 2014
2013-11-08T00:00:00ZAdiabatic and stable adiabatic timesBradford, Kyle B.http://hdl.handle.net/1957/400782013-07-08T17:17:02Z2013-05-15T00:00:00ZAdiabatic and stable adiabatic times
Bradford, Kyle B.
While the stability of time-homogeneous Markov chains have been extensively studied through the concept of mixing times, the stability of time-inhomogeneous Markov chains has not been studied as in depth. In this manuscript we will introduce special types of time-inhomogeneous Markov chains that are defined through an adiabatic transition. After doing this, we define the adiabatic and the stable adiabatic times as measures of stability these special time-inhomogeneous Markov chains. To construct an adiabatic transition one needs to make a transitioning convex combination of an initial and final probability transition matrix over the time interval [0, 1] for two time-homogeneous, discrete time, aperiodic and irreducible Markov chains. The adiabatic and stable adiabatic times depend on how this convex combinations transitions. In the most general setting, we suggested that as long as P : [0, 1] --> P[superscript ia][subscript n] is a Lipschitz continuous function with respect to the ‖ ·‖₁ matrix norm, then the adiabatic time is bounded above by a function of the mixing time of the final probability transition matrix [equation] For the stable adiabatic time, the most general result we achieved was for nonlinear adiabatic transitions P[subscript ø (t)] = (1-ø (t))P₀+ ø(t)P₁ where ø is a Lipschitz continuous functions that is piecewise defined over a finite partition of the interval [0, 1] so that on each subinterval ø is a bi-Lipschitz continuous function. In this setting we asymptotically bounded the stable adiabatic time by the largest mixing of P[subscript ø(t)] over all t∈[0, 1]. We found that [equation] We also have some additional results at bound the stable adiabatic time in this manuscript, but they are included to show the different attempts we took and highlight how important it is to pick the right variables to compare. We also provide examples to queueing and statistical mechanics.
Graduation date: 2013
2013-05-15T00:00:00ZContinued fractions and the divisor at infinity on a hyperelliptic curve : examples and order boundsDaowsud, Katthaleeyahttp://hdl.handle.net/1957/386632013-05-20T19:59:50Z2013-04-25T00:00:00ZContinued 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
2013-04-25T00:00:00ZSome results in probability from the functional analytic viewpointGelbaum, Zachary A.http://hdl.handle.net/1957/385922013-05-15T23:04:11Z2013-04-17T00:00:00ZSome 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