In this thesis, we discuss the problem of estimating a characteristic function and its derivatives. We obtain estimates which are consistent and asympototically normal, and uniformly consistent with probability one. The methods employed here are similar to the methods used in estimating a probability density function and its derivatives (see [7], [9] for references). / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (Ph.D.) - University of St Andrews, May 2010.

Dissertation (Ph.D.)--Auburn University, 2009. / Abstract. Vita. Includes bibliographic references (p.30).

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 50-51). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 50-51). Also available on the Internet.

In this thesis we develop and analyze a Galerkin-Characteristic method to integrate the potential vorticity equations of a baroclinic ocean. The method proposed is a two stage inductive algorithm. In the first stage the material derivative of the potential vorticity is approximated by combining Galerkin-Characteristic and Particle methods. This yield a computationally efficient algorithm for this stage. Such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the foot of the characteristic curves of the advective component of the equations. The algorithm is unconditionally stable and conservative for Δt = O(h). The error analysis with respect to L² -norm shows that the algorithm converges with order O(h); however, in the maximum norm it is proved that for sufficiently smooth functions the foot of the characteristic curves are superconvergent points of order O(h⁴ /Δt). The second stage of the algorithm is a projection of the Lagrangian representation of the flow onto the Cartesian space-time Eularian representation coordinated with Crank-Nicholson Finite Elements. The error analysis for this stage with respect to L²-norm shows that the approximation component of the global error is O(h²) for the free-slip boundary condition, and O(h) for the no-slip boundary condition. These estimates represent an improvement with respect to other estimates for the vorticity previously reported in the literature. The evolutionary component of the global error is equal to K(Δt² + h), where K is a constant that depends on the derivatives of the advective quantity along the Characteristic. Since the potential vorticity is a quasi-conservative quantitiy, one can conclude that K is in general small. Numerical experiments illustrate our theoretical results for both stages of the method. / Science, Faculty of / Mathematics, Department of / Graduate

We first review recent work on stable and multistable random processes and their localisability. Then most of the thesis concerns a new approach to these topics based on characteristic functions. Our aim is to construct processes on R, which are α(x)-multistable, where the stability index α(x) varies with x. To do this we first use characteristic functions to define α(x)-multistable random integrals and measures and examine their properties. We show that an α(x)-multistable random measure may be obtained as the limit of a sequence of measures made up of α-stable random measures restricted to small intervals with α constant on each interval. We then use the multistable random integrals to define multistable random processes on R and study the localisability of these processes. Thus we find conditions that ensure that a process locally ‘looks like’ a given stochastic process under enlargement and appropriate scaling. We give many examples of multistable random processes and examine their local forms. Finally, we examine the dimensions of graphs of α-stable random functions defined by series with α-stable random variables as coefficients.

Thesis (Ph.D.)--McMaster University, 2005. / Supervisor: Manfred Kolster. Includes bibliographical references (leaves 93-96).

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