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Vector interpolation functions which approximate electromagnetic vector fields are constructed in this thesis. These vector functions are to be used when the solution of Maxwell's equations involves an irrotational or solenoidal vector field. In addition the functions are chosen so that they can easily be used in the implementation of a finite element method. / Four bases are constructed. The first two span the spaces of solenoidal or irrotational two component vector polynomials of order one in two variables whereas the other two span the spaces of solenoidal or irrotational three component vector polynomials of order one in three variables. The vector polynomials are then used within the finite element method to approximate the two component current density J and electric field E over a conducting plate and the three component current density in a three dimensional wire.

Thesis (Ph.D.)--McMaster University, 2005. / Advisor: Ian Hambleton. Includes bibliographical references (leaves 74-75). Also available via World Wide Web.

This thesis deals with the topic of min-max formulations of robust model predictive control problems. The sets involved in guaranteeing robust feasibility of the min-max program in the presence of state constraints are of particular interest, and expanding the applicability of well understood solvers of linearly constrained quadratic min-max programs is the main focus. To this end, a generalisation for the set of uncertainty is considered: instead of fixed bounds on the uncertainty, state- and input-dependent bounds are used. To deal with state- and input dependent constraint sets a framework for a particular class of set-valued maps is utilised, namely parametrically convex set-valued maps. Relevant properties and operations are developed to accommodate parametrically convex set-valued maps in the context of robust model predictive control. A quintessential part of this work is the study of fundamental properties of piecewise polyhedral set-valued maps which are parametrically convex, we show that one particular property is that their combinatorial structure is constant. The study of polytopic maps with a rigid combinatorial structure allows the use of an optimisation based approach of robustifying constrained control problems with probabilistic constraints. Auxiliary polytopic constraint sets, used to replace probabilistic constraints by deterministic ones, can be optimised to minimise the conservatism introduced while guaranteeing constraint satisfaction of the original probabilistic constraint. We furthermore study the behaviour of the maximal robust positive invariant set for the case of scaled uncertainty and show that this set is continuously polytopic up to a critical scaling factor, which we can approximate a-priori with an arbitrary degree of accuracy. Relevant theoretical statements are developed, discussed and illustrated with examples.

The models for volatility, autoregressive conditional heteroscedastic (ARCH) and generalized autoregressive conditional heteroscedastic (GARCH) are discussed. Stationarity condition and forecasting for simple ARCH(1) and GARCH(1,1) models are given. The model for discrete time series is proposed to be negative binomial integer-valued GARCH model, which is a generalization of the Poisson INGARCH model. The stationarity conditions and the autocorrelation function are given. For parameter estimation, three methodologies are presented with a focus on maximum likelihood approach. Simulation study on a sample size of 100 and 500 are carried out and the results are presented. An application of the model to a real time series with numerical example is given indicating that the proposed methodology performs better than the Poisson and double Poisson model-based methods.

Please read the abstract in the front section of this document / Thesis (PhD (Mathematics))--University of Pretoria, 2006. / Mathematics and Applied Mathematics / unrestricted

The Banach spaces L(X, Y), K(X, Y), Lw*(X*, Y), and Kw*(X*, Y) are studied to determine when they contain the classical Banach spaces co or l ∞. The complementation of the Banach space K(X, Y) in L(X, Y) is discussed as well as what impact this complementation has on the embedding of co or l∞ in K(X, Y) or L(X, Y). Results concerning the complementation of the Banach space Kw*(X*, Y) in Lw*(X*, Y) are also explored and how that complementation affects the embedding of co or l ∞ in Kw*(X*, Y) or Lw*(X*, Y). The l p spaces for 1 ≤ p < ∞ are studied to determine when the space of compact operators from one l p space to another contains co. The paper contains a new result which classifies these spaces of operators. Results of Kalton, Feder, and Emmanuele concerning the complementation of K(X, Y) in L(X, Y) are generalized. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis as well as a new proof of the fact that l ∞ is prime.

The purpose of this paper is two-fold; we shall first establish a complete existential theory of functions of one real variable with respect to continuity, uniform continuity, absolute continuity, bounded variation, and Lipschitz condition, and second we shall study set-functions in a similar manner, except that the properties to be considered will be continuity, absolute continuity, bounded variation, and additivity.

Recently, was developed a new theory of the Jost function, within which, it was split in two terms involving on one side, singlevalued analytic functions of the energy, and on the other, factors responsible for the existence of the branching-points. For the single-valued part of the Jost function, a procedure for the powerseries expansion around an arbitrary point on the energy plane was suggested. However, this theory lacks a rigorous proof that these parts are entire functions of the energy. It also gives an intuitive (not rigorous) derivation of the domain where they are entire. In the present study, we ll this gap by using a method derived from the method of successive approximations. / Dissertation (MSc)--University of Pretoria, 2013. / gm2014 / Physics / unrestricted

PhD / Department of Mathematics and Applied Mathematics