The function $sin x$ as one of the six trigonometric functions is fundamental in nearly every branch of mathematics, and its applications. In this thesis, we study an integral equation related to that of $sin x$: $mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}] mbox{~and~} p>1$ $$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$ We find that the function $S_{p}(x)$ is well defined. Its properties are also similar to those of $sin x$ : differentiation, identities, periodicity, asymptotic expansions, $cdots$, etc. For example, we have $$|S_{p}(x)|^{p}+|S'_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S'_{p}(x)|^{p-2}S'_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$ We call $S_{p}(x)$ the generalized sine function. Similarly, we define the generalized cosine function $C_{p}(x)$ by $|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for $xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and derive its properties. Thus we obtain two sets of trigonometric functions: egin{itemize} item[(i)]$~S_{p}(x),~ S'_{p}(x),~ T_{p}(x)=frac{S_{p}(x)}{S'_{p}(x)},~RT_{p}(x)=frac{S'_{p}(x)}{S_{p}(x)},~ SE_{p}(x)=frac{1}{S'_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$ item[(ii)]$~C_{p}(x),~ C'_{p}(x),~RCT_{p}(x)=-frac{C'_{p}(x)}{C_{p}(x)},~ CT_{p}(x)=-frac{C_{p}(x)}{C'_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~ CS_{p}(x)=-frac{1}{C'_{p}(x)}mbox{~¡C~}$ end{itemize}These two sets of functions have similar differentiation formulas, identities and periodic properties as the classical trigonometric functions. They coincide when $p=2$. Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.

The applications of generalized inverses of matrices appear in many fields like applied mathematics, statistics and engineering [2]. In this thesis, we discuss generalized inverses of matrices over Ore polynomial rings (also called Ore matrices). We first introduce some necessary and sufficient conditions for the existence of {1}-, {1,2}-, {1,3}-, {1,4}- and MP-inverses of Ore matrices, and give some explicit formulas for these inverses. Using {1}-inverses of Ore matrices, we present the solutions of linear systems over Ore polynomial rings. Next, we extend Roth's Theorem 1 and generalized Roth's Theorem 1 to the Ore matrices case. Furthermore, we consider the extensions of all the involutions ψ on R(x), and construct some necessary and sufficient conditions for ψ to be an involution on R(x)[D;σ,δ]. Finally, we obtain two different explicit formulas for {1,3}- and {1,4}-inverses of Ore matrices. The Maple implementations of our main algorithms are presented in the Appendix. / May 2017

In a generalized linear mixed model (GLMM), the random effects are typically uncorrelated and assumed to follow a normal distribution. However, findings from recent studies on how the misspecification of the random effects distribution affects the estimated model parameters are inconclusive. In the thesis, we extend the randomization approach for deriving linear models to the GLMM framework. Based on this approach, we develop an algorithm for estimating the model parameters of the randomization-based GLMM (RBGLMM) for the completely randomized design (CRD) which does not require normally distributed random effects. Instead, the discrete uniform distribution on the symmetric group of permutations is used for the random effects. Our simulation results suggest that the randomization-based algorithm may be an alternative when the assumption of normality is violated. In the second part of the thesis, we consider an RB-GLMM for the randomized complete block design (RCBD) with random block effects. We investigate the effect of misspecification of the correlation structure and of the random effects distribution via simulation studies. In the simulation, we use the variance covariance matrices derived from the randomization approach. The misspecified model with uncorrelated random effects is fitted to data generated from the model with correlated random effects. We also fit the model with normally distributed random effects to data simulated from models with different random effects distributions. The simulation results show that misspecification of both the correlation structure and of the random effects distribution has hardly any effect on the estimates of the fixed effects parameters. However, the estimated variance components are frequently severely biased and standard errors of these estimates are substantially higher.

In this thesis, we consider the recent definition of gravitational energy at the quasi-local level provided by Mu-Tao Wang and Shing-Tung Yau. Their definition poses a variational question predicated on isometric embedding of Riemannian surfaces into the Minkowski space; as such, there is a naturally associated Euler-Lagrange equation, which is a fourth-order system of partial differential equations for the embedding functions. We prove a perturbation result for solutions of this Euler-Lagrange equation.

This dissertation presents the Generalized Finite Element Method (GFEM) for the scalar Helmholtz equation, which describes the time harmonic acoustic wave propagation problem. We introduce several handbook functions for the Helmholtz equation, namely the planewave, wave-band, and Vekua functions, and we use these handbook functions to enrich the Finite Element space via the Partition of Unity Method to create the GFEM space. The enrichment of the approximation space by these handbook functions reduces the pollution effect due to wave number and we are able to obtain a highly accurate solution with a much smaller number of degrees-of-freedom compared with the classical Finite Element Method. The q-convergence of the handbook functions is investigated, where q is the order of the handbook function, and it is shown that asymptotically the handbook functions exhibit the same rate of exponential convergence. Hence we can conclude that the selection of the handbook functions from an admissible set should be dictated only by the ease of implementation and computational costs. Another issue addressed in this dissertation is the error coming from the artificial truncation boundary condition, which is necessary to model the Helmholtz problem set in the unbounded domain. We observe that for high q, the most significant component of the error is the one due to the artificial truncation boundary condition. Here we propose a method to assess this error by performing an additional computation on the extended domain using GFEM with high q.

Habilitationsschrift, Bonn, 1971; extra t.p. inserted. / Includes bibliographical references (p. [92-93]).

Thesis (M. Sc. (Mathematics))--Rhodes University, 2002.

published_or_final_version / Mathematics / Master / Master of Philosophy

In modern power system, the energy conversion includes a large number of the energy processors, and demands high quality, small, lightweight, reliable and efficient power procedures. The existing linear power regulators can only handle low power levels and demonstrate a low efficiency in the power processing. Pulse-width modulated (PWM) converters demonstrate high turn on and turn off losses, and increase in the electromagnetic interference (EMI). Resonant power conversion becomes more suitable in the renewable energy and energy harvesting applications. Since the resonant conversion requires operating in high frequency, the electrical components such as transformers, filter inductors and capacitors become much smaller and lighter. This can result in reducing size and cost. In addition, use of soft switching technique in the resonant conversion reduced the switching losses and EMI level. In this research project, a DC/DC resonant converter has been designed and modelled in PSCAD/EMTDC. The functionality of DC/DC resonant converter is validated in a hardware implementation of the small scale DC system.

In modern power system, the energy conversion includes a large number of the energy processors, and demands high quality, small, lightweight, reliable and efficient power procedures. The existing linear power regulators can only handle low power levels and demonstrate a low efficiency in the power processing. Pulse-width modulated (PWM) converters demonstrate high turn on and turn off losses, and increase in the electromagnetic interference (EMI). Resonant power conversion becomes more suitable in the renewable energy and energy harvesting applications. Since the resonant conversion requires operating in high frequency, the electrical components such as transformers, filter inductors and capacitors become much smaller and lighter. This can result in reducing size and cost. In addition, use of soft switching technique in the resonant conversion reduced the switching losses and EMI level. In this research project, a DC/DC resonant converter has been designed and modelled in PSCAD/EMTDC. The functionality of DC/DC resonant converter is validated in a hardware implementation of the small scale DC system.