The main objective of this dissertation is the development of CUSUM procedures based on signed and unsigned sequential ranks. These CUSUMs can be applied to detect changes in the location or dispersion of a process. The signed and unsigned sequential rank CUSUMs are distribution-free and robust against the effect of outliers in the data. The only assumption that these CUSUMs require is that the in-control distribution is symmetric around a known location parameter. These procedures specifically do not require the existence of any higher order moments. Another advantage of these CUSUMs is that Monte Carlo simulation can readily be applied to deliver valid estimates of control limits, irrespective of what the underlying distribution may be. Other objectives of this dissertation include a brief discussion of the results and refinements of the CUSUM in the literature. We justify the use of a signed sequential rank statistic. Also, we evaluate the relative efficiency of the suggested procedure numerically and provide three real-world applications from the engineering and financial industries. / MSc (Risk Analysis), North-West University, Potchefstroom Campus, 2015

Thesis (PhD (Mathematics))--University of Stellenbosch, 2010. / Contains bibliography. / ENGLISH ABSTRACT: See full text for summary. / AFRIKAANSE OPSOMMING: Sien volteks vir opsomming

In the first part of the thesis we define an automorphism φn for each star graph Stn of degree n-1, which yields permutations of labels for the edges of Stn taken from the set of integers {1,..., [n/2c]}. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known two-labelled Hamilton cycle H1 2(n) in Stn. The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies in computer science. All our results improve on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs.

In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied.

Symmetric generalized eigenvalue problems arise in many physical applications and frequently only a few of the eigenpairs are of interest. Typically, the problems are large and sparse, and therefore traditional methods such as the QZ algorithm may not be considered. Moreover, it may be impractical to apply shift-and-invert Lanczos, a favored method for problems of this type, due to difficulties in applying the inverse of the shifted matrix. With these difficulties in mind, Golub and Ye developed an inverse free Krylov subspace algorithm for the symmetric generalized eigenvalue problem. This method does not rely on shift-and-invert transformations for convergence acceleration, but rather a preconditioner is used. The algorithm suffers, however, in the presence of multiple or clustered eigenvalues. Also, it is only applicable to the location of extreme eigenvalues. In this work, we extend the method of Golub and Ye by developing a block generalization of their algorithm which enjoys considerably faster convergence than the usual method in the presence of multiplicities and clusters. Preconditioning techniques for the problems are discussed at length, and some insight is given into how these preconditioners accelerate the method. Finally we discuss a transformation which can be applied so that the algorithm extracts interior eigenvalues. A preconditioner based on a QR factorization with respect to the B-1 inner product is developed and applied in locating interior eigenvalues.

This thesis is dedicated to the study of the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for the existence of Einstein metrics with totally geodesic fibers in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We investigate the existence of such Einstein metrics for invariant bisymmetric fibrations of maximal rank, i.e., when both the base and the fiber are symmetric spaces and the base is an isotropy irreducible space of maximal rank. We find this way new Einstein metrics. For such spaces we describe explicitly the isotropy representation in terms subsets of roots and compute the eigenvalues of the Casimir operators of the fiber along the horizontal direction. Results for compact simply connected 4-symmetric spaces of maximal rank follow from this. Also, new invariant Einstein metrics are found on Kowalski n-symmetric spaces.

This work considers the effect of symmetries on analysing bifurcations in dynamical systems. We consider an example of a laser with strong optical feedback which is modelled using coupled non-linear differential equations. A stationary point can be found in space, which can then be continued in parameter space using software such as AUTO. This software will then detect and continue bifurcations which indicate change in dynamics as parameters are varied. Due to symmetries in the equations, using AUTO may require the system of equations to be reduced in order to study periodic orbits of the original system as (relative) equilibria of the reduced system. Reasons for this are explored as well as considering how the equations can be changed or reduced to remove the symmetry. Invariant and Equivariant theory provide the tools for reducing the system of equations to the orbit space, allowing further analysis of the lasers dynamics.

Bakgrund och problem: Underprissättning har länge studerats, och fenomenet har konstaterats förekomma i en absolut majoritet av de studerade länderna. Trots detta råder ingen konsensus kring vilka teorier som förklarar underprissättning. Teorier för underprissättning är i huvudsak indelade i teorier baserade på symmetrisk eller asymmetrisk information, och de empiriska resultat som presenterats målar en spretig bild med resultat som går både emot och ger stöd för flera teorier. Mot bakgrund av den motstridighet som föreligger för teoretiska förklaringar till underprissättning har vi valt att studera ämnet noggrannare. Syfte: Studiens syfte är att, med utgångspunkt i motsägande empiriska resultat, utreda vilka olika teoretiska incitament som kan förklara underprissättning för att skapa en bättre förståelse för den komplexitet som föreligger inom teoribildningen för underprissättning. Metod: Vi applicerar i studien en kvantitativ forskningsmetodik där vi, i linje med internationella referensstudier, väljer att använda en multipel regressionsanalys med minstakvadrat-metoden för att kunna testa vilka teorier som är applicerbara för att förklara underprissättning. Vi approximerar teorierna via variabler med tillhörande hypoteser för att mäta dess individuella och ackumulerade effekt på underprissättning. Via hypotesprövning möjliggör vi en koppling mellan det statistiska resultatet och respektive teori. Slutsats: Studiens resultat stödjer såväl teorier baserade på symmetrisk som asymmetrisk information vilket illustrerar komplexiteten i fenomenet underprissättning. Med utgångspunkt i resultaten av den statistiska modellen förs en analys där teorier frikopplats från symmetri respektive asymmetri, och istället analyserats simultant. Denna analys presenterar en alternativ syn på teori kring underprissättning, där teorier tillhörande symmetrisk respektive asymmetrisk information inte är ömsesidigt uteslutande, vilket introducerar ett nytt sätt att se på teoretiska förklaringar för underprissättning. / Background and problem: Underpricing has been subject to studies for many years, and the phenomena have been observed on a vast majority of studied countries. Despite the fact that studies have been conducted, there is no consensus as to which theories might best explain underpricing. Theories explaining underpricing are mainly divided into either symmetric or asymmetric information, and the empirical results presented testifies to the theoretical discrepancy, with support for and against several theories. The theoretical discrepancy is the main reason as to why we choose to closer examine the phenomena. Purpose: The purpose of this study is, based upon contradictory empirical results, to investigate as to which different theoretical incentives can explain IPO underpricing in order to create a better understanding of the complexity regarding theory for IPO underpricing. Methodology: In the study we apply a quantitative research method in which we, similar to international reference studies, use a multiple regression analysis with an ordinary least squares approach in order to test which theories are applicable in order to explain underpricing. We approximate the theories using variables with belonging hypotheses to quantify the variables individual and accumulated effect on underpricing. Using hypothesis testing we enable a connection between the statistical results and theories explaining underpricing. Conclusions: The study’s result support both theories based upon symmetric and asymmetric information which illustrates the complexity of the phenomenon of underpricing. Based on the statistical results the authors conduct an analysis in which theories are discarded from the context of symmetric and asymmetric information and are instead analyzed simultaneously. This analysis presents an alternative view of underpricing theories, in which theories belonging to symmetric as well as asymmetric information no longer are mutually excluding, which introduces a new approach to theoretical explanations for IPO underpricing.

The problem of the quadrupole interaction occurring in a vibrating-rotating C₃v symmetric top molecule has been studied in detail. The quadrupole interaction has been treated as another perturbation term to a general frequency expression accounting for the vibrating-rotating interaction of the molecule so that a complete frequency formula is obtained for both interactions, and from which hyperfine spectral components are predicted and measured. The hyperfine transitions in the ground, and v₈=1 and v₈=2 excited vibrational states of the ¹³C isotopes of methyl cyanide have been investigated in the frequency range 17-72 GHz, primarily in the low J transitions (0≤J≤3). The study of the ground state of isotope i3CH3i3CN, and the v₈=1, v₈=2 excited vibrational states for all the isotopes have been conducted here for the first time. A substantial perturbation has been discovered and discussed at the ΔJ=3→4 transitions within the Kl=1 sets in the v₈=1 mode for isotopes ¹³CH₃CN and CH₃¹³CN. A total of 716 hyperfine transitions have been assigned from measurements, only 7 of which have been measured previously. A total of 84 molecular constants have been reported; 70 of these constants are derived for the first time from microwave data.

The optical design for a dual modality endoscope based on piezo scanning fiber technology is presented including a novel technique to combine forward-viewing navigation and side viewing OCT. Potential applications include navigating body lumens such as the fallopian tube, biliary ducts and cardiovascular system. A custom cover plate provides a rotationally symmetric double reflection of the OCT beam to deviate and focus the OCT beam out the side of the endoscope for cross-sectional imaging of the tubal lumen. Considerations in the choice of the scanning fiber are explored and a new technique to increase the divergence angle of the scanning fiber to improve system performance is presented. Resolution and the necessary scanning density requirements to achieve Nyquist sampling of the full image are considered. The novel optical design lays the groundwork for a new approach integrating side-viewing OCT into multimodality endoscopes for small lumen imaging.