Inverse problems aim at estimating the unknown excitations or properties of a physical system based on available measurements of the system response. For example, wave tomography is used in geophysics for seismic waveform inversion; in biomedical engineering, optical tomography is used to detect breast cancer tissue; in structural engineering, inversion techniques are used for health monitoring and damage detection in structural safety evaluation. Inverse solvers depend on the type of measurement data the unknown parameters to be estimated. The work in this thesis focuses on structural parameter identification based on static and dynamic measurements. As an integral part of the formulated inverse solver, the associated forward problem is studded and deeply investigated. In reality, the data are associated with uncertainties caused by measurement devices or unfriendly environmental conditions during data acquisition. Traditional approaches use probability theory and model uncertainties as random variables. This approach has its own limitation due to a prior assumption on the probability structure of uncertainty. This is usually too optimistic or not realistic. However, in practice, it is usually difficult to reliably assess the statistical nature of uncertainties. Instead, only bounds on the uncertain variables and some partial information about their probabilities are known. The main source of uncertainty is due to the accuracy of measuring devices; these are designed to operate within specific allowable tolerances, as defined by National Institute of Standards and Technology (NIST). Tolerances are performance requirements that fix the limit of allowable error or departure from true performance or value. Thus closed intervals are the most realistic way to model uncertainty in measurements. In this work, uncertainties in measurement data are modeled as interval variables bounded by their endpoints. It is proven that interval analysis provides guaranteed enclosure of the exact solution set regardless of the underlying nature of the associated uncertainties. This work presents a solution of inverse problems under measurements uncertainty within the framework of Interval Finite Element Methods (IFEM) and adjoint-based optimization techniques. The solution consists of a two-step algorithm: first, an estimate of the parameters is obtained by means of a deterministic iterative solver. Then, the algorithm switches to a full interval solution, using the previous deterministic estimate as an initial guess. In general, the solution of an inverse problem requires iterative solutions of the forward problem. Efficient and accurate interval forward solutions in static and dynamic domains have been developed. In particular, overestimation due to interval dependency has been drastically reduced using a new decomposition of the load, stiffness, and mass matrices. Further improvements in the available interval iterative solvers have been achieved. Conjugate gradient and Newton-Raphson methods to gether with an inexact line search are used in the newly formulated optimization procedure. Moreover Tikhonov regularization is used to improve the conditioning of the ill-posed inverse problem. The developed interval solution for the inverse problem under uncertainty has been tested in a wide range of applications in static and dynamic domains. By comparing current solutions with other available methods in the literature, it is proven that the developed method provides guaranteed sharp bounds on the exact solution sets at a low computational cost. In addition, it contains those solutions provided by probabilistic approaches regardless of the used probability distributions. In conclusion, the developed method provides a powerful tool for the analysis of structural inverse problem under uncertainty.

Over the years the coming together of Brazil, Russia, India and China into the BRIG forum has been treated as a moment in time when emerging economies started making their mark in the global arena. Similar patterns of extraordinary productivity and growth as well as tangible development have been the driving force that predicts these countries as being the world economic leaders of the future. South Africa has now come into BRIG, which necessitated the changes in the acronym to BRIGS, an alleged hegemon of Southern Africa and the continent's economic power house. This country joined BRIG group, adding to the world's emerging economies and as a representative of the African continent. When South Africa became a member, its main aim was to develop its economy and align itself socio-politically with the emerging economies of the world. These emerging economies have shown some positive growth. The emergence of BRICS has recently captured the commanding heights across all borders in the 'South' and the 'North'. It is evident that South Africa has striven to be recognised worldwide, by its foreign policy, stable politics and its sophisticated economic policy. South-South multilateral agreements have been widely discussed in International Political Economy, but little has been said about the impact they make in the socio-economic development of the emerging countries. The issue of multilateral agreements concerning BRIGS is still relatively unexplored in comparison to other multilateral agreements that South Africa has ventured into. Globally the BRIGS multilateral agreement has generated different views concerning its future expectations of promoting growth for its member states, particularly South Africa. The dissertation has discovered and highlighted the importance of South Africa as a member of BRIGS, and how its engagement in the multilateral agreement will benefit the country with opportunities that lie ahead. This paper also looked at the insinuation that South Africa may be used by its powerful BRIGS partners and thereby lose its continental dominance. The study scrutinises and allows open space for further investigation to be made about the BRIGS multilateral agreement and its impact on socio-economic development in the Republic of South Africa. Many argue that South Africa's entry in BRIGS is a dangerous and risky move while others prefer to say it is a wise decision taken by the country. However the action taken by South Africa is a fait accompli and the country should plan ahead and understand the needs of its national interests and foreign policy to achieve its goals. / Thesis (M. Soc Sc (International Relations) North-West University, Mafikeng Campus, 2013

published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy

In this work a new foundation for the study of positive braids in Artin's braid groups is given. The basic braids considered are the set SBn of positive permutation braids, defined as those positive braids where each pair of arcs cross at most once. These are shown to be in 1-1 correspondence with the permutations in S . A canonical n form for positive braids as products of braids in SB is given, ton gether with an explicit algorithm for writing every positive braid in canonical form and a practical test for use in the algorithm. This is a useful approach to braid theory because permutations can be particularly easily handled. Applications of this canonical form are: (1) An easily handled approach to Garside's solution of the word problem in B . n (2) An algorithm to decide whether (/1 ) k is a factor of a positive n braid; this happens if and only if at most k canonical factors have equal to /1 n (where /1 n is the positive braid with each pair of arcs cross exactly once). (3) A proof that a positive braid is a factor of (/1 ) k if and only if n its canonical form has at most k factors. (4) An improvement of Garside's solution of the conjugacy problem, this is by reducing the summit set to a much smaller invariant class under conjugation (super summit set). This includes a necessary and sufficient condition for positive braid to contain /1 n up to conjugation. The linear generators of the Hecke algebras used by Morton. and/ Short are in 1-1 correspondence with the elements of SB. The n canonical forms above give a quick proof that the number of strands in a twist positive braid (one of the form (/1 )2mp for positive braid n P and for positive integer m) is the braid index of the closure of that braid, which was first proved by Franks and Williams. It is also shown that if the 2-variable link invariant P L (v, z) for an oriented link L has width k in the variable v, then it is the same as the polynomial of a closed k-braid, for k = 1, 2. A complete list of 3-braids of width 2, which close to knots, is given. It is also shown that twist positive 3-braids do not admit exchange moves (in the sense of Birman). Consequently the conjugacy class of a twist positive 3-braid representative is a complete link invariant, provided that Birman's conjecture about Markov's moves and exchange moves holds. Lorenz knots and links are studied as an example of positive links. It is proved that a positive permutation braid 1T is a Lorenz braid if and only if all braid words which equal 1T have the same single starting letter. A semicanonical form for a minimal braid representative of a Lorenz link is given. It is proved that every algebraic link of c components is a Lorenz link, for c = 1, 2. (The case for knots was first proved by Birman and Williams). Consequently a necessary and sufficient condition for a knot to be algebraic is given, together with a canonical form for a minimal braid representative for every algebraic knot. To some extent the relation between Lorenz knots and their companions is studied. It is shown that Lorenz knots and links of braid index 3 are determined by conjugacy classes in B 3. A complete list of 3 -braids which close to Lorenz knots and links is given and a complete list of pure 4-braids which close to Lorenz links is also given. It is shown that Lorenz knots and links of braid index 3 are determined by their Alexander polynomials. As a further analogy with the properties of algebraic links it is shown that the linking pattern of a Lorenz link L with pure braid representative and braid index t ~4, determines a unique braid representative for L and so determines L.

The work presented in this thesis is primarily concerned with solving structural problems in peptides using a combination of classical chemical methods, mass spectrometry and nuclear magnetic resonance spectroscopy (NMR). Chapter One gives brief descriptions of the applications of fast atom bombardment mass spectrometry (FABMS) and NMR to the solution of structural problems in peptides. Chapter Two of this thesis describes the elucidation of the structure of vancomycin-type antibiotic, OA-7653, isolated from <i>Streptomyces hygroscopicus</i> subsp. <i>hiwasaensis</i> subsp. nov. Nishida. Chapter Three examines the interaction of vancomycin and OA-7653 with peptide cell-wall analogues, N-acetyl-D-alanyl-Dalanine and di-N-acetyl-L-lysyl-D-alanyl-D-alanine, in aqueous solutions by NMR and ultraviolet (UV) difference spectroscopy. In Chapter Four, the technique of FAB mapping was applied to study the amino acid sequence of actinidin, a proteolytic enzyme isolated from the fruit of <i>Actinidiaz chinensis</i> (commonly known as kiwi fruit). Also included in this chapter are mass spectrometric studies on model peptides to examine the conditions for accurate quantitation of phosphate content of peptides. Finally, structural and quantitation studies performed on phosphorylated peptides from the enzyme glycogen synthase are described.

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