Degree: Master of Science Department: Science / The superharmonic characterization of the value function is proved, under the assumption that an optimal stopping time exists. The fair price of an American contingent claim is established as an optimal stopping problem. The price of the perpetual Russian option is derived, using the dual martingale measure to reduce the dimension of the problem. American barrier options are discussed, and the solution to the perpetual American up-and-out put is derived. The price of the American put on a finite time horizon is shown to be the price of the European put plus an early exercise premium, through the use of a local time-space formula. The optimal stopping boundary is characterised as the unique increasing solution of a non-linear integral equation. Finally, the integral representation of the price of an American floating strike Asian call with arithmetic averaging is derived.

M.Sc.(Med.), Faculty of Health Sciences, University of the Witwatersrand, 2009 / Introduction Foot involvement is common in rheumatoid arthritis (RA). Foot pain, instability and deformity affect ambulation and impacts on health-related quality of life. The aim of this study was to determine the nature, extent and functional impact of rheumatoid foot problems in established RA. Patients and Methods One hundred RA patients were studied. Functional status was evaluated using the modified Health Assessment Questionnaire (mHAQ) and Foot Health Status Questionnaire (FHSQ). Foot deformity and footwear suitability was assessed using the Foot Problems Survey (FP Survey) and Footwear Suitability Scale (FWS Scale). Results In this predominantly female group of 95%, with a mean (± SD) disease duration of 12.2 (7.9) and moderate functional disability [mHAQ: 1.3 (0.6)], the FP Survey showed all patients had one or more foot deformity. Foot function was impaired with a mean (± SD) FHSQ score of 41.3 (12.4) and the FWS Scale showed that 93% wore unsuitable footwear. A strong correlation was observed of the global FHSQ (r=-0.5489, p<0.0001), its pain domain (r=-0.472, p<0.0001) and foot function domain (r=-0.599, p<0.0001), with the global mHAQ score. Despite the high frequency of foot problems observed only 27% had visited a podiatrist. Conclusion In conclusion foot problems and foot function disability is common in Black South African patients with established RA. Furthermore the strong correlation between mHAQ and FHSQ showed that foot functional disability was a major driver of overall functional disability in RA.

Thesis advisor: Vladimir Bandera / The purpose of this dissertation is to analyze the balance-of-payments problems of a particular debtor country by focusing on the effects of rapid external debt accumulation and of expanding debt servicing obligations on external balance. Brazil from 1948 to 1963 Is a logical choice for a study of this type since severe debt servicing problems were experienced throughout this period, particularly between 1955 and 1963. The Intention is not to portray Brazil as a typical debtor country since its problems were more intense and more immediate than related problems in most other debtor countries in the process of development. / Thesis (PhD) — Boston College, 1967. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Economics.

We are interested in solving the time-harmonic inverse acoustic scattering problem for planar sound-soft obstacles. In this work, we introduce four methods for solving inverse scattering problems. The first method is a variation of the method introduced by Johansson and Sleeman. This method solves the inverse problem when we have the far field pattern given for only one incident wave. It is an iterative method based on a pair of integral equations used to obtain the far field pattern of a known single object. The method proposed in this thesis has a better computational performance than the method of Johansson and Sleeman. The second method we present is a multi-frequency method called the recursive linearization algorithm. This method solves the inverse problem when the far field pattern is given for multiple frequencies. The idea of this method is that from an initial guess, we solve the single frequency inverse problem for the lowest frequency. We use the result obtained as the initial guess to solve the problem for the next highest frequency. We repeat this process until we use the data from all frequencies. To solve the problem at each frequency, we use the first method proposed. To improve the quality of the reconstruction of the shadowed part of the object, we solve the inverse scattering problem of reconstructing an unknown sound-soft obstacle in the presence of known scatterers. We show that depending on the position of the scatterers, we may be able to obtain very accurate reconstructions of the entire unknown object. Next, we introduce a method for solving the inverse problem of reconstructing a convex sound-soft obstacle, given measures of the far field pattern at two frequencies that are not in the resonance region of the object. This method is based on the use of an approximation formula for the far field pattern using geometric optics. We are able to prove that for the reconstruction of the circle of radius $R$ and center at the origin, the size of the interval of convergence of this method is proportional to the inverse of the wavenumber. This procedure is effective at reconstructing the illuminated part of the object; however, it requires an initial guess close to the object for frequencies out of the resonance region. Finally, we propose a globalization technique to obtain a better initial guess to solve the inverse problem at frequencies out of the resonance region. In this technique, given the far field pattern of a convex object at two frequencies out of the resonance region, we use our extrapolation operator to generate synthetic data for low frequencies. We apply the recursive linearization algorithm, using as a single frequency solver the method that is based on geometric optics. We obtain an approximation of the object that can be used as the initial guess to apply the recursive linearization algorithm using the first method introduced as the single frequency solver.

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General routing problems deal with transporting some commodities and/or travelling along the axes of a given network in some optimal manner. In the modern world such problems arise in several contexts such as distribution of goods, transportation of commodities and/or people etc.In this thesis we focus on two classical routing problems, namely the Travelling Salesman Problem (TSP) and the Vehicle Routing Problem (VRP). The TSP can be described as a salesman travels from his home city, visits each of the other ( n - 1) cities exactly once and returns back to the home city such that the total distance travelled by him is minimised. The VRP may be stated as follows: A set of n customers (with known locations and demands for some commodity) is to be supplied from a single depot using a set of delivery vehicles each with a prescribed capacity. A delivery route starts from the depot, visits some customers and returns back to the depot. The VRP is to determine the delivery routes for each vehicle such that the total distance travelled by all the vehicles is minimised.These routing problems are simple to state in terms of describing them in words. But they are very complex in terms of providing a suitable mathematical formulation and a valid procedure to solve them. These routing problems are simple to state in terms of describing them in words. But they are very complex in terms of providing a suitable mathematical formulation and a valid procedure to solve them. These problems belong to the class of NP-hard (Non- deterministic Polynomial) problems. With the present knowledge, it is believed that the problems in NP-hard are unlikely to have any good algorithms to arrive at optimal solutions to a general problem. Hence researchers have focused their effort on; (i) developing exact algorithms to solve as large size problems as possible; (ii) developing heuristics to produce ++ / near optimal solutions.The exact algorithms for such problems have not performed satisfactorily as they need an enormous amount of computational time to solve moderate size problems. For instance, in the literature, TSP of size 225-city, 4461-city and 7397-city were solved using computational time of 1 year, 1.9 years and 4 years respectively (Junger et al., 1995). Thus heuristics, in particular the probabilistic methods such as tabu search, play a significant role in obtaining near optimal solutions. In the literature there is very little comparison between the various exact algorithms and heuristics. (Very often the real-life problems are too large and no optimal solution can be found in a reasonable time.)One of the problems with a probabilistic heuristic is that different implementations (runs) of the same probabilistic heuristic on a given problem may produce distinct solutions of different quality. Thus the desired quality and reproducibility of the solution cannot be ensured. Furthermore, the performance of the heuristics on the benchmark problems provide no Guarantee of the quality of solutions that can be obtained for the problem faced by a researcher. Most of the documentation on the performance of heuristics in literature problems provides no information regarding the computational effort (CPU time) spent in obtaining the claimed solution, reproducibility of the claimed solution and the hardware environment of the implementation. This thesis focuses on some of these deficiencies.Most of the heuristics for general combinatorial optimisation problems are based on neighbourhood search methods. This thesis explores and provides a formal setup for defining neighbourhood structures, definitions of local optimum and global optimum. Furthermore it highlights the dependence and drawbacks of such methods on the neighbourhood structure.It is necessary to emphasise ++ / the need for a statistical analysis of the output to be part of any such probabilistic heuristic. Some of the statistical tools used for the two probabilistic heuristics for TSP and VRP are developed. Furthermore, these heuristics axe part of a bigger class called tabu search heuristics for combinatorial optimisation problems. Hence it includes an overview of the TSP, VRP and tabu search methods in Chapters 2, 3 and 4 respectively. Subsequently in Chapters 5, 6, 7 and 8 ideas of neighbourhood structure, local optimum etc. are developed and the required statistical analysis for some heuristics on the TSP and VRP is demonstrated. In Chapter 9, the conclusion of this thesis is drawn and the direction of future work is outlined. The following is a brief outline of the contribution of this thesis.In Chapter 5, the ideas of neighbourhood structure, local optimum, global optimum and probabilistic heuristics for any combinatorial optimisation problem sare developed. The drawbacks of the probabilistic heuristics for such problems axe highlighted. Furthermore, the need to select the best heuristic on the basis of testing a statistical hypothesis and related statistical analysis is emphasised.Chapter 6 illustrates some of the ideas presented in Chapter 5 using the GENIUS algorithm proposed for the TSP. Statistical analysis is performed for 36 variations of GENIUS algorithm based on different neighbourhood parameters, different types of insertion methods used and two types of constructions of starting solutions. The analysis is performed on 27 literature problems with size ranging from 100 cities to 532 cities and 20 randomly generated problems with size ranging from 100 cities to 480 cities. In both cases the best heuristic is selected. Furthermore, the fitting of the Weibull Distribution to the objective function values of the heuristic solutions provides an estimate of the ++ / optimal objective function value and a corresponding confidence interval for both the literature and randomly generated problems. In both cases the estimate of the optimal objective function values are within 8.2% of the best objective function value known.Since the GENIUS algorithm proved to be efficient, a hybrid heuristic for the TSP combining the branch and bound method and GENIUS algorithm to solve large dimensional problems is proposed. The algorithm is tested on both the literature problems with sizes ranging from 575 cities to 724 cities and randomly generated problems with sizes ranging from 500 cities to 700 cities. Though problems could not be solved to optimality within the 10 hours time limit, solutions were found within 2.4% of the best known objective function value in the literature.In Chapter 7, a similar statistical analysis for the TABUROUTE algorithm proposed for the VRP is conducted. The analysis is carried out based on the different neighbourhood parameters and tested using 14 literature problems with sizes ranging from 50 cities to 199 cities and 49 randomly generated problems with sizes ranging from 60 cities to 120 cities. In both sets of the problems, the statistical tests accepted the hypothesis that there is no significant difference in the solution produced between the various parameters used for the TABUROUTE algorithm. By fitting the Weibull distribution to the objective function values of the local optimal solutions, the optimal objective function value and a corresponding confidence intervals for each problem is estimated. These estimates give values that are to within 6.1% and 18.3% of the best known values for the literature problems and randomly generated problems respectively.In Chapter 8, the general neighbourhood search method for a general combinatorial optimisation problem is presented. Very often, the neighbourhood structure ++ / can be defined suitably only on a superset S of the set of feasible solutions S. Thus the solutions in SS are infeasible. Several questions axe posed regarding the computational complexity of the solution space of a problem. These concepts are illustrated on the 199-city CDVRP, using the TABUROUTE algorithm.In addition, the idea of complexity of the solution space based on the samples collected over the 140 runs is explored. Some of the data collected include the number of solutions with distance and/or capacity feasible, the number of feasible neighbourhood solutions encountered for one run, etc. Questions such asHow many solutions are there for the 199-city problem ?How many numbers of local minima solutions are there for the 199-city problem?What is the size of the feasible region for the 199-city problem?are answered. Finally, the conclusion is drawn that this problem cannot be used as a benchmark based on the size of the feasible region and too many local minima.The conclusion of this thesis and directions of future work are discussed in Chapter 9. There are two appendices presented at the end of the thesis. Appendix A presents the details of the Friedman test, the expected utility function test and the estimation of the optimal objective function value based on the Weibull distribution. Appendix B presents a list of tables from Chapters 6, 7 and 8.

This thesis considers three related structures on finite sets and outstanding conjectures on two of them. Several new problems and conjectures are stated.A union-closed collection of sets is a collection of sets which contains the union of each pair of sets in the collection. A completely separating system of sets is a collection of sets in which for each pair of elements of the universal set, there exists a set in the collection which contains the first element but not the second, and another set which contains the second element but not the first. An antichain (Sperner Family) is a collection of distinct sets in which no set is a subset of another set in the collection. The size of an antichain is the number of sets in the collection. The volume of an antichain is the sum of the cardinalities of the sets in the collection. A flat antichain is an antichain in which the difference in cardinality between any two sets in the antichain is at most one.The two outstanding conjectures considered are:The union-closed sets conjecture - In any union-closed collection of non-empty sets there is an element of the universal set in at least half of the sets in the collection;The flat antichain conjecture - Given an antichain with size s and volume V, there is a flat antichain with the same size and volume.Union-closed collections are considered in two ways. Improvements are made to the previously known bounds concerning the minimum size of a counterexample to the union-closed sets conjecture. Results are derived on the minimum size of a union-closed collection generated by a given number of k-sets. An ordering on sets is described, called order R and it is conjectured that choosing a collection of m k-sets in order R will always minimise the size of the union-closed collection generated by m k-sets.Several variants on completely separating systems of sets are considered. A ++ / determination is made of the minimum size of such collections, subject to various constraints on the collections. In particular, for each n and k, exact values or bounds are determined for the minimum size of completely separating systems on a n-set in which each set has cardinality k.Antichains are considered in their relationship to completely separating systems and the flat antichain conjecture is shown to be true in certain cases.

<p>Abstract</p><p>Title Cashews by SMS – an implementation in Mozambique</p><p>Problem</p><p>Innovation is described by Tidd, Bessant and Pavitt (2005) as the core</p><p>process within organisations associated with renewal and as generic</p><p>activity associated with survival and growth. Yet many organisations</p><p>fail to realise the benefits of adopting an innovation. Which the theory will show this is most likely due to a problem with one certain phase in the innovation process: the implementation.</p><p>Purpose</p><p>The purpose with this academic paper is by a practical example</p><p>illustrate the risks and problems one can come across in an</p><p>implementation and the consequences of this. We also intend to give</p><p>suggestion on how it is possible to restart an implementation process</p><p>when the process once has failed.</p><p>Research questions</p><p>Why has marketAlerts failed to be implemented in Mozambique?</p><p>How should IPEX resume the implementation of marketAlerts?</p><p>Methodology</p><p>Ethnographical approach.</p><p>Conclusion</p><p>Our conclusion is that the Institute for Export Promotion (IPEX) has</p><p>managed to adopt marketAlerts but has failed to implement it in their</p><p>daily work mainly due to the fact that they only completed the</p><p>acquiring phase. The failure is due to a combination of hierarchy, lack of interest and absents of routines for sending marketAlerts. In order for IPEX to make the best use of marketAlerts we believe that they have to go back and start from the executing phase and implement the service once again.</p>

Spectral integration methods have been introduced for constant-coefficient two-point boundary value problems by Greengard, and pseudospectral integration methods for Volterra integral equations have been investigated by Kauthen. This thesis presents an approach to variable-coefficient two-point boundary value problems which employs pseudospectral integration methods to solve an equivalent integral equation. This thesis covers three topics in the application of spectral integration methods to two-point boundary value problems. The first topic is the development of the spectral integration concept and a derivation of the spectral integration matrices. The derivation utilizes the discrete Chebyshev transform and leads to a stable algorithm for generating the integration matrices. Convergence theory for spectral integration of C[subscript k] and analytic functions is presented. Matrix-free implementations are discussed with an emphasis on computational efficiency. The second topic is the transformation of boundary value problems to equivalent Fredholm integral equations and discretization of the resulting integral equations. The discussion of boundary condition treatments includes Dirichlet, Neumann, and Robin type boundary conditions. The final topic is a numerical comparison of the spectral integration and spectral differentiation approaches to two-point boundary value problems. Numerical results are presented on the accuracy and efficiency of these two methods applied to a set of model problems. The main theoretical result of this thesis is a proof that the error in spectral integration of analytic functions decays exponentially with the number of discretization points N. It is demonstrated that spectrally accurate solutions to variable-coefficient boundary value problems can be obtained in O(NlogN) operations by the spectral integration method. Numerical examples show that spectral integration methods are competitive with spectral differentiation methods in terms of accuracy and efficiency. / Graduation date: 2000