This thesis consists of seven chapters. Chapter 1 is an introduction to the infinite products. Here we provide a proof for representing sine function as an infinite product. This chapter also describes the notation used throughout the thesis as well as the method used to prove the identities. Each of the other chapters may be read independently, however some chapters assume familiarity with the Jacobi triple product identity. Chapter 2 is about the Jacobi triple product identity as well as several implications of this identity. In Chapter 3 the quintuple product identity and some of its special cases are derived. Even though there are many known proofs of this identity since 1916 when it was first discovered, the proof presented in this chapter is new. Some beautiful formulas in number theory are derived at the end of this chapter. The simplest two dimensional example of the Macdonald identity, A2, is investigated in full detail in Chapter 4. Ian Macdonald first outlined the proof for this identity in 1972 but omitted many of the details hence making his work hard to follow. In Chapters 5 and 6 we somewhat deviate from the method which uses the two specializations to evaluate the constant term and prove Winquist's identity and Macdonald's identity for G2. Some of the work involved in proving G2 identity is new. Finally in Chapter 7 we discuss the work presented with some concluding remarks as well as underlining the possibilities for the future research. Throughout the thesis we point to the relevant papers in this area which might provide different strategies for proving above identities.

Mathematical models for plant location problems form an important class of integer and mixed-integer linear programs. The Two-Stage Capacitated Plant Location Problem (TSCPLP), the subject of this thesis, consists of a three level structure: in the first or upper-most level are the production plants, the second or central level contains the distribution depots, and the third level is the customers. The decisions to be made are: the subset of plants and depots to open; the assignment of customers to open depots, and therefore open plants; and the flow of product from the plants to the depots, to satisfy the customers' service or demand requirements at minimum cost. The formulation proposed for the TSCPLP is unique from previous models in the literature because customers can be served from multiple open depots (and plants) and the capacity of both the set of plants and the set of depots is restricted. Surrogate constraints are added to strengthen the bounds from relaxations of the problem. The need for more understanding of the strength of the bounds generated by this procedure for the TSCPLP is evident in the literature. Lagrangian relaxations are chosen based more on ease of solution than the knowledge that a strong bound will result. Lagrangian relaxation has been applied in heuristics and also inserted into branch-and-bound algorithms, providing stronger bounds than traditional linear programming relaxations. The current investigation provides a theoretical and computational analysis of Lagrangian relaxation bounds for the TSCPLP directly. Results are computed through a Lagrangian heuristic and CPLEX. The test problems for the computational analysis cover a range of problem size and strength of capacity constraints. This is achieved by scaling the ratio of total depot capacity to customer demand and the ratio of total plant capacity to total depot capacity on subsets of problem instances. The analysis shows that there are several constraints in the formulation that if dualized in a Lagrangian relaxation provide strong bounds on the optimal solution to the TSCPLP. This research has applications in solution techniques for the TSCPLP and can be extended to some transformations of the TSCPLP. These include the single-source TSCPLP, and the multi-commodity TSCPLP which accommodates for multiple products or services.

In this thesis I study the effects of gap junctions on pattern formation in a neural field model for working memory. I review known results for the base model (the “Amari model”), then see how the results change for the “gap junction model”. I find steady states of both models analytically and numerically, using lateral inhibition with a step firing rate function, and a decaying oscillatory coupling function with a smooth firing rate function. Steady states are homoclinic orbits to the fixed point at the origin. I also use a method of piecewise construction of solutions by deriving an ordinary differential equation from the partial integro-differential formulation of the model. Solutions are found numerically using AUTO and my own continuation code in MATLAB. Given an appropriate level of threshold, as the firing rate function steepens, the solution curve becomes discontinuous and stable homoclinic orbits no longer exist in a region of parameter space. These results have not been described previously in the literature. Taking a phase space approach, the Amari model is written as a four-dimensional, reversible Hamiltonian system. I develop a numerical technique for finding both symmetric and asymmetric homoclinic orbits. I discover a small separate solution curve that causes the main curve to break as the firing rate function steepens and show there is a global bifurcation. The small curve and the global bifurcation have not been reported previously in the literature. Through the use of travelling fronts and construction of an Evans function, I show the existence of stable heteroclinic orbits. I also find asymmetric steady state solutions using other numerical techniques. Various methods of determining the stability of solutions are presented, including a method of eigenvalue analysis that I develop. I then find both stable and transient Turing structures in one and two spatial dimensions, as well as a Type-I intermittency. To my knowledge, this is the first time transient Turing structures have been found in a neural field model. In the Appendix, I outline numerical integration schemes, the pseudo-arclength continuation method, and introduce the software package AUTO used throughout the thesis.

Consider distinct latin squares, L and M, of order n. Then the pair (T1, T2) where T1 = L \M and T2 = M \ L is called a latin bitrade. Furthermore T1 and T2 are referred to as latin trades, in which T2 is said to be a disjoint mate of T1 (and vice versa). Drápal (1991) showed that, under certain conditions, a partition of an equilateral triangle of side length n, where n is some integer, into smaller, integer length sided equilateral triangles gives rise to a latin trade within the latin square based on the addition table for the integers modulo n. A partial latin square P of order n is said to be completable if there exists a latin square L of order n such that P ⊆ L. If there is only one such possible latin square, L, of order n then P is said to be uniquely completable and P is called a defining set of L. Furthermore, if C is a uniquely completable partial latin square such that no proper subset of C is uniquely completable, C is said to be a critical set or a minimal defining set. These concepts, namely latin trades and defining sets in latin squares, are intimately connected by the following observation. If L is a latin square and D ⊆ L is a defining set, then D intersects all latin bitrades for which one mate is contained in L. In Part I of this thesis Dr´apal’s result is applied to investigate the structure of certain defining sets in latin squares. The results that are obtained are interesting in themselves; furthermore, the geometric approach to the problem yields additional appealing results. These geometric results are discussed in Chapters 3, 4, 5 and 6. They pertain to partitioning regions (polygons in R2 that satisfy certain obvious conditions) into equilateral, integer length sided, triangles, such that no point, in the region, is a corner of more than three distinct triangles. In Chapter 2 one of the main two theorems on defining sets is established, as is a method for using the above geometric results to prove the nonexistence of certain types of defining sets. In Part II of this thesis, intersection problems, for latin squares and Steiner triple systems, are considered. The seminal works, for problems of these types, are due to Lindner and Rosa (1975) and Fu (1980). A natural progression, from the established literature, for intersection problems between elements in a pair of latin squares or Steiner triple systems is to problems in which the intersection is composed of a number of disjoint configurations (isomorphic copies of some specified partial triple system). In this thesis solutions to two intersection problems for disjoint configurations are detailed. An m-flower, (F,F), is a partial triple system/configuration, such that: F = {{x, yi, zi} | {yi, zi} ∩ {yj , zj} = ∅, for 0 ≤ i, j ≤ m − 1, i 6= j}; and F = UX∈FX. The first such problem considered in this thesis asks for necessary and sufficient conditions for integers k and m ≥ 2 such that a pair of latin squares of order n exists that intersect precisely in k disjoint m-flowers. The necessary terminology, constructions, lemmas and proof for this result are contained in Chapters 7, 8 and 9. The second such problem considered in this thesis asks for necessary and sufficient conditions for integers k such that a pair of Steiner triple systems of order u exists that intersect precisely in k disjoint 2-flowers. This result relies on the solution to the latin square problem and an additional result from Chapter 9. The further constructions and lemmas used to prove this result are detailed in Chapter 10.

This is a dissertation about the Halle method, or the visual pedagogies of the Pietist Orphanage as they were developed in the German university town of Halle from 1695 until 1730. A “Pietist” was someone who was affiliated with an evangelical reform movement first initiated by Philipp Jakob Spener in the 1670s. A long and deeply entrenched historiographical tradition has portrayed the Halle proponents of this movement—especially their leader August Hermann Francke—as zealous, yet practical, Lutheran reformers who were forced to directly confront the ideals of early Enlightenment in conjunction with the state-building mandate of Brandenburg-Prussia. This has led to a persistent tendency to see Halle Pietists as “others” who cultivated their collective identity in opposition to so-called Enlightenment intellectuals, like Christian Wolff, at the same time as they exerted a marked influence on these same persons. As a result of this dichotomous portrayal over the years, the impact of the Halle method on educational reform, and on the meanings eighteenth-century Europeans attached to philanthropy more generally, has been misunderstood. I argue that the Pietist Orphanage holds the key to remedying several problems that have impeded our ability to understand the significance of Pietist pedagogy and philanthropy. This was a site specifically designed to introduce children to the conciliatory knowledge-making strategies of the first Berlin Academy of Science members and their associates. These strategies championed the status of the heart as an assimilatory juncture point and were refined in the schools of the Pietist Orphanage, which itself functioned as a visual showplace that viewers could observe in order to edify and improve themselves. It was the material expression of Halle Pietists’ commitment to a “third way” and marked their attempt to assimilate experience and cognition, theology and philosophy, absolutism and voluntarism. The dissertation examines several personalities who had a direct bearing on this conciliatory project: namely E. W. von Tschirnhaus, Johann Christoph Sturm, Leonhard Christoph Sturm, Gottfried Wilhelm Leibniz and Christian Wolff. It also examines how the method was applied in the Halle Orphanage schools and extended elsewhere.

This is a dissertation about the Halle method, or the visual pedagogies of the Pietist Orphanage as they were developed in the German university town of Halle from 1695 until 1730. A “Pietist” was someone who was affiliated with an evangelical reform movement first initiated by Philipp Jakob Spener in the 1670s. A long and deeply entrenched historiographical tradition has portrayed the Halle proponents of this movement—especially their leader August Hermann Francke—as zealous, yet practical, Lutheran reformers who were forced to directly confront the ideals of early Enlightenment in conjunction with the state-building mandate of Brandenburg-Prussia. This has led to a persistent tendency to see Halle Pietists as “others” who cultivated their collective identity in opposition to so-called Enlightenment intellectuals, like Christian Wolff, at the same time as they exerted a marked influence on these same persons. As a result of this dichotomous portrayal over the years, the impact of the Halle method on educational reform, and on the meanings eighteenth-century Europeans attached to philanthropy more generally, has been misunderstood. I argue that the Pietist Orphanage holds the key to remedying several problems that have impeded our ability to understand the significance of Pietist pedagogy and philanthropy. This was a site specifically designed to introduce children to the conciliatory knowledge-making strategies of the first Berlin Academy of Science members and their associates. These strategies championed the status of the heart as an assimilatory juncture point and were refined in the schools of the Pietist Orphanage, which itself functioned as a visual showplace that viewers could observe in order to edify and improve themselves. It was the material expression of Halle Pietists’ commitment to a “third way” and marked their attempt to assimilate experience and cognition, theology and philosophy, absolutism and voluntarism. The dissertation examines several personalities who had a direct bearing on this conciliatory project: namely E. W. von Tschirnhaus, Johann Christoph Sturm, Leonhard Christoph Sturm, Gottfried Wilhelm Leibniz and Christian Wolff. It also examines how the method was applied in the Halle Orphanage schools and extended elsewhere.

This project was formulated with the purpose of advancing knowledge of the invasion dynamics of rats within archipelagos in New Zealand. The concentration on islands reflected the conservation focus of this project - islands are the last refuges for many native New Zealand species that cannot survive in the wild on the mainland. This project can be divided into four areas: 1. Data collection: There was no intent for innovation here, but a deeper understanding of the environments in which rats are born, breed, migrate, and die was developed. 2. Development of tools for data exploration: • A user-friendly point-and-click graphical interface for the R program was designed to allow any user to easily explore simple genetic characteristics of the data. • A novel method for exploring the genetic similarity between individuals was developed and showcased with real data, proving successful in cases of both high and low genetic differentiation, and in detecting likely individual migrants. 3. Improvement of a method for estimating migration: • An attempt was made to improve the Markov chain Monte Carlo procedure underlying this method. • The migration model used by the method was significantly improved, so that it could cope with any level of migration. Previously, results from situations where migration rates were high were invalid. 4. Investigated topics of ecological interest: • Field measurements of rats were used to show that Norway rats tend to have larger masses than ship rats, southern rats are generally larger than northern rats, but the effect on mass of living on an island as opposed to the mainland depends on the latitude. It was also shown that relative tail length is a good species discriminator. • Multiple paternity was confirmed for both Norway and ship rats. This breeding characteristic might form part of the explanation for why rats are such successful invaders. During the project, case studies involving rats on Big South Cape Island, Great Barrier Island and in the Bay of Islands were used to highlight the methods developed, and provided some unexpected and fascinating results.

Gravitational waves are predicted by general relativity theory. Their existence could be confirmed by astronomical observations, but until today they have not yet been measured directly. A measurement would not only confirm general relativity, but also allow for interesting astronomical observations. Great effort is currently being expended to facilitate gravitational radiation measurement, most notably through earth-bound interferometers (such as LIGO and Virgo), and the planned space-based LISA interferometer. Earth-bound interferometers have recently taken up operation, so that a detection might be made at any time, while the space-borne LISA interferometer is scheduled to be launched within the next decade.Among the most promising signals for a detection are the waves emitted by the inspiral of a binary system of stars or black holes. The observable gravitational-wave signature of such an event is determined by properties of the inspiralling system, which may in turn be inferred from theobserved data. A Bayesian inference framework for the estimation of parameters of binary inspiral events as measured by ground- and space-based interferometers is described here. Furthermore, appropriate computational methods are developed that are necessary for its application in practice. Starting with a simplified model considering only 5 parameters and data from a single earth-bound interferometer, the model is subsequently refined by extending it to 9 parameters, measurements from several interferometers, and more accurate signal waveform approximations. A realistic joint prior density for the 9 parameters is set up. For the LISA application the model is generalised so that the noise spectrum is treated as unknown as well and can be inferred along with the signal parameters. Inference through the posterior distribution is facilitated by the implementation of Markov chain Monte Carlo (MCMC) methods. The posterior distribution exhibits many local modes, and there is only a small "attraction region" around the global mode(s), making it hard, if not impossible, for basic MCMC algorithms to find the relevant region in parameter space. This problem is solved by introducing a parallel tempering algorithm. Closer investigation of its internal functionality yields some insight into a proper setup of this algorithm, which in turn also enables the efficient implementation for the LISA problem with its vastly enlarged parameter space. Parallel programming was used to implement this computationally expensive MCMC algorithm, so that the code can be run efficiently on a computer cluster. In this thesis, a Bayesian approach to gravitational wave astronomy is shown to be feasible and promising.

This project was formulated with the purpose of advancing knowledge of the invasion dynamics of rats within archipelagos in New Zealand. The concentration on islands reflected the conservation focus of this project - islands are the last refuges for many native New Zealand species that cannot survive in the wild on the mainland. This project can be divided into four areas: 1. Data collection: There was no intent for innovation here, but a deeper understanding of the environments in which rats are born, breed, migrate, and die was developed. 2. Development of tools for data exploration: • A user-friendly point-and-click graphical interface for the R program was designed to allow any user to easily explore simple genetic characteristics of the data. • A novel method for exploring the genetic similarity between individuals was developed and showcased with real data, proving successful in cases of both high and low genetic differentiation, and in detecting likely individual migrants. 3. Improvement of a method for estimating migration: • An attempt was made to improve the Markov chain Monte Carlo procedure underlying this method. • The migration model used by the method was significantly improved, so that it could cope with any level of migration. Previously, results from situations where migration rates were high were invalid. 4. Investigated topics of ecological interest: • Field measurements of rats were used to show that Norway rats tend to have larger masses than ship rats, southern rats are generally larger than northern rats, but the effect on mass of living on an island as opposed to the mainland depends on the latitude. It was also shown that relative tail length is a good species discriminator. • Multiple paternity was confirmed for both Norway and ship rats. This breeding characteristic might form part of the explanation for why rats are such successful invaders. During the project, case studies involving rats on Big South Cape Island, Great Barrier Island and in the Bay of Islands were used to highlight the methods developed, and provided some unexpected and fascinating results.

Gravitational waves are predicted by general relativity theory. Their existence could be confirmed by astronomical observations, but until today they have not yet been measured directly. A measurement would not only confirm general relativity, but also allow for interesting astronomical observations. Great effort is currently being expended to facilitate gravitational radiation measurement, most notably through earth-bound interferometers (such as LIGO and Virgo), and the planned space-based LISA interferometer. Earth-bound interferometers have recently taken up operation, so that a detection might be made at any time, while the space-borne LISA interferometer is scheduled to be launched within the next decade.Among the most promising signals for a detection are the waves emitted by the inspiral of a binary system of stars or black holes. The observable gravitational-wave signature of such an event is determined by properties of the inspiralling system, which may in turn be inferred from theobserved data. A Bayesian inference framework for the estimation of parameters of binary inspiral events as measured by ground- and space-based interferometers is described here. Furthermore, appropriate computational methods are developed that are necessary for its application in practice. Starting with a simplified model considering only 5 parameters and data from a single earth-bound interferometer, the model is subsequently refined by extending it to 9 parameters, measurements from several interferometers, and more accurate signal waveform approximations. A realistic joint prior density for the 9 parameters is set up. For the LISA application the model is generalised so that the noise spectrum is treated as unknown as well and can be inferred along with the signal parameters. Inference through the posterior distribution is facilitated by the implementation of Markov chain Monte Carlo (MCMC) methods. The posterior distribution exhibits many local modes, and there is only a small "attraction region" around the global mode(s), making it hard, if not impossible, for basic MCMC algorithms to find the relevant region in parameter space. This problem is solved by introducing a parallel tempering algorithm. Closer investigation of its internal functionality yields some insight into a proper setup of this algorithm, which in turn also enables the efficient implementation for the LISA problem with its vastly enlarged parameter space. Parallel programming was used to implement this computationally expensive MCMC algorithm, so that the code can be run efficiently on a computer cluster. In this thesis, a Bayesian approach to gravitational wave astronomy is shown to be feasible and promising.