Modelling of volcanic ashfall has been attempted by volcanologists but very little work has been done by mathematicians. In this thesis we show that mathematical models can accurately describe the distribution of particulate materials that fall to the ground following an eruption. We also report on the development and analysis of mathematical models to calculate the ash concentration in the atmosphere during ashfall after eruptions. Some of these models have analytical solutions. The mathematical models reported on in this thesis not only describe the distribution of ashfall on the ground but are also able to take into account the effect of variation of wind direction with elevation. In order to model the complexity of the atmospheric flow, the atmosphere is divided into horizontal layers. Each layer moves steadily and parallel to the ground: the wind velocity components, particle settling speed and dispersion coefficients are assumed constant within each layer but may differ from layer to layer. This allows for elevation-dependent wind and turbulence profiles, as well as changing particle settling speeds, the last allowing the effects of the agglomeration of particles to be taken into account.

Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.

Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He work extensively in a branch of mathematics called "q-series". Around 1913, he found an important formula which now is known as Ramanujan's 1ψ1summation formula. The aim of this thesis is to investigate Ramanujan's 1ψ1summation formula and explore its applications to number theory and combinatorics. First, we consider several classical important results on elliptic functions and then give new proofs of these results using Ramanujan's 1ψ1 summation formula. For example, we will present a number of classical and new solutions for the problem of representing an integer as sums of squares (one of the most celebrated in number theory and combinatorics) in this thesis. This will be done by using q-series and Ramanujan's 1ψ1 summation formula. This in turn will give an insight into how Ramanujan may have proven many of his results, since his own proofs are often unknown, thereby increasing and deepening our understanding of Ramanujan's work.

Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.

We propose an improved technique to calculate the linear response of a single and multiple plates models due to ocean waves. The single plate model is the basis for the multiple plates model which we take to be a periodic array of identical plates. For the single plate model we solve the plate displacement by the Finite Element Method (FEM) and the water potential by the Boundary Element Method (BEM). The displacement is expanded in terms of the basis functions of the FEM. The boundary integral equation representing the potential is approximated by these basis functions. The resulting integral operator involving the free-surface Green's function is solved using an elementary integration scheme. Results are presented for the single plate model. We then use the same technique to solve for the periodic array of plates problem because the single and the periodic array plates model differ only in the expression of the Green's function. For the periodic array plate model the boundary integral equation for the potential involves a periodic Green's function which can be obtained by taking an infinite sum of the free-surface Green's function for the single plate model. The solution for the periodic array plate is derived in the same way as the single plate model. From this solution we then calculate the waves scattered by this periodic array.

The purpose of fisheries management is to achieve a sustainable development of the activity, so that future generations can also benefit from the resource. However, the optimal harvesting strategy usually maximizes an economically important objective function formed by the harvester which can lead to the extinction of the resource population. Therefore, sustainability has been far more difficult to achieve than is commonly thought; fish populations are becoming increasingly limited and catches are declining due to overexploitation. The aim of this research is to determine an optimal harvesting strategy which fulfills the economic objective of the harvester while maintaining the population density over a pre-specified minimum viable level throughout the harvest. We develop and investigate the harvesting model in both deterministic and stochastic settings. We first employ the Expected Net Present Value approach and determine the optimal harvesting policy using various optimization techniques including optimal control theory and dynamic programming. Next we use real options theory, model fish harvesting as a real option, and compute the value of the harvesting opportunity which also yields the optimal harvesting strategy. We further extend the stochastic problem to include price elasticity of demand and present results for di¤erent values of the coefficient of elasticity.

Microplitis croceipes, one of the nectar feeding parasitoid wasps, has been found to associatively learn chemical cues through feeding. The experiments on M. croceipes are performed and recorded by a Sony camcorder in the USDA-ARS Biological Control Laboratory in Tifton, GA, USA. The experimental videos have shown that M. croceipes can respond to Coffee odour in this study. Their detection capabilities and the behaviour of M. croceipes with different levels of coffee odours were studied. First, the data that are related to trained M. croceipes behaviour was extracted from the experimental videos and stored in a Microsoft Excel database. The extracted data represent the behaviour of M. croceipes trained to 0.02g and then exposed to 0.001g, 0.005g, 0.01g, 0.02g and 0.04g of coffee. Secondly, indices were developed to uniquely characterise the behaviour of trained M. croceipes under different coffee concentrations. Thirdly, a preliminary model and its parameters were developed to classify the response of trained wasps when exposed to these five different coffee odours. In summary, the success of this thesis demonstrates the usefulness of data analysis for interpreting experimental data, developing indices, as well as understanding the design principles of a simple model based on trained wasps.

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.