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SOLA International, a company which manufactures optical lenses, attended the 2000 Mathematics-in-Industry Study Group (MISG) with a wish list. Topping this list was the creation of a mathematical model of a lens, which given the lens geometry and material properties, could predict the deformation of the lens when it was subjected to an impact, such as that experienced in the fracture tests lenses must pass before being approved for sale. The first steps towards such a model were taken at MISG. At MISG, a lens was modelled simply as a thin uniform thickness plate, undergoing small, linear deformations. In the first section of my thesis I extend this model by considering variable thickness plates and larger, nonlinear deformations. For this extended model I have confirmed that the result obtained at MISG, that the contact between a plate and a spherical indentor occurs at a single point, still holds. The second part of this thesis looks at the dynamic deformation, or vibration, of plates. I have developed numerical solution methods for the large amplitude vibration equations with and without the in-plane inertia terms, based on a finite difference scheme. A comparison of these solutions confirms the often used assumption that the in-plane inertia may be neglected. I have also implemented a number of solution methods from the literature, which use separation of variables techniques. Comparing these with the numerical solutions, we find that the numerical solutions better capture the multi-modal nature of the vibration - showing multiple cycles in the approximate period. Having achieved an understanding of the types of forces involved in plate deformation and vibration I consider shell theory in the final section of my thesis. While time constraints meant no dynamic results could be obtained, general nonlinear deep shell equations have been derived. The static version of these equations has then been solved, with the development of a new solution technique which combines a Taylor expansion to approximate the behaviour at the shell centre with a numerical shooting method. Various shallow shell simplifications of the deep shell equations are then discussed and solved. By comparison of the solutions obtained for the deep and shallow, linear and nonlinear equations I have been able to determine which theories apply to which geometries. A complete model of a lens needs to take into account the shape, its thickness and curvature and the material from which it is made. From the work done in this thesis we have been able to determine that a lens model would require the nonlinear theory. Whether the deep shell theory is necessary is debatable as the geometry of a typical lens falls in the grey area, where either theory could be used depending on the accuracy required. For a very accurate model, deep shell theory would be necessary; if an approximate solution obtained quickly was more useful then I suggest the use of a particular set of shallow shell equations. A full lens model would require variable thickness shell theory and the solution of the dynamic equations, neither of which has been achieved here, but the solution techniques I have developed would be applicable to these theories.

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.