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This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions. The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting. The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method. Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions. As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.

The response of piles and two-pile groups to lateral loading has been studied by field tests and computationally. Due to the lack of field test data and because of uncertainty concerning the pile/soil system it has been suggested that further experimental studies of pile groups under lateral loading should be undertaken. The research was conducted through a series of tests on vertical single piles and two-pile groups at various spacing and pile cap overhang heights, to identify the lateral stiffness, bending moment and axial force distribution. Attempts were also made to measure the in-situ total lateral soil pressure on the pile walls. Piles were designed to behave as "long" pile since most piles used in the U.K. are long and flexible. Piles were instrumented with strain gauges for measurement of bending moments and axial forces. Field tests were conducted in a sand trench using 4.0m long piles. A stiff steel pile cap was used to connect head of the two piles firmly together. Linear elastic back analyses of single pile tests were carried out to estimate the soil modulus profile with depth. Thereafter comparisons were made between the field test results on two-pile groups, published analyses and also a three dimensional finite element analysis. Tests results showed that the lateral stiffness of a two-pile groups tends towards a limit as spacing increases. A similar result was found from predictive and finite element analyses. The ratio between the maximum pile shaft bending moment and horizontal force varied between dry and wet season, being greater in the latter. The ratio between maximum reverse bending moment and horizontal load increased as the pile spacing and the overhang increased. Similar results results were found by finite element analysis. One of the main achievements in this research was the measurement of the axial forces in the vertical piles due to lateral loading. It was found that as the pile spacing increased and pile cap overhang height decreasd the peak axial forces per unit load decreased. Similar results were obtained by three dimensional finite element analysis.

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