This thesis contains five chapters. The first two are devoted to the background which consists of integration, Fourier analysis, distributions and linear operators in Hilbert spaces. The third chapter is a generalization of a work done by Albrecht-Spain in 2000. We give a shorter proof of the main theorem they proved for bounded operators and we generalize it to unbounded operators. We give a counterexample that shows that the result fails to be true for another class of operators. We also say why it does not hold. In chapters four and five, the idea is the same, that is to find classes of unbounded, real-valued V<i>s </i>for which  + <i>V</i> is self-adjoint on <i>D</i>(), where  is the wave operator. Throughout these two chapters we will see how different the Laplacian and the wave operator can be.

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We prove the group configuration theorem in simple theories, a very abstract result reconstructing a group (action) from a certain independence-theoretic configuration of points, and argue that such a result gives rise to 'geometric simplicity theory' (analogues of methods and results of geometric stability theory). The proof involves studying the behaviour of multivalued algebraic structures like polygroups and polyspaces, a development of the theory of independence for almost hyperimaginaries, and a sophisticated blowup procedure. Some of the corollaries of the group configuration theorem we obtain include finding the group associated to a polygroup in a simple theory, interpreting a vector space over a finite field inside a one-based <i>w-</i>categorical theory of SU-rank 1, and showing how pseudolinearity implies one-basedness under the assumption of w-categoricity.

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Complex-valued representation of a two-component real-valued time series yields additional physical insights that are lost otherwise. The spectral representation theorem allows us to study covariance stationary complex-valued random sequences in the frequency domain, and this is known as rotary spectral analysis. It is a widely-used technique for studying elliptical motions in ocean currents, wind etc. An important and useful parameter in rotary spectral analysis of scalar complex-valued time series is the rotary coefficient. It measures the tendency of vectors to rotate in a clockwise or counter-clockwise manner. We derive the theoretical distribution of the rotary coefficient estimator and apply our results to ocean current speed and direction measurements at six depths in the Labrador Sea. Canonical correlation techniques are commonly employed in the analysis of a pair of vector-valued random variables. We introduce a framework to extend classical multivariate analysis techniques such as canonical correlation analysis, partial least squares, and multivariate linear regression, to define coherence - a measure of correlation in the frequency domain. In the statistical analysis of complex-valued time series, we refer to a time series as proper/improper according to whether it is uncorrelated/correlated with its complex conjugate. In earlier work, complex-valued signals were assumed to be proper for the simple reason that it led to a simpler algebra. However, the loss in performance caused by overlooking the potential impropriety of such data is realized to be significant, and therefore, when the data is improper, information contained in the complementary covariance structure must be considered. Since impropriety in the time domain may not necessarily correspond to impropriety at all frequencies, we propose a generalized likelihood ratio test which may be used to test propriety of a discrete time complex-valued process at a given frequency. Finally, the idea of vector circulant embedding is exploited to yield a frequency domain bootstrap methodology. With the help of three example parameters involved in the study of multi-channel complex-valued time series, we illustrate how our method allows us to draw statistical inference such as confidence intervals. Our method can prove useful in cases where no theoretical distributional results are available, or to check the effect of nuisance parameter estimates where theoretical results are available.

In this thesis two novel approaches to pricing of barrier and American options are developed in the setting of local volatility models with jumps: the moments method and the Markov chain method. The moments method is a valuation approach for barrier options that is based on a characterisation of the exit location measure and the expected occupation measure of the price process of the underlying in terms of the corresponding moments. It is shown how the value of barrier-type derivatives can be expressed using these moments, which are in turn shown to be characterised by an infinite-dimensional linear system. By solving finite-dimensional linear programming problems, which are obtained by restricting to moments of a finite degree, upper and lower-bounds are found for the values of the options in question. The Markov chain method for the valuation of American options is based on an approximation of the underlying price process by a continuous-time Markov chain. The value-function of the American option driven by the approximating chain is identified by solving the associated optimal stopping problem. In particular, a novel explicit characterisation of the optimal exercise boundary is derived in terms of the generator of the Markov chain. Using this characterisation it is shown that the optimal exercise boundary and the corresponding value-function can be evaluated efficiently. For both of the presented methods convergence results are established. The methods are implemented for a range of local volatility models with jumps, and a number of numerical examples are discussed in detail to illustrate the scope of the methods.

It has been conjectured that Fano manifolds correspond to certain Laurent polynomials under Mirror Symmetry. This correspondence predicts that the regularized quantum period of a Fano manifold coincides with the classical period of a Laurent polynomial mirror. This correspondence is not one-to-one, as many different Laurent polynomials can have the same classical period; it should become one-to-one after imposing the correct equivalence relation on Laurent polynomials. In this thesis we introduce what we believe to be the correct notion of equivalence: this is algebraic mutation of Laurent polynomials. We also consider combinatorial mutation, which is the transformation of lattice polytopes induced by algebraic mutation of Laurent polynomials supported on them. We establish the basic properties of algebraic and combinatorial mutations and give applications to algebraic geometry, most notably to the classification of Fano manifolds up to deformation. Our main focus is on the surface case, where the theory is particularly rich.

A fundamental challenge of marine ecology is to understand climate change induced range shifts. At the level of individual ecosystems, the impacts of these range shifts will result from the complex changes in community composition. This forces us to consider not simply individual species, but the species that may replace them and play similar roles in an ecosystem. Here we consider the important zooplankton species Calanus finmarchicus and C. helgolandicus. Recent climate change has resulted in the gradual replacement of C. finmarchicus by C. helgolandicus in the North Sea. However, the ability of C. helgolandicus to fully replace C. finmarchicus has been questioned by some researchers. We therefore sought to fill a key knowledge gap. The comparative differences between the two species have never been critically reviewed. Further, the relative geographic distributions of both species have never been related to inter-species differences in biology or ecology. This thesis has two key elements. First, we critically review and synthesize existing knowledge of inter-species differences between C. finmarchicus and C. helgolandicus, overturning many assumptions common in the literature. Second, we produce a unified population model of both species across the North Atlantic, which relates differences in geographic distribution to inter-species differences in biology. This model is then used to highlight the limits of current understanding of mortality, and of the vital importance of improved quantitative knowledge of overwintering if we are to understand the future evolution of both species' distributions.