The interaction of multiple bodies and water waves : with the application to the motion of ice floes : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand

To understand the propagation of water waves through arrays of floating or (fully or partially) submerged bodies it is necessary to know how these bodies interact with each other under the influence of ambient waves. However, the conventional full diffraction calculation of the scattered wavefields of many interacting bodies requires a considerable computational effort. In this thesis, a method is developed which makes it possible to quickly calculate the wave scattering of many interacting floating or (fully or partially) submerged, vertically non-overlapping bodies of arbitrary geometry in water of infinite depth. It extends Kagemoto and Yue's analysis for axisymmetric bodies in finite depth. The idea of this method is to expand the water velocity potential into its cylindrical eigenfunctions such that, the scattered potentials of the bodies are defined by a set of coefficients only. Representing the scattered wavefield of each body as an incident wave upon all other bodies, a linear system of equations for the coefficients of the scattered wavefields is derived. Diffraction transfer matrices which relate the coefficients of the incoming wavefield upon a single body to those of its scattered wavefield play an important role in the process. The calculation of the diffraction transfer matrices for bodies of arbitrary shape requires the representation of the infinite depth free surface Green's function in the eigenfunctions of an outgoing wave. This eigenfunction expansion will be derived from the equivalent finite depth Green's function. An important application of this interaction method is the propagation of ocean waves through fields of ice floes which can be modelled as floating flexible thin plates. Meylan's method of solution is used to calculate the motion of a single ice floe from which the solutions for multiple interacting ice floes are computed. While the interaction theory will be derived for general floating or submerged bodies, particular examples are always given for the case of ice floes. Results are presented for ice floes of different geometries and in different arrangements and convergence tests comparing the finite and the infinite depth method are conducted with two square interacting ice floes where full diffraction calculations serve as references

Perspective: border security in the age of globalization: how can we protect ourselves without losing the benefits of openness?

Border security has become increasingly important since 9-11. Yet the benefits of globalization depend on moving people and goods across national boundaries. How can we improve border security without losing the benefits of openness?National security

Convergence of numerical methods for stochastic differential equations in mathematical finance

Many stochastic differential equations that occur in financial modelling do not satisfy the standard assumptions made in convergence proofs of numerical schemes that are given in textbooks, i.e., their coefficients and the corresponding derivatives appearing in the proofs are not uniformly bounded and hence, in particular, not globally Lipschitz. Specific examples are the Heston and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia model with rational coefficient functions. Simple examples show that, for example, the Euler-Maruyama scheme may not converge either in the strong or weak sense when the standard assumptions do not hold. Nevertheless, new convergence results have been obtained recently for many such models in financial mathematics. These are reviewed here. Although weak convergence is of traditional importance in financial mathematics with its emphasis on expectations of functionals of the solutions, strong convergence plays a crucial role in Multi Level Monte Carlo methods, so it and also pathwise convergence will be considered along with methods which preserve the positivity of the solutions.Comment: Review Pape

Nondispersive two-electron wave packets in driven helium

We provide a detailed quantum treatment of the spectral characteristics and of the dynamics of nondispersive two-electron wave packets along the periodically driven, collinear frozen planet configuration of helium. These highly correlated, long-lived wave packets arise as a quantum manifestation of regular islands in a mixed classical phase space, which are induced by nonlinear resonances between the external driving and the unperturbed dynamics of the frozen-planet configuration. Particular emphasis is given to the dependence of the ionization rates of the wave packet states on the driving field parameters and on the quantum mechanical phase space resolution, preceded by a comparison of 1D and 3D life times of the unperturbed frozen planet. Furthermore, we study the effect of a superimposed static electric field component, which, on the grounds of classical considerations, is expected to stabilize the real 3D dynamics against large (and possibly ionizing) deviations from collinearity.Comment: 31 pages, 18 figures, submitted to European Physical Journal

Rayleigh-Jeans condensation of pumped magnons in thin film ferromagnets

We show that the formation of a magnon condensate in thin ferromagnetic films can be explained within the framework of a classical stochastic non-Markovian Landau-Lifshitz-Gilbert equation where the properties of the random magnetic field and the dissipation are determined by the underlying phonon dynamics. We have numerically solved this equation for a tangentially magnetized yttrium-iron garnet film in the presence of a parallel parametric pumping field. We obtain a complete description of all stages of the nonequilibrium time evolution of the magnon gas which is in excellent agreement with experiments. Our calculation proves that the experimentally observed condensation of magnons in yttrium-iron garnet at room temperature is a purely classical phenomenon which should be called Rayleigh-Jeans rather than Bose-Einstein condensation.Comment: 5 pages, 4 figures, 7 pages supplemental material with 2 additional figure

Efficient Minimization of Decomposable Submodular Functions

Many combinatorial problems arising in machine learning can be reduced to the problem of minimizing a submodular function. Submodular functions are a natural discrete analog of convex functions, and can be minimized in strongly polynomial time. Unfortunately, state-of-the-art algorithms for general submodular minimization are intractable for larger problems. In this paper, we introduce a novel subclass of submodular minimization problems that we call decomposable. Decomposable submodular functions are those that can be represented as sums of concave functions applied to modular functions. We develop an algorithm, SLG, that can efficiently minimize decomposable submodular functions with tens of thousands of variables. Our algorithm exploits recent results in smoothed convex minimization. We apply SLG to synthetic benchmarks and a joint classification-and-segmentation task, and show that it outperforms the state-of-the-art general purpose submodular minimization algorithms by several orders of magnitude.Comment: Expanded version of paper for Neural Information Processing Systems 201

The effect of prototyping material on verbal and non-verbal behaviours in collaborative design tasks

This paper reports a study of 23 controlled experiments, with a total of 99 individual tasks, between pairs of designers collaborating to solve a simple design task using four different types of prototyping media. The aim of the study was to correlate verbal and non-verbal behaviours across different types of media with a range of measurement indicators. Using innovative movement trail images we show how collaborative sketching activity results in attenuated use of interpersonal collaborative space when compared with cardboard, clay, and Lego, which provoked intensive collaboration. Furthermore, the sketching (control) condition resulted in pre-conceived ideas being executed when compared with the three-dimensional media, where ideas emerged through collaboration. This finding suggests that increased creativity in design can result through the careful choice of prototyping media at the beginning of the design process

Regular infinite dimensional Lie groups

Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisingly far the geometry of principal bundles: parallel transport exists and flat connections integrate to horizontal foliations as in finite dimensions. As consequences we obtain that Lie algebra homomorphisms intergrate to Lie group homomorphisms, if the source group is simply connected and the image group is regular.Comment: AmSTeX, using diag.tex with fonts lams?.ps, 38 page