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Perhaps one of the most amazing events that occurs in nature, is in the emergence and growth of biological life. Emergence speaks of the well-coined phrase Primordial ooze from which the chemical building blocks of life first gave rise to the complicated molecular structure of Deoxyribonucleic acid (DNA), that has the mind boggling task of encoding every chemical and physical attribute and trait of the organism for which it is encoded. This incredible feat of nature is only equalled by the ability of single fertilized cell (zygote) to undergo a seemingly magical transformation through enlargement, growth and change to give rise to a fully formed animal (or plant). The study and body of knowledge of this latter process is called Developmental Biology, and it seeks to define and explain all of the intricate sub-stages and bio-chemical, molecular and physical processes along the time-line of this transformation, that is from fertilization to birth, hatching or germination and beyond. One might consider, and quite reasonably, that the variety of different processes leading to the development of a complete biological organism would be so vast as to render the problem untenable. Indeed the almost inconceivable amount of genetic information contained within the nucleus of the simplest of cells would seem to corroborate this assumption. However when one takes a more holistic view, we can see that the development of any complex biological organism can be reduced to a set of five distinct processes, all of which are orchestrated to define structures from a body of cells. Viewed in this light the generation of any complex multi-cellular organism, be it small or large, must involve: cell-division, differentiation, pattern formation, change in form and growth [1]. To mediate and orchestrate these different processes during the development of the embryo are a enumerable number of bio-chemicals that are produced within the cells that can diffuse into the surrounding environment, activating (and de-activating) inter/intra-cellular signalling pathways that trigger further productions and possibly one or more of the processes suggested above. One such case of this, and which is of particular interest in this thesis, is in the role of morphogens in the growth of vertebrate embryos, where it is known that interacting morphogen gradients can give rise to spatially stable concentrations [2] that are known to be involved in organ growth [3], primitive streak formation [4] and the extension and patterning of the primary body axis [5, 6, 7]. In this thesis we are considering one such problem involving these mechanisms/processes, during the primary body axis extension in the chick embryo. During this phase of development the early brain is beginning to form and the central nervous system (CNS) is beginning to extend unilaterally in a posterior direction defining the main anteroposterior (head to tail) body axis; in simple terms one may see this as the generation of the spinal cord and surrounding structures. Extension of this axis is known to be orchestrated by a small cellular structure located at the posterior-most tip of the extension, encompassing what is known as the primary organising centre in the chick embryo: Hensen’s node. This structure including the node is known to move independently/autonomously of the rest of the embryo and as it does so the cells in the region are growing and proliferating, and ultimately differentiating and leaving this region to literally fuel the axial extension. This broad description leads us to the heart of our thesis, and which will preoccupy the rest of this dissertation. We postulate that the motile behaviour of the group is as a result of biochemical gradients to which the group is attracted toward areas of highest concentration or towards areas of lowest concentration of some as yet unnamed morphogen. That is we assume that the group moves as a result of a chemotaxis. Furthermore, the growth and subsequent differentiation of cells exiting the group, contributing to the growth of the CNS, are also regulated by the same morphogen. Therefore we propose that a singular bio-chemical mechanism can account for the motile and growth behaviour observed during CNS extension.

In this thesis we present a study of the glueball spectrum in lattice QCD, using 2 + 1 flavours of Asqtad fermions and the one-loop Symanzik improved Lüscher-Weisz gauge action. The ensembles for this study have been generated with very high statistics ($N_traj\sim 20000−30000$) to enable us to resolve the notoriously noisy glueball states as accurately as possible. We introduce the theoretical construction of lattice QCD and of staggered fermions in particular before describing how one goes about generating ensembles of gauge fields for such analysis. Here we briefly present our tuning results performed using the improved RHMC algorithm used to generate the finer ensemble. We then present the methods by which one measures glueballs on the lattice before presenting our measurements and analysis for the scalar glueball. Here we discuss several complications which one may face, as we have, performing spectroscopy on the lattice. We finally present determinations of the $0^{++}$ and the excited $0^{++}$ glueball masses, as well as tentative continuum extrapolations. We have also measured the pseudoscalar and tensor glueball states on our lattices and we present these results. Where possible we present a comparison with previous lattice measurements of the glueball spectrum obtained using both dynamical quarks and the quenched approximation in order to gauge the scale of unquenching effects on the glueball spectrum. The study of the scalar glueball forms part of a wider physics project by the UKQCD collaboration which aims to study the flavour-singlet sector using 2 + 1 flavours of dynamical fermions with unprecedented statistics. We briefly present motivation and an outline of the measurements performed as part of this wider project.

Complex interacting systems permeate the modern world. Many diverse natural, social and human made systems—ranging from food webs to human contact networks, to the Internet—can be studied in the context of network science. This thesis is a compendium of research in applied network science, investigating structural and dynamical patterns behind the formation of networks and processes supported on them. Trophic food webs—networks of who eats whom in an ecosystem—have fascinated network scientists since data from field observations of the gut content of species first became available. The empirical patterns in food webs reveal a rich hierarchy of feeding patterns. We study how global structure of food webs relates to species immediate diet over a range of 46 different ecosystems. Our finding suggest that food webs fall broadly into two different families based on the extent of species tendency towards omnivory. Drawing inspiration from food webs, we investigate how trophic networks support spreading processes on them. We find that the interplay of dynamics and network structure determines the extent and duration of contagion. We uncover two distinct modes of operation—short-lived outbreaks with high incidence and endemic infections. These results could be important for understanding spreading phenomena such as epidemics, rumours, shocks to ecosystems and neuronal avalanches. Finally, we study the emergence of structural order in random network models. Random networks serve as null models to empirical networks to help uncover significant non-random patterns but are also interesting to study in their own right. We study the effect of triadic ties in delaying the formation of extensive giant components— connected components taking over the majority of the network. Our results show that, depending on the network formation process, order in the form of a giant component can emerge even with a significant number of triadic ties.

The work presented in this thesis develop algorithms to solve inverse problems where source terms are unknown. The algorithms are developed 011frameworks provided by domain decomposition methods and the numerical schemes use finite volume and finite difference discretisations. Three algorithms are developed in the context of a metal cutting problem. The algorithms require measurement data within the physical body in order to retrieve the temperature field and the unknown source terms. It is shown that the algorithms can retrieve both the temperature field and the unknown source accurately. Applicability of the algorithms to other problems is shown by using one of the algorithms to solve a welding problem. Presence of untreated noisy measurement data can severely affect the accuracy of the retrieved source. It is illustrated that a simple noise treatment procedure such as a least squares method can remedy this situation. The algorithms are implemented 011parallel computing platforms to reduce the execution time. By exploiting domain and data parallelism within the algorithms significant performance improvements are achieved. It is also shown that by exploiting mathematical properties such as change of nonlinearity further performance improvements can be made.

Solar sailing is the use of a thin and lightweight membrane to reflect sunlight and obtain a thrust force on the spacecraft. That is, a sailcraft has a potentially-infinite specific impulse and, therefore, it is an attractive solution to reach mission goals otherwise not achievable, or very expensive in terms of propellant consumption. The recent scientific interest in near-Earth asteroids (NEAs) and the classification of some of those as potentially hazardous asteroids (PHAs) for the Earth stimulated the interest in their exploration. Specifically, a multiple NEA rendezvous mission is attractive for solar-sail technology demonstration as well as for improving our knowledge about NEAs. A preliminary result in a recent study showed the possibility to rendezvous three NEAs in less than ten years. According to the NASA’s NEA database, more than 12,000 asteroids are orbiting around the Earth and more than 1,000 of them are classified as PHA. Therefore, the selection of the candidates for a multiple-rendezvous mission is firstly a combinatorial problem, with more than a trillion of possible combinations with permutations of only three objects. Moreover, for each sequence, an optimal control problem should be solved to find a feasible solar-sail trajectory. This is a mixed combinatorial/optimisation problem, notoriously complex to tackle all at once. Considering the technology constraints of the DLR/ESA Gossamer roadmap, this thesis focuses on developing a methodology for the preliminary design of a mission to visit a number of NEAs through solar sailing. This is divided into three sequential steps. First, two methods to obtain a fast and reliable trajectory model for solar sailing are studied. In particular, a shape-based approach is developed which is specific to solar-sail trajectories. As such, the shape of the trajectory that connects two points in space is designed and the control needed by the sailcraft to follow it is analytically retrieved. The second method exploits the homotopy and continuation theory to find solar-sail trajectories starting from classical low-thrust ones. Subsequently, an algorithm to search through the possible sequences of asteroids is developed. Because of the combinatorial characteristic of the problem and the tree nature of the search space, two criteria are used to reduce the computational effort needed: (a) a reduced database of asteroids is used which contains objects interesting for planetary defence and human spaceflight; and (b) a local pruning is carried out at each branch of the tree search to discard those target asteroids that are less likely to be reached by the sailcraft considered. To reduce further the computational effort needed in this step, the shape-based approach for solar sailing is used to generate preliminary trajectories within the tree search. Lastly, two algorithms are developed which numerically optimise the resulting trajectories with a refined model and ephemerides. These are designed to work with minimum input required by the user. The shape-based approach developed in the first stage is used as an initial-guess solution for the optimisation. This study provides a set of feasible mission scenarios for informing the stakeholders on future mission options. In fact, it is shown that a large number of five-NEA rendezvous missions are feasible in a ten-year launch window, if a solar sail is used. Moreover, this study shows that the mission-related technology readiness level for the available solar-sail technology is larger than it was previously thought and that such a mission can be performed with current or at least near-term solar sail technology. Numerical examples are presented which show the ability of a solar sail both to perform challenging multiple NEA rendezvous and to change the mission en-route.

We study approaches for finding good solutions, and lower bounds, for three difficult combinatorial optimisation problems. The supply ship travelling salesman problem is a simplification of a situation faced by a naval logistics coordinator who must direct a support vessel tasked with resupplying ships in a fleet. It is a generalisation of the travelling salesman problem in which the nodes are in motion, each following some predetermined route. We apply dynamic programming state-space relaxation techniques, producing lower bounds for the problem that are 73% to 84% of the best solution, on average. We also apply heuristics to find good solutions to this NP-hard problem, showing that restricted dynamic programming approaches outperform simple 2-opt and 3-opt local search procedures for instances with 20 nodes. We introduce the supply ship scheduling problem, another roblem inspired by a support vessel environment. We wish to minimise the number of mobile machines required to process a set of jobs; each job is in a different stationary location and features a fixed start time. Jobs may be simultaneously processed by multiple machines, obtaining a speed-up in processing time. We represent the problem as a directed graph and use the minimum flow in a transformed network to determine the minimum number of machines. We present a neighbourhood structure based on the maximum cut, applying it within descent and tabu search procedures. We construct a restricted dynamic programming based approach, but this is outperformed by the tabu search algorithm. The task allocation problem, arising in distributed computing, is to assign a set of tasks to a set of processors so that the overall cost is minimised. Costs are incurred from processor usage, interprocessor communication and task execution. We construct, and try to improve, semidefinite programming relaxations to find lower bounds for variants of this NP-hard problem. We develop a branch-and-bound approach to find optimal solutions, but this is only effective for small instances.

In ‘Groups acting on connected cubes and Kazhdan’s property T’, [29], Niblo and Roller showed that any CAT(0) cube complex embeds combinatorially and quasi-isometrically in the Hilbert space `2(H) where H is the set of hyperplanes. This Hilbert space may be viewed as the completion of an infinite product of trees. In this thesis, we consider the question of the existence of quasi-isometric maps from CAT(0) cube complexes to finite products of trees, restricting our attention to folding maps as used in [29]. Following an overview of the properties of CAT(0) cube complexes, we first prove that there exists CAT(0) square complexes which do not fold into a product of trees with fewer than k factors for arbitrary k, giving examples which admit co-compact proper actions by right-angled Coxeter groups. We also show that there exists a CAT(0) square complex which does not fold into any finite product of trees. We then identify a class of group actions on CAT(0) cube complexes for which the existence of such an action implies the existence of a quasi-isometric embedding of that group in a finite product of finitely branching trees. We apply this result to surface groups, certain 3-manifold groups and more generally to Coxeter groups which do not contain affine triangle subgroups.

Reasoning about actions and change based on common sense knowledge is one of the most important and difficult tasks in the artificial intelligence research area. A series of such tasks are identified which motivate the consideration and application of reasoning formalisms. There follows a discussion of the broad issues involved in modelling time and constructing a logical language. In general, worlds change over time. To model the dynamic world, the ability to predict what the state of the world will be after the execution of a particular sequence of actions, which take time and to explain how some given state change came about, i.e. the causality are basic requirements of any autonomous rational agent. The research work presented herein addresses some of the fundamental concepts and the relative issues in formal reasoning about actions and change. In this thesis, we employ a new time structure, which helps to deal with the so-called intermingling problem and the dividing instant problem. Also, the issue of how to treat the relationship between a time duration and its relative time entity is examined. In addition, some key terms for representing and reasoning about actions and change, such as states, situations, actions and events are formulated. Furthermore, a new formalism for reasoning about change over time is presented. It allows more flexible temporal causal relationships than do other formalisms for reasoning about causal change, such as the situation calculus and the event calculus. It includes effects that start during, immediately after, or some time after their causes, and which end before, simultaneously with, or after their causes. The presented formalism allows the expression of common-sense causal laws at high level. Also, it is shown how these laws can be used to deduce state change over time at low level. Finally, we show that the approach provided here is expressive.

This thesis introduces a new method for solving the linear regression problem where the number of observations n is smaller than the number of variables (predictors) v. In contrast to existing methods such as ridge regression, Lasso and Lars, the proposed method uses the idea of graphical models and provides unbiased parameter estimates under certain conditions. In addition, the new method provides a detailed graphical conditional correlation structure for the predictors, whereby the real causal relationship between predictors can be identified. Furthermore, the proposed method is extended to form a hybridisation with the idea of ridge regression to improve efficiency in terms of computation and model selection. In the extended method, less important variables are regularised by a ridge type penalty, and a search for models in the space is made for important covariates. This significantly reduces computational cost while giving unbiased estimates for the important variables as well as increasing the efficiency of model selection. Moreover, the extended method is used in dealing with the issue of portfolio selection within the Markowitz mean-variance framework, with n < v. Various simulations and real data analyses were conducted for comparison between the two novel methods and the aforementioned existing methods. Our experiments indicate that the new methods outperform all the other methods when n<v.