The global phenomena observed in complex systems are not inherent in the individual constituents but arise from their local interactions. It is often useful to model such systems as networks, which retain these important local interactions while abstracting the detail away. Adaptive networks, in particular, are well suited to the modelling of such systems. The feedback loop between state and topology can give rise to the self-organising patterns observed in complex systems. In the mathematical exploration of such models the central challenge is often to map the problem onto a tractable set of equations. This thesis aims to address this challenge by coupling heterogeneous moment expansions and generating functions in order to model adaptive networks. In the first part of the thesis, we use heterogeneous moment expansions to describe stochastic agent-based models by infinite-dimensional systems of ordinary differential equations (ODEs). We then convert the infinite-dimensional systems of ODEs into low-dimensional systems of partial differential equations (PDEs) using generating functions. We finally solve the PDEs using methods from the literature. When analytic solutions are not possible we provide a method to obtain an accurate approximation to the solution using a Taylor series expansion of the generating function. In the second part of the thesis, we use the methodology of heterogeneous expansions and generating functions to address the challenge of designing networks that self-organise towards target degree distributions. Beginning with a set of processes acting on a network and a target steady-state degree distribution, we investigate the generating function PDEs produced under their action. From this we determine the combination of process rates required to produce such a target distribution. Where the first half of the thesis focuses on modelling real-world systems using adaptive networks, the second half instead looks at reverse engineering such systems in order to produce some desired global phenomenon.

Let X and Y be two non-negative, integer-valued random variables, such that X > Y, and let Z=X-Y. When the conditional distribution of Y/X=n is used for making inferences about the distribution of X or the distribution of Y, this model is called a conditionality model. Rao (Classical and Contagious discrete distributions 1963), introduced a new version of the conditionality model; he called this a damage model. In this model X represents an observation which is produced by some natural process and which may be partially damaged; Y/X=n is the destructive process. Thus Y stands for what we actually observe of X (the remaining part of X). Rao and Rubin (Sankhya 1964) obtained a characterization for the Poisson distribution using damage model theory and a condition which has come to be known as the Rao-Rubin condition. In this thesis an extension of the Rao-Rubin characterization which has been suggested by the work of Shanbhag (J.A.P. 1977) has been used to characterize many well-known discrete distributions as the distribution of X or as the distribution of Y/X=n when the other one of the two is given. This model is extended to provide characterizations for truncated distributions. A new model is suggested enabling us to characterize finite discrete distributions, truncated and untruncated. Bivariate and Multivariate extensions of all the results obtained in the Univariate case are derived. Finally, the damage model is examined in the more general situation where either the distribution of X or the distribution of Y/X=n is a compound distribution. Some interesting characterizations are provided by this situation. Many of the results existing in the literature in this field are found to be special cases of our results.

This thesis concerns the iteration of transcendental self-maps of the punctured plane C* := C \ {O}, that is, functions f : C* --> C* that are holomorphic on C* and for which both zero and infinity are essential singularities. We focus on the escaping set of such functions, which consists of the points whose orbit accumulates to zero and/or infinity under iteration. The escaping set is closely related to the structure of the phase space due to its connection with the Julia set. We introduce the.concept of essential itinerary of an escaping point, which is a sequence that describes how its orbit accumulates to the essential singularities, and plays a very important role throughout the thesis. This allows us to partition the escaping set into uncountably many non-empty subsets of points that escape in non-equivalent ways, the boundary of each of which is the Julia set. We combine the iterates of the maximum and minimum modulus functions to define the fast escaping set for functions in this class and, for such functions, construct orbits with several types of annular itinerary, including fast escaping and arbitrarily slowly escaping points. Next we proceed to study in detail the class 13* of bounded-type transcendental self-maps of C*, for which the escaping set is a sub- set of the Julia set, so such functions do not have escaping Fatou components. We show that, for finite compositions of transcendental self-maps of C* of finite order (and hence in 'B*), every escaping point can be joined to one of the essential singularities by a curve of points that escape uniformly. Moreover, we prove that, for every essential itinerary, the corresponding escaping set contains a Cantor bouquet and, in particular, uncountably many such curves. Finally, in the last part of the thesis we direct our attention to the functions that do have escaping Fatou components. We give the first explicit examples of transcendental self-maps of C* with Baker domains and escaping wandering domains and use approximation theory to construct functions with escaping Fatou components that have any prescribed essential itinerary. This thesis concerns the iteration of transcendental self-maps of the punctured plane C* := C \ {O}, that is, functions f : C* --> C* that are holomorphic on C* and for which both zero and infinity are essential singularities. We focus on the escaping set of such functions, which consists of the points whose orbit accumulates to zero and/ or infinity under iteration. The escaping set is closely related to the structure of the phase space due to its connection with the Julia set. We introduce the, concept of essential itinerary of an escaping point, which is a sequence that describes how its orbit accumulates to the essential singularities, and plays a very important role throughout the thesis. This allows us to partition the escaping set into uncountably many non-empty subsets of points that escape in non-equivalent ways, the boundary of each of which is the Julia set. We combine the iterates of the maximum and minimum modulus functions to define the fast escaping set for functions in this class and, for such functions, construct orbits with several types of annular itinerary, including fast escaping and arbitrarily slowly escaping points. Next we proceed to study in detail the class B* of bounded-type transcendental self-maps of C*, for which the escaping set is a sub- set of the Julia set, so such functions do not have escaping Fatou components. We show that, for finite compositions of transcendental self-maps of C* of finite order (and hence in B*), every escaping point can be joined to one of the essential singularities by a curve of points that escape uniformly. Moreover, we prove that, for every essential itinerary, the corresponding escaping set contains a Cantor bouquet and, in particular, uncountably many such curves. Finally, in the last part of the thesis we direct our attention to the functions that do have escaping Fatou components. We give the first explicit examples of transcendental self-maps of C* with Baker domains and escaping wandering domains and use approximation theory to construct functions with escaping Fatou components that have any prescribed essential itinerary.

Although design optimisation has been well explored using mesh-based approaches, little work has been performed with meshless simulation. Equally, design optimisation has been well explored for methods capable of representing a single design topology, but much less well explored are methods that allow for optimisation of design geometry and topology. To allow for topology changes in design, a new volume-based parameterisation method is proposed which uses the fraction of volume that is solid in an underlying parameterisation grid as the design variables. This technique can be easily used with a low number of design variables, making it usable with agent based global optimisation methods and black box solvers. Optimisation of a NACA 0012 aerofoil in transonic flow is performed with the new parameterisation method and multi-body aerofoil configurations are obtained for optimisation with supersonic flow. Optimising the design of a pivoting, fluid filled tank, shows that the damping of the tank motions can be affected by the tank geometry, which suggests that the wing fuel tanks can be designed to alleviate the flutter instability. It is shown that the effect of fuel is to raise the flutter boundary so the concept of optimising tank design is explored by optimisation of the external tank geometry and by optimising interior baffle configuration. Orifices for vascular self healing networks in composites are optimised to increase mass flow rate. Additionally, the flow of self healing resin into a representative composite crack geometry is modelled using a smoothed particle hydrodynamics solver which incorporates surface tension. The design of a coastal defence structure is also automated through an optimisation process with the fluid behaviour being modelled by smoothed particle hydrodynamics. These optimisation cases have produced novel designs but also, importantly, demonstrate the versatility of the volume based shape parameterisation and the importance of topological change in fluids optimisation.

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