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The aim of this doctoral thesis is to present a body of work aimed at reducing the time spent by animators manually constructing keyframed animation. To this end we present a number of state of the art machine learning techniques applied to the domain of character animation. Data-driven tools for the synthesis and production of character animation have a good track record of success. In particular, they have been adopted thoroughly in the games industry as they allow designers as well as animators to simply specify the high-level descriptions of the animations to be created, and the rest is produced automatically. Even so, these techniques have not been thoroughly adopted in the film industry in the production of keyframe based animation [Planet, 2012]. Due to this, the cost of producing high quality keyframed animation remains very high, and the time of professional animators is increasingly precious. We present our work in four main chapters. We first tackle the key problem in the adoption of data-driven tools for key framed animation - a problem called the inversion of the rig function. Secondly, we show the construction of a new tool for data-driven character animation called the motion manifold - a representation of motion constructed using deep learning that has a number of properties useful for animation research. Thirdly, we show how the motion manifold can be extended as a general tool for performing data-driven animation synthesis and editing. Finally, we show how these techniques developed for keyframed animation can also be adapted to advance the state of the art in the games industry.

This thesis proposes a new model for motivic analysis which, being based on the metaphor of a web or network and expanded using the mathematical field of graph theory, balances the polar concerns prevalent in analytical writing to date: those of static, out-of-time category membership and dynamic, in-time process. The concepts that constitute the model are presented in the third chapter, both as responses to a series of analytical observations (using the worked example of Beethoven’s Piano Sonata in F minor, Op. 2, No. 1), and as rigorously defined mathematical formalisms. The other chapters explore in further detail the disciplines and methodologies on which this model impinges, and serve both to motivate, and to reflect upon, its development. Chapter 1 asks what it means to make mathematical statements about music, and seeks to disentangle mathematics (as a tool or language) from science (as a method), arguing that music theory’s aims can be met by the former without presupposing its commonly assumed inextricability from the latter. Chapter 2 provides a thematic overview of the field of motivic theory and analysis, proposing four archetypal models that combine to underwrite much thought on the subject before outlining the problems inherent in a static account and the creative strategies that can be used to construct a dynamic account. Finally, Chapter 4 applies these strategies, together with Chapter 3’s model and the piece’s extensive existing scholarly literature, to the analysis of the first and last movements of Mahler’s Sixth Symphony. The central theme throughout – as it relates to mathematical modelling, music theory, and music analysis – is that of potential, invitation, openness, and dialogic engagement.

The aim of this thesis was to accurately simulate the dynamic nature of known porous organic cage molecules, with a view to understanding the diffusion of different gases through their pore structures. Initial calculations showed that, due to their unique chemical structure, no ‘off-the-shelf’ force field (FF) was accurate enough to describe their dynamic motion. Therefore, a cage specific force field, (CSFF), was developed to be transferable across the first three cage systems, CC1-CC3. CSFF was subsequently used to rationalise the ‘on’/‘off’ porosity observed in two different polymorphs of CC1. A combination of computational simulations, including simulated surface area calculations, grand canonical Monte Carlo (GCMC) adsorption isotherms, and molecular dynamics (MD) simulations, for hydrogen and nitrogen in CC1α and CC1β, helped to confirm experimental results, as well as to provide further insight into why the polymorphism of CC1 alters the porosity of the molecule. In addition, CSFF was used to study the diffusion of a range of gases through crystalline CC3. Seven gases were chosen: hydrogen, nitrogen, carbon dioxide, methane, sulfur hexafluoride, krypton and xenon. A detailed understanding of the diffusivity within CC3 was accomplished by combining MD simulations with new methodologies and techniques, for example analysis of the dynamic connectivity. This helped to rationalise why CC3 showed good experimental uptake of gas, as well as highlighting potential separation capabilities. In summary, the development of CSFF has made it possible to simulate the diffusion of small gases through porous organic imine cages, and it has been shown that this diffusion is dependent on the relative size of the gas to the cage window, assuming that there is a suitable diffusion pathway. Using MD simulations, we have unlocked phenomena such as gas selectivity, rare-event hopping and the diffusion of gases to regions previously thought inaccessible. This has aided the rationalisation of existing experimental observations, and is a significant step forward for a priori prediction of porous organic cage systems, and their properties. This work has also led to new experiments that were prompted by my simulations. Finally, a new way to visualise the connectivity of a system has been introduced. This is achieved by monitoring how the surface area evolves with respect to time, during a MD simulation. This suggests how the pore channels of various systems, previously thought too small for gas adsorption, are actually suitable candidates for separations.