Titlebook: Analysis of Quantised Vortex Tangle; Alexander John Taylor Book 2017 Springer International Publishing AG 2017 Filamentary Tangle.Wave Cha

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PDES-STEP Standards: A Questionrandom walks, and comparing with other systems of filamentary tangle. These measures include densities, curvatures and torsions, whose behaviours can be compared to known results on wave chaos, as well as the fractalities and scaling relations which can take particular universal values, indicating that vortices behave randomly on large scales.

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Free Form Fabrication-Process Overview we introduce and explain the mathematical techniques necessary for topological analysis, summarising the relevant areas of knot theory, introducing the knot invariant quantities whose values can practically be calculated, and discussing further numerical methods to safely simplify individual vortex curves before calculating topological results.

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Topological Methods, we introduce and explain the mathematical techniques necessary for topological analysis, summarising the relevant areas of knot theory, introducing the knot invariant quantities whose values can practically be calculated, and discussing further numerical methods to safely simplify individual vortex curves before calculating topological results.

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Juliet Lodge (Senior Lecturer in Politics)epts, the historical context for considering such systems, the mathematical formalisms for wave chaos and the study of singularities, and the geometrical and topological techniques that will be used to analyse their conformations.

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Juliet Lodge (Senior Lecturer in Politics)and their extensions to each model system. In particular, we introduce a recursive algorithm that can focus numerical effort only around the vortex curves, efficiently but accurately capturing their small-scale geometry and topology.

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Book 2017This latter phenomenon occurs on ‘vortex lines‘ in three dimensions.  In general wave superpositions modelling e.g. chaotic cavity modes, these vortex lines form dense tangles that have never been visualised on the large scale before, and cannot be analysed mathematically by any known techniques. 

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Manufacturing Analysis of Machined Partsatistics of the filamentary tangle, and demonstrating that topological quantities can be particularly sensitive to boundary conditions. We further discuss some specific topological problems that arise when considering how knotted structures may form.

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