Analysis of Quantised Vortex Tangle. (2016)

• Record Type: Book
• Other Names: Taylor, Alexander John
• Contents: Supervisor's Foreword; Abstract; Acknowledgements; Contents; 1 Introduction; 1.1 General Introduction; 1.2 What Does It Mean to Be Tangled?; 1.3 Phase Singularities in Two and Three Dimensions; 1.4 Random Waves and Vortex Tangle; 1.5 Knotting and Linking as a Statistical Tool; 1.6 Other Random Tangles and Random Walks; 1.7 Variations on Random Waves; 1.7.1 The 3-Torus (Periodic Boundary Conditions); 1.7.2 The 3-Sphere; 1.7.3 The Quantum Harmonic Oscillator; 1.8 Thesis Outline; References; 2 Numerical Methods; 2.1 Numerically Locating Vortices; 2.2 Tracking Vortices with a Resampling Algorithm. 2.3 Vortex Tracking in Different Systems2.4 Geometrical Details in Different Systems; 2.4.1 The 3-Torus; 2.4.2 The 3-Sphere; 2.4.3 The Quantum Harmonic Oscillator; 2.5 Summary of Numerical Results; 2.5.1 The 3-Torus; 2.5.2 The 3-Sphere; 2.5.3 The Quantum Harmonic Oscillator; References; 3 Geometry and Scaling of Vortex Lines; 3.1 Introduction to Geometrical Statistics; 3.2 Local Geometry of Random Wave Vortices; 3.3 Statistics of Curvature and Torsion; 3.4 Geometric Correlations; 3.5 Three Measures of Fractality; 3.6 Scaling Relations of Random Tangle. 3.7 Crossing Number and Writhe of Vortex Loops3.8 Discussion; References; 4 Topological Methods; 4.1 A Practical Introduction to Knot Theory; 4.2 Calculating Invariants; 4.2.1 The Alexander Polynomial; 4.2.2 The Determinant and Roots of the Alexander Polynomial; 4.2.3 The Jones and HOMFLY-PT Polynomials; 4.2.4 Hyperbolic Volume; 4.2.5 Minimal Crossing Number; 4.2.6 The Linking Number; 4.2.7 The Fary-Milnor Theorem; 4.3 Simplifying Space Curves; 4.4 Topological Complexity; References; 5 Knotting and Linking of Vortex Lines; 5.1 Finding Knots; 5.1.1 The 3-torus; 5.1.2 The 3-sphere. 5.1.3 The Quantum Harmonic Oscillator5.2 Probabilities of Knotting; 5.3 Comparison of Knot Type Statistics; 5.4 Finding Links; 5.4.1 The 3-torus; 5.4.2 The 3-sphere ; 5.4.3 The Quantum Harmonic Oscillator; 5.5 Probabilities of Linking; 5.6 Knotting in Lines with Non-trivial Homology; 5.7 Postselection Relations; 5.8 The Smallest Knot; 5.9 Discussion; References; 6 Conclusions; Appendices.
• Publisher Details: Cham, SWITZERLAND : Springer
• Publication Date: 2016
• Extent: 1 online resource (206)
• Subjects: 530; Physics; Mathematics -- Study and teaching; Mathematics -- Study and teaching; Science -- Mathematical Physics; Mathematics -- Topology; Mathematics -- Probability & Statistics -- General; Mathematical physics; Topology; Probability & statistics; Topology; Mathematical statistics; Electronic books
• Languages: English
• ISBNs: 3319485563; 9783319485560
• Related ISBNs: 3319485555
• Notes: Note: Print version record.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.