LEGAL INFRINGEMENTS Clause Samples

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Main results and layout of the thesis . . . . . . . . . . . . . . . . . . 2 1.2.1 Part I: Generating minimally transitive groups . . . . . . . . 2 1.2.2 Part II: Generating transitive groups . . . . . . . . . . . . . . 3 1.3 Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Group actions and transitivity . . . . . . . . . . . . . . . . . 7 2.1.2 Transitive actions . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Induced modules for finite groups . . . . . . . . . . . . . . . . . . . . 12 2.3 Further results from representations . . . . . . . . . . . . . . . . . . 14 2.4 Number Theory: The prime counting function . . . . . . . . . . . . . 16 I Generating minimally transitive groups 17 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Some observations on minimally transitive groups . . . . . . . . . . . 19 3.3 Crown-based powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 Indices of proper subgroups in finite simple groups . . . . . . . . . . 22 3.5 The proof of Theorem 1.2.1 24 Chapter 4 Minimally transitive groups of degree 2m3 28 4.1 Introduction 28 4.2 Subgroups of index 2m3 in direct products of nonabelian simple groups 29 4.3 The proof of Theorem 1.2.4 34 5.1 Introduction 40 5.2 Partially ordered sets 41 5.3 Preliminary results on induced modules for finite groups 42 5.3.1 Composition factors in induced modules 42

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: ............................................................. Date: .......................................................... 1.1 Background 1 1.2 Existing Approaches 3 1.3 Thesis Organization and Main Contributions 4 2.1 Functional Magnetic Resonance Imaging 5 2.1.1 A Brief Introduction to MRI 5 2.1.2 Hemodynamic Response 9 2.2 Pre-processing of fMRI Data 10 2.3 Statistical Modeling of fMRI Data 11 2.4 Heritability and Twin Studies 14 2.5 Inference on Heritability 15 2.5.1 Terminology on Heritability Analysis 16 2.5.2 Permutation Test and FWE Correction 18 2.5.3 Bootstrapping Confidence Interval 20 2.5.4 Summary Statistics 22 3.1 The General Linear Model 24 3.2 Brief Review of the Existing Methods 27 3.2.1 ▇▇▇▇▇▇▇▇’▇ Method 27 3.2.2 Bayesian Restricted Maximum Likelihood 28 3.2.3 Structural Equation Modeling 29 3.3 Frequentist Restricted Maximum Likelihood 30 3.3.1 Restricted Likelihood Maximum 31 3.3.2 ▇▇▇▇▇▇ Scoring Algorithm 32

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . Local Analogues to z ∼ 5 ▇▇▇▇▇ Break Galaxies Contents List of Tables v List of Figures vi Acknowledgments viii Declarations x Abstract xi Chapter 1 Introduction 1 1.1 The First Galaxies & The Epoch of Reionization . . . . . . . . . . . 2 1.2 How to Identify High-Redshift Galaxies . . . . . . . . . . . . . . . . 5 1.2.1 The ▇▇▇▇▇ Break, or Dropout, Technique . . . . . . . . . . . 5 1.2.2 Other techniques . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 ▇▇▇▇▇ Break Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . 8

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . 3.3 L21 Co2MnSi and B2 CoMn0.5Si0.5 . . . . . . . . . . . . . . . . . . . 18 Chapter 4 Half-metallicity of CoFexMn0.5−xGaySi0.5−y 21 5.5 Mean free path 38 List of Figures‌ 2.1 Schematic diagrams of the muffin-tin potential 9 2.2 Construction of the coherent potential 12 3.1 Crystal structures of Heusler alloys 16 3.2 Schematic representations of density of states for metals, ferromag- nets and half-metals 16 3.3 Spin polarised density of states for Co with the HCP and FCC structure 19 3.4 Spin polarised density of states for L21 Co2MnSi and B2 CoMn0.5Si0. 4.1 Band gap analysis for minority spin 22 4.2 Spin polarised density of states plot of CoFe0Mn0.5Ga0.025Si0.475 and CoFe0.5Mn0Ga0.5Si0 23 4.3 DOS difference analysis 24 4.4 Combination of the band gap and density of states difference at Fermi level information 25 5.1 Primitive cell of simple cubic lattice 33 5.2 ▇▇▇▇▇ spectral function of L21 Co2MnSi and B2 CoMn0.5Si0.5 34 5.3 Spin up and down ▇▇▇▇▇ spectral function of B2 CoMn0.5Si0.5 sliced by the Γ-X×Γ-X plane through the Fermi surface. 35 5.4 Minority spin ▇▇▇▇▇ spectral function at ky = 0 of B2 CoMn0.5Si0.5 sliced by the Γ-X×Γ-X plane through the Fermi surface at s = (sf − 0.004)Ry, (sf − 0.002)Ry, sf Ry, (sf + 0.002)Ry and (sf + 0.004)Ry . 36 5.5 Fermi velocity analysis for minority spin 37 5.6 Fermi velocity analysis for majority spin 38 5.7 The real data and the fitting of the cropped ▇▇▇▇▇ spectral function 39 5.8 Mean free path analysis for minority spin 40 5.9 Mean free path analysis for majority spin 40 6.1 Combination of the band gap analysis, density of states difference at the Fermi level and the mean free path calculations information . . . 42 Acknowledgments‌ I would like to pay special thanks to ▇▇▇▇. ▇. ▇. ▇▇▇▇▇▇▇▇ for her wise supervision throughout my MRes studies, not only that but also for supporting me with my job hunting. I am also very grateful to ▇▇. ▇. Bell and the half-metals team at York university for providing me some insights into experimental work being done. My thoughts also go to my friends, too many to be named, but ...

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Date: . . . . . . . . . . . . . . . . . Chapter 1 Introduction and preliminaries 1 1.1 Introduction, motivation and history 1 1.2 Basic concepts 5 1.3 Weighted projective space 7 1.4 Cyclic quotient singularities 9 1.5 The ▇▇▇▇▇▇▇ syzygies theorem 14 2.1 Preliminaries 18 2.2 Main result and proof 23 2.3 Examples and applications 32 Chapter 3 Generalising the formula to arbitrary rational functions 44 3.1 Definitions, examples and observations in the curve orbifold locus case 44 3.2 The “isolated” case 48 3.3 The curve locus case 51 3.4 Generalisations of the “curve” locus case 60 4.1 Main theorem and examples 70 4.2 First parsing 80 4.3 Contributions from orbipoints 81 4.4 Contributions from orbicurves 82 4.5 End of proof 91 5.1 Understanding the curve orbifold locus more completely 94 5.2 Towards general results with arbitrary orbifold locus 97 5.3 General formulae without the symmetric assumption 102 Acknowledgments‌ First and foremost massive thanks are due to my supervisor ▇▇▇▇▇ ▇▇▇▇ for guidance, support and help throughout the course of my PhD, especially when things (perhaps inevitably) became difficult. His deep subject knowledge, ability to generate ideas and sheer enthusiasm for the topic have all been invaluable, as has his ability to navigate through the administrative loops when required. I am also grateful to ▇▇▇▇▇ ▇▇▇▇▇ for a number of useful discussions, and ▇▇▇▇▇▇▇▇▇ ▇▇▇▇ for sharing her insight in these areas. Throughout my time at War- wick I have been fortunate enough to be backed by a team of colleagues and visiting experts, and I would like to thank all of them for the assistance they provided. My work at Warwick was supported by an EPSRC grant. I was fortunate enough to be able to visit places as far away as Moscow, Seoul and Shanghai in the course of my studies and would like to thank ▇▇▇▇ ▇▇▇▇▇▇▇▇▇, ▇▇▇▇▇▇▇ ▇▇▇ and ▇▇▇▇ ▇▇▇▇ for inviting me and making sure I was suitably looked after. Thanks also must go to my friends ▇▇▇▇▇, ▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇ for taking time out of their busy lives to help me feel welcome and show me around. My journey as a budding Mathematician star...

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: ......................................................…… Date: .......................................................... 1.7.3 The ▇▇▇▇▇▇ Group of Field 21 1.7.4 ▇▇▇▇▇▇-▇▇▇▇▇ Obstruction 23 1.8 ▇▇▇▇▇▇▇’▇ Theorem 25 Chapter 2 Properties of HS(K) 31 2.1 Background 31 2.2 First Results 35 2.2.1 Relation between H0(K) and r(S, K) 39 2.3 HS(K) and Hyperplane Sections 40 2.4 Universal Equivalence 40 2.5 Weak Approximation and HS(K) 43 2.6 HS(R) 44 2.7 HS(Qp) 46 3.1 Main Theorem 51 4.1 Setup 57 4.2 Lonely Points 58 4.3 Some Geometry 65 4.4 Point Generation 68 5.1 The ▇▇▇▇▇▇▇-▇▇▇▇ rank over Fq 84 I could not put into words my gratitude for my supervisor, ▇▇▇▇▇ ▇▇▇- ▇▇▇, for his support, invaluable comments, and guidance the past four years. Without him, my research wouldn’t have been so fulfilling and exciting as it has been, or even possible. Furthermore, I would also like to thank my teachers throughout the years, especially ▇▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇▇▇▇▇ and ▇▇▇▇▇ ▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇▇. They are part of some of the most treasured memories I have kept from my journey in Mathematics so far. I am also grateful to ▇▇▇▇▇▇▇ ▇▇▇▇▇, for his help, guidance, and the fact that he was there to give me the tools to formulate crucial geometric arguments of my thesis. To my supportive partner, ▇▇▇▇▇▇, I would like to offer my deepest gratitude; for her peer review, emotional support, and the unmeasurable deal of help with LATEX. This thesis would not have been possible without my family in Greece, ▇▇▇▇▇▇▇▇, ▇▇▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇, as they provided more financial support they could afford. Also, their understanding to the challenges that writing a thesis poses could not go unmentioned. My special thanks also go to my friends ▇▇▇▇, ▇▇▇▇▇▇▇▇ and ▇▇▇▇▇. They may have been far away, but our communication encouraged me and made this time in my life much easier. I would also like to thank ▇▇▇▇▇▇▇ and ▇▇▇▇▇▇▇▇, two peers and dear friends that I had the luck of meeting during the first year of my studies in Warwick. Their help and friendship has been invaluable. At this point I should thank my dear friends, ▇▇▇▇▇ and ▇▇▇▇▇▇▇. Living in Coventry would have been a challenge, but they filled it with fu...

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis.

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: ......................................................…… Date: .......................................................... Revised July 2014 1. ▇▇▇▇▇▇▇▇ on Knowledge and the Enquirer 2 2. Enquiry and the Value of Knowledge 16

LEGAL INFRINGEMENTS. I understand that neither the University of Warwick nor the British Library have any obligation to take legal action on behalf of myself, or other rights holders, in the event of infringement of intellectual property rights, breach of contract or of any other right, in the thesis. Student’s signature: ......................................................…… Date: .......................................................... 1.1 Literature Review 3 1.2 Outline and results of the thesis 11 2.1 Basic Definitions 19 2.1.1 Calculus on R 19 2.1.2 Calculus on a ▇▇▇▇▇▇ Space 20 2.2 Classical Partial Differential Equations 20 2.3 Stochastic Differential Equations 21 2.4 Stochastic Representation of Solutions to Partial Differential Equations 25 2.5 The Fokker–▇▇▇▇▇▇ Equation 26 2.6 Surface Partial Differential Equations 29 3.1 The Problem 32 3.2 Formulation of the Problem in arc-length parameter. 34 3.3 Formulation of the Problem along flow lines 37 3.4 Scaling Properties of the PDE 37 4.1 Sub–Critical Regime 40 4.1.1 Vanishing and boundedness of the solution 40 4.1.2 Power law for the behaviour of the solution near the singularity 44 4.2 Critical Regime 48 4.2.1 Vanishing of the solution and a power law 50 4.3 Super–Critical Regime 56 4.3.1 Boundedness of the solution 56 4.3.2 Behaviour of the solution at the singularity 60 5.1 Sub–Critical Regime 67 5.2 Critical Regime 74 5.2.1 Short–Time Existence 74 5.2.2 Bounds on the trajectories of the SDE 77 5.3 Super–Critical Regime 78 6.1 Rougher Initial Conditions 81 6.1.1 Sub–Critical Regime 82 6.2 Deterministic Perturbation 84 6.2.1 Sub–Critical Regime 84 6.3 Stochastic Perturbation 86 6.3.1 A Heuristic exploration into the noise 86 6.3.2 Sub–Critical Regime 88 7.1 Parameterisation of the Problem 95 7.2 Scaling properties of the PDEs 101 7.2.1 Scaling of the heat equation 101 7.2.2 Scaling of the density equation 102 8.1 Sub–Critical Regime 104 8.2 Critical Regime 113 8.3 Super–Critical Regime 119 9.1 Formulation of the Continuation of the Solution 129 9.2 Existence of a continued Stochastic Process 132 10.1 The Problem 143 10.1.1 Formulation of the Problem in arc-length parameter. 144 10.2 Analysis of the Problem 146 11.1 Open problems raised in the thesis 149 11.1.1 Problem I: Before the singularity 149 11.1.2 Problem I: After the singularity 149 11.1.3 Perturbation of Problem I 150 11.1.4 Problem II: Before the singularity 150 11.1.5 Problem II: After the singularity 152 11.1.6 Problem III 152 11.2 Future research prob...