| Preface | p. ix |
| Introduction | p. 1 |
| Perspective | p. 1 |
| A Recent Referee Speaks | p. 2 |
| The Original Motivation | p. 2 |
| The Essential Entities | p. 3 |
| Simple Examples and a Picture | p. 6 |
| Applications to-Date | p. 9 |
| Organization of this Book | p. 10 |
| Commentary | p. 12 |
| Exercises | p. 14 |
| The Original Motivation: Operator Semigroups | p. 15 |
| Perspective | p. 15 |
| Abstract Initial Value Problems | p. 16 |
| The Hille-Yosida-Phillips-Lumer Theorem | p. 17 |
| The Rellich-Kato-Nelson-Gustafson Theorem | p. 17 |
| The Multiplicative Perturbation Theorem | p. 18 |
| When are Positive Operator Products Positive? | p. 20 |
| Nonnegative Contraction Semigroups | p. 21 |
| Commentary | p. 22 |
| Exercises | p. 23 |
| The Essentials of Antieigenvalue Theory | p. 25 |
| Perspective | p. 25 |
| Convexity Properties of Norm Geometry | p. 26 |
| The Min-Max Theorem | p. 27 |
| The Euler Equation | p. 33 |
| Higher Antieigenvalues and Antieigenvectors | p. 39 |
| The Triangle Inequality | p. 46 |
| Extended Operator Trigonometry | p. 47 |
| Commentary | p. 49 |
| Exercises | p. 50 |
| Applications in Numerical Analysis | p. 53 |
| Perspective | p. 53 |
| Gradient Descent: Kantorovich Bound is Trigonometric | p. 54 |
| Minimum Residual Ax = b Solvers | p. 56 |
| Richardson Relaxation Schemes (e.g. SOR) | p. 57 |
| Very Rich Trigonometry Underlies ADI | p. 60 |
| Domain Decomposition Multilevel Schemes | p. 61 |
| Preconditioning and Condition Numbers | p. 63 |
| Commentary | p. 55 |
| Exercises | p. 67 |
| Applications in Wavelets, Control, Scattering | p. 69 |
| Perspective | p. 59 |
| The Time Operator of Wavelets | p. 70 |
| Frame Operator Trigonometry | p. 74 |
| Wavelet Reconstruction is Trigonometric | p. 76 |
| New Basis Trigonometry | p. 78 |
| Trigonometry of Lyapunov Stability | p. 85 |
| Multiplicative Perturbation and Irreversibility | p. 86 |
| Commentary | p. 88 |
| Exercises | p. 89 |
| The Trigonometry of Matrix Statistics | p. 91 |
| Perspective | p. 91 |
| Statistical Efficiency | p. 91 |
| The Euler Equation versus the Inefficiency Equation | p. 101 |
| Canonical Correlations and Rayleigh Quotients | p. 105 |
| Other Statistics Inequalities | p. 107 |
| Prediction Theory: Association Measures | p. 112 |
| Antieigenmatrices | p. 115 |
| Commentary | p. 116 |
| Exercises | p. 118 |
| Quantum Trigonometry | p. 123 |
| Perspective | p. 123 |
| Bell-Wigner-CHSH Inequalities | p. 125 |
| Trigonometric Quantum Spin Identities | p. 129 |
| Quantum Computing: Phase Issues | p. 132 |
| Penrose Twisters | p. 135 |
| Elementary Particles | p. 142 |
| Trigonometry of Quantum States | p. 144 |
| Commentary | p. 152 |
| Exercises | p. 153 |
| Financial Instruments | p. 155 |
| Perspective | p. 155 |
| Some Remarks on Mathematical Finance | p. 161 |
| Quantos: Currency Options | p. 167 |
| Multi-Asset Pricing: Spread Options | p. 172 |
| Portfolio Rebalancing | p. 175 |
| American Options with Random Volatility | p. 177 |
| Risk Measures for Incomplete Markets | p. 179 |
| Commentary | p. 181 |
| Exercises | p. 182 |
| Other Directions | p. 183 |
| Perspective | p. 183 |
| Operators | p. 183 |
| Angles | p. 186 |
| Optimization | p. 189 |
| Equalities | p. 191 |
| Geometry | p. 194 |
| Applications | p. 199 |
| Commentary | p. 200 |
| Exercises | p. 200 |
| Linear Algebra | p. 203 |
| Matrix Analysis | p. 203 |
| Operator Theory | p. 204 |
| Hints and Answers to Exercises | p. 205 |
| Bibliography | p. 229 |
| Index | p. 241 |
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