Contents -- Preface -- 1. Introduction -- Perspective -- 1.1 A Recent Referee Speaks -- 1.2 The Original Motivation -- 1.3 The Essential Entities -- 1.4 Simple Examples and a Picture -- 1.5 Applications to-Date -- 1.6 Organization of this Book -- Commentary -- 1.7 Exercises -- 2. The Original Motivation: Operator Semigroups -- Perspective -- 2.1 Abstract Initial Value Problems -- 2.2 The Hille-Yosida-Phillips-Lumer Theorem -- 2.3 The Rellich-Kato-Nelson-Gustafson Theorem -- 2.4 The Multiplicative Perturbation Theorem -- 2.5 When are Positive Operator Products Positive? -- 2.6 Nonnegative Contraction Semigroups -- Commentary -- 2.7 Exercises -- 3. The Essentials of Antieigenvalue Theory -- Perspective -- 3.1 Convexity Properties of Norm Geometry -- 3.2 The Min-Max Theorem -- 3.3 The Euler Equation -- 3.4 Higher Antieigenvalues and Antieigenvectors -- 3.5 The Triangle Inequality -- 3.6 Extended Operator Trigonometry -- Commentary -- 3.7 Exercises -- 4. Applications in Numerical Analysis -- Perspective -- 4.1 Gradient Descent: Kantorovich Bound is Trigonometric -- 4.2 Minimum Residual Ax = b Solvers -- 4.3 Richardson Relaxation Schemes (e.g. SOR) -- 4.4 Very Rich Trigonometry Underlies ADI -- 4.5 Domain Decomposition Multilevel Schemes -- 4.6 Preconditioning and Condition Numbers -- Commentary -- 4.7 Exercises -- 5. Applications in Wavelets, Control, Scattering -- Perspective -- 5.1 The Time Operator of Wavelets -- 5.2 Frame Operator Trigonometry -- 5.3 Wavelet Reconstruction is Trigonometric -- 5.4 New Basis Trigonometry -- 5.5 Trigonometry of Lyapunov Stability -- 5.6 Multiplicative Perturbation and Irreversibility -- Commentary -- 5.7 Exercises -- 6. The Trigonometry of Matrix Statistics -- Perspective -- 6.1 Statistical Efficiency -- 6.2 The Euler Equation versus the Inefficiency Equation -- 6.3 Canonical Correlations and Rayleigh Quotients.

6.4 Other Statistics Inequalities -- 6.5 Prediction Theory: Association Measures -- 6.6 Antieigenmatrices -- Commentary -- 6.7 Exercises -- 7. Quantum Trigonometry -- Perspective -- 7.1 Bell-Wigner-CHSH Inequalities -- 7.2 Trigonometric Quantum Spin Identities -- 7.3 Quantum Computing: Phase Issues -- 7.4 Penrose Twistors -- 7.5 Elementary Particles -- 7.6 Trigonometry of Quantum States -- Commentary -- 7.7 Exercises -- 8. Financial Instruments -- Perspective -- 8.1 Some Remarks on Mathematical Finance -- 8.2 Quantos: Currency Options -- 8.3 Multi-Asset Pricing: Spread Options -- 8.4 Portfolio Rebalancing -- 8.5 American Options with Random Volatility -- 8.6 Risk Measures for Incomplete Markets -- Commentary -- 8.7 Exercises -- 9. Other Directions -- Perspective -- 9.1 Operators -- 9.2 Angles -- 9.3 Optimization -- 9.4 Equalities -- 9.5 Geometry -- 9.6 Applications -- Commentary -- 9.7 Exercises -- Appendix A Linear Algebra -- A.1 Matrix Analysis -- A.2 Operator Theory -- Appendix B Hints and Answers to Exercises -- Chapter 1. -- Chapter 2. -- Chapter 3. -- Chapter 4. -- Chapter 5. -- Chapter 6. -- Chapter 7. -- Chapter 8. -- Chapter 9. -- Bibliography -- Index.