| Preface |
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ix | |
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1 | (14) |
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1 | (1) |
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1.1 A Recent Referee Speaks |
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2 | (1) |
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1.2 The Original Motivation |
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2 | (1) |
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1.3 The Essential Entities |
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3 | (3) |
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1.4 Simple Examples and a Picture |
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6 | (3) |
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9 | (1) |
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1.6 Organization of this Book |
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10 | (4) |
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12 | (2) |
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14 | (1) |
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2 The Original Motivation: Operator Semigroups |
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15 | (10) |
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15 | (1) |
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2.1 Abstract Initial Value Problems |
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16 | (1) |
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2.2 The Hille-Yosida-Phillips-Lumer Theorem |
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17 | (1) |
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2.3 The Rellich-Kato-Nelson-Gustafson Theorem |
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17 | (1) |
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2.4 The Multiplicative Perturbation Theorem |
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18 | (2) |
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2.5 When are Positive Operator Products Positive? |
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20 | (1) |
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2.6 Nonnegative Contraction Semigroups |
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21 | (2) |
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22 | (1) |
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23 | (2) |
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3 The Essentials of Antieigenvalue Theory |
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25 | (28) |
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25 | (1) |
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3.1 Convexity Properties of Norm Geometry |
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26 | (1) |
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27 | (6) |
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33 | (6) |
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3.4 Higher Antieigenvalues and Antieigenvectors |
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39 | (7) |
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3.5 The Triangle Inequality |
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46 | (1) |
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3.6 Extended Operator Trigonometry |
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47 | (3) |
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49 | (1) |
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50 | (3) |
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4 Applications in Numerical Analysis |
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53 | (16) |
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53 | (1) |
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4.1 Gradient Descent: Kantorovich Bound is Trigonometric |
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54 | (2) |
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4.2 Minimum Residual Ax = b Solvers |
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56 | (1) |
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4.3 Richardson Relaxation Schemes (e.g. SOR) |
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57 | (3) |
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4.4 Very Rich Trigonometry Underlies ADI |
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60 | (1) |
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4.5 Domain Decomposition Multilevel Schemes |
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61 | (2) |
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4.6 Preconditioning and Condition Numbers |
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63 | (4) |
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65 | (2) |
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67 | (2) |
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5 Applications in Wavelets, Control, Scattering |
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69 | (22) |
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69 | (1) |
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5.1 The Time Operator of Wavelets |
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70 | (4) |
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5.2 Frame Operator Trigonometry |
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74 | (2) |
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5.3 Wavelet Reconstruction is Trigonometric |
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76 | (2) |
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5.4 New Basis Trigonometry |
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78 | (7) |
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5.5 Trigonometry of Lyapunov Stability |
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85 | (1) |
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5.6 Multiplicative Perturbation and Irreversibility |
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86 | (3) |
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88 | (1) |
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89 | (2) |
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6 The Trigonometry of Matrix Statistics |
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91 | (32) |
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91 | (1) |
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6.1 Statistical Efficiency |
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91 | (10) |
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6.2 The Euler Equation versus the Inefficiency Equation |
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101 | (4) |
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6.3 Canonical Correlations and Rayleigh Quotients |
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105 | (2) |
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6.4 Other Statistics Inequalities |
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107 | (5) |
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6.5 Prediction Theory: Association Measures |
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112 | (3) |
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115 | (3) |
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116 | (2) |
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118 | (5) |
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123 | (32) |
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123 | (2) |
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7.1 Bell-Wigner-CHSH Inequalities |
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125 | (4) |
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7.2 Trigonometric Quantum Spin Identities |
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129 | (3) |
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7.3 Quantum Computing: Phase Issues |
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132 | (3) |
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135 | (7) |
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142 | (2) |
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7.6 Trigonometry of Quantum States |
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144 | (9) |
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152 | (1) |
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153 | (2) |
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155 | (28) |
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155 | (6) |
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8.1 Some Remarks on Mathematical Finance |
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161 | (6) |
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8.2 Quantos: Currency Options |
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167 | (5) |
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8.3 Multi-Asset Pricing: Spread Options |
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172 | (3) |
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8.4 Portfolio Rebalancing |
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175 | (2) |
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8.5 American Options with Random Volatility |
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177 | (2) |
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8.6 Risk Measures for Incomplete Markets |
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179 | (3) |
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181 | (1) |
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182 | (1) |
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183 | (20) |
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183 | (1) |
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183 | (3) |
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186 | (3) |
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189 | (2) |
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191 | (3) |
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194 | (5) |
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199 | (1) |
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200 | (1) |
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200 | (3) |
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Appendix A Linear Algebra |
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203 | (2) |
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203 | (1) |
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204 | (1) |
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Appendix B Hints and Answers to Exercises |
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205 | (24) |
| Bibliography |
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229 | (12) |
| Index |
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241 | |
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