| Description |
1 online resource |
| Contents |
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 -- 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. 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 -- 4. Applications in numerical analysis. 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 -- 5. Applications in wavelets, control, scattering. 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 -- 6. The trigonometry of matrix statistics. 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 -- 7. Quantum trigonometry. 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 -- 8. Financial instruments. 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 -- 9. Other directions. 9.1. Operators. 9.2. Angles. 9.3. Optimization. 9.4. Equalities. 9.5. Geometry. 9.6. Applications |
| Summary |
Karl Gustafson is the creater of the theory of antieigenvalue analysis. Its applications spread through fields as diverse as numerical analysis, wavelets, statistics, quantum mechanics and finance. Antieigenvalue analysis, with its operator trigonometry, is a unifying language which enables new and deeper geometrical understanding of essentially every result in operator theory and matrix theory, together with their applications. This book will open up its methods to a wide range of specialists |
| Bibliography |
Includes bibliographical references and index |
| Subject |
Eigenvalues.
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Mathematical analysis.
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Numerical analysis.
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Quantum theory.
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Statistics.
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Wavelets (Mathematics)
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| Form |
Electronic book
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| ISBN |
9789814366298 |
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9814366293 |
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