Preface |

1. Linear and Nonlinear Transformations |

1.1 Introduction |

1.2 Vector Spaces; Linear Transformations |

1.3 Eigenvalues; Eigenvectors |

1.4 Inner-product Spaces |

1.5 Self-adjoint Transformations |

1.6 The Infinite-dimensional Case |

1.7 Operators |

1.8 Applications |

1.9 Banach Spaces and Linear Functionals |

1.10 Fixed-point Theorems and Applications |

1.11 Lebesgue Integration, a Survey |

2. Nonlinear Algebraic and Transcendental Equations |

2.1 Introduction |

2.2 The Newton-Raphson Method |

2.3 The Method of Steepest Descent |

2.4 Saddle-point Method or Steepest-descent Method of Complex Integration |

3. Nonlinear Optimization; Nonlinear Programming and Systems of Inequalities |

3.1 Introduction |

3.2 Maxima, Minima, Quadratic Forms, and Convex Functions |

3.3 Nonlinear Programming |

3.4 Linear Programming |

3.5 Characterization of the Optimum on the Boundary; Saddle Points; Duality |

3.6 Construction of Solutions |

3.7 Optimization Problems with Infinitely Many Constraints |

4. Nonlinear Ordinary Differential Equations |

4.1 Introduction |

4.2 Some Nonlinear Equations |

4.3 Existence and Uniqueness for First-order Systems |

4.4 Linear Equations--Oscillatory Motion, Stability |

4.5 Nonlinear Equations--Perturbation Method |

4.6 Phase-plane Analysis--Stability Behavior in the Small (Systems of Two Equations) |

4.7 Limit Cycles--Stability Behavior in the Large (Systems of Two Equations) |

4.8 Topological Considerations: Indices and the Existence of Limit Cycles (Systems of Two Equations) |

4.9 Periodic Solutions of Systems with Periodic Coefficients |

4.10 Periodic Solutions of Nonlinear Systems with Periodic Coefficients |

4.11 Lyapunov Stability |

4.12 General Methods of Solution |

5. Introduction to Automatic Control and the Pontryagin Principle |

5.1 Introduction |

5.2 Stability and a Class of Control Equations |

5.3 Pontryagin's Maximum Principle |

5.4 Functional Analysis and Optimum Control |

6. Linear and Nonlinear Prediction Theory |

6.1 Introduction |

6.2 The Discrete Stationary Case |

6.3 Construction of the Discrete-case Estimate |

6.4 The Discrete Prediction Problem |

6.5 The Prediction Error |

6.6 The Special Case of Rational Densities |

6.7 The Continuous Stationary Case |

6.8 Construction of the Continuous-case Estimate |

6.9 The Continuous Prediction Problem |

6.10 Examples |

6.11 Conditional Expectation |

6.12 The General Estimation Problem |

6.13 Polynomial Estimation |

6.14 The Karhunen-Loeve Expa |

6.15 Dynamical Systems with Control Variables |

Appendix; Index |