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 |