The nature of mathematical modeling

Cover......Page __sk_0000.djvu
Copyright......Page __sk_0004.djvu
Contents......Page __sk_0005.djvu
Preface......Page __sk_0011.djvu
1 Introduction......Page __sk_0013.djvu
1.1 Selected References......Page __sk_0015.djvu
Part One: Analytical Models......Page __sk_0017.djvu
2.1 Linear Differential Equations......Page __sk_0021.djvu
2.2 Systems of Differential Equations and Normal Modes......Page __sk_0024.djvu
2.3 Laplace Transforms......Page __sk_0025.djvu
2.4 Perturbation Expansions......Page __sk_0029.djvu
2.5 Discrete Time Equations......Page __sk_0030.djvu
2.6 z-Transforms......Page __sk_0031.djvu
2.7 Selected References......Page __sk_0033.djvu
2.8 Problems......Page __sk_0034.djvu
3.1 The Origin of Partial Differential Equations......Page __sk_0036.djvu
3.2 Linear Partial Differential Equations......Page __sk_0038.djvu
3.3 Separation of Variables......Page __sk_0039.djvu
3.3.1 Rectangular Coordinates......Page __sk_0040.djvu
3.3.2 Cylindrical Coordinates......Page __sk_0041.djvu
3.3.3 Spherical Coordinates......Page __sk_0043.djvu
3.4 Transform Techniques......Page __sk_0044.djvu
3.6 Problems......Page __sk_0045.djvu
4.1.1 Euler\'s Equation......Page __sk_0046.djvu
4.1.2 Integrals and Missing Variables......Page __sk_0048.djvu
4.1.3 Constraints and Lagrange Multipliers......Page __sk_0049.djvu
4.2.2 Analytical Mechanics: Hamilton\'s Principle......Page __sk_0050.djvu
4.2.3 Symmetry: Noether\'s Theorem......Page __sk_0051.djvu
4.3 Rigid Body Motion......Page __sk_0052.djvu
4.5 Problems......Page __sk_0055.djvu
5.1 Random Variables......Page __sk_0056.djvu
5.1.1 Joint Distributions......Page __sk_0058.djvu
5.1.2 Characteristic Functions......Page __sk_0060.djvu
5.2 Stochastic Processes......Page __sk_0062.djvu
5.2.1 Distribution Evolution Equations......Page __sk_0063.djvu
5.2.2 Stochastic Differential Equations......Page __sk_0067.djvu
5.3 Random Number Generators......Page __sk_0068.djvu
5.3.1 Linear Congruential......Page __sk_0069.djvu
5.3.2 Linear Feedback......Page __sk_0071.djvu
5.5 Problems......Page __sk_0072.djvu
Part Two: Numerical Models......Page __sk_0075.djvu
6.1 Numerical Approximations......Page __sk_0079.djvu
6.2 Runge-Kutta Methods......Page __sk_0082.djvu
6.3 Beyond Runge-Kutta......Page __sk_0084.djvu
6.5 Problems......Page __sk_0088.djvu
7 Finite Differences: Partial Differential Equations......Page __sk_0090.djvu
7.1 Hyperbolic Equations: Waves......Page __sk_0091.djvu
7.2 Parabolic Equations: Diffusion......Page __sk_0093.djvu
7.3 Elliptic Equations: Boundary Values......Page __sk_0096.djvu
7.5 Problems......Page __sk_0103.djvu
8.1 Weighted Residuals......Page __sk_0105.djvu
8.2 Rayleigh-Ritz Variational Methods......Page __sk_0111.djvu
8.3 Selected References......Page __sk_0112.djvu
8.4 Problems......Page __sk_0113.djvu
9 Cellular Automata and Lattice Gases......Page __sk_0114.djvu
9.1 Lattice Gases and Fluids......Page __sk_0115.djvu
9.2 Cellular Automata and Computing......Page __sk_0119.djvu
9.3 Selected References......Page __sk_0121.djvu
9.4 Problems......Page __sk_0122.djvu
Part Three: Observational Models......Page __sk_0123.djvu
10 Function Fitting......Page __sk_0127.djvu
10.1 Model Estimation......Page __sk_0128.djvu
10.2 Least Squares......Page __sk_0129.djvu
10.3 Linear Least Squares......Page __sk_0130.djvu
10.3.1 Singular Value Decomposition......Page __sk_0131.djvu
10.4 Nonlinear Least Squares......Page __sk_0134.djvu
10.4.1 Levenberg-Marquardt Method......Page __sk_0136.djvu
10.5 Estimation, Fisher Information, and the Cramér-Rao Inequality......Page __sk_0137.djvu
10.7 Problems......Page __sk_0139.djvu
11.1 Orthogonal Transforms......Page __sk_0140.djvu
11.2 Fourier Transforms......Page __sk_0141.djvu
11.3 Wavelets......Page __sk_0143.djvu
11.4 Principal Components......Page __sk_0148.djvu
11.6 Problems......Page __sk_0150.djvu
12.1.1 Padé Approximants......Page __sk_0151.djvu
12.1.2 Splines......Page __sk_0153.djvu
12.2 Orthogonal Functions......Page __sk_0154.djvu
12.3 Radial Basis Functions......Page __sk_0157.djvu
12.4 Overfitting......Page __sk_0159.djvu
12.5 Curse of Dimensionality......Page __sk_0160.djvu
12.6 Neural Networks......Page __sk_0162.djvu
12.6.1 Back Propagation......Page __sk_0164.djvu
12.7 Regularization......Page __sk_0165.djvu
12.9 Problems......Page __sk_0167.djvu
13 Optimization and Search......Page __sk_0168.djvu
13.1 Multidimensional Search......Page __sk_0169.djvu
13.2 Local Minima......Page __sk_0173.djvu
13.3 Simulated Annealing......Page __sk_0174.djvu
13.4 Genetic Algorithms......Page __sk_0176.djvu
13.5 The Blessing of Dimensionality......Page __sk_0178.djvu
13.6 Selected References......Page __sk_0179.djvu
13.7 Problems......Page __sk_0180.djvu
14.1 Histogramming, Sorting, and Trees......Page __sk_0181.djvu
14.2 Fitting Densities......Page __sk_0184.djvu
14.3 Mixture Density Estimation and Expectation-Maximization......Page __sk_0186.djvu
14.4 Cluster-Weighted Modeling......Page __sk_0190.djvu
14.6 Problems......Page __sk_0197.djvu
15.1 Matched Filters......Page __sk_0198.djvu
15.2 Wiener Filters......Page __sk_0199.djvu
15.3 Kalman Filters......Page __sk_0201.djvu
15.4 Nonlinearity and Entrainment......Page __sk_0207.djvu
15.5 Hidden Markov Models......Page __sk_0209.djvu
15.7 Problems......Page __sk_0215.djvu
16 Linear and Nonlinear Time Series......Page __sk_0216.djvu
16.1 Linear Time Series......Page __sk_0217.djvu
16.2 The Breakdown of Linear Systems Theory......Page __sk_0219.djvu
16.3 State-Space Reconstruction......Page __sk_0220.djvu
16.4 Characterization......Page __sk_0225.djvu
16.4.1 Dimensions......Page __sk_0226.djvu
16.4.2 Lyapunov Exponents......Page __sk_0228.djvu
16.4.3 Entropies......Page __sk_0229.djvu
16.5 Forecasting......Page __sk_0232.djvu
16.7 Problems......Page __sk_0236.djvu
Appendix 1 Graphical and Mathematical Software......Page __sk_0237.djvu
A1.1.1 Programming Environments......Page __sk_0238.djvu
A1.1.2 Interactive Environments......Page __sk_0240.djvu
A1.2.1 Postscript......Page __sk_0242.djvu
A1.2.2 X Windows......Page __sk_0246.djvu
A1.2.3 OpenGL......Page __sk_0252.djvu
A1.2.4 Java......Page __sk_0256.djvu
A1.3 Problems......Page __sk_0261.djvu
A2.1 OSI, TCP/IP, and All That......Page __sk_0262.djvu
A2.2 Socket I/O......Page __sk_0263.djvu
A2.3 Parallel Programming......Page __sk_0266.djvu
Appendix 3 Benchmarking......Page __sk_0269.djvu
A4.2 Ordinary Differential and Difference Equations......Page __sk_0271.djvu
A4.3 Partial Differential Equations......Page __sk_0278.djvu
A4.4 Variational Principles......Page __sk_0281.djvu
A4.5 Random Systems......Page __sk_0283.djvu
A4.6 Finite Differences: Ordinary Differential Equations......Page __sk_0288.djvu
A4.7 Finite Differences: Partial Differential Equations......Page __sk_0293.djvu
A4.8 Finite Elements......Page __sk_0301.djvu
A4.9 Cellular Automata and Lattice Gases......Page __sk_0304.djvu
A4.10 Function Fitting......Page __sk_0314.djvu
A4.11 Transforms......Page __sk_0317.djvu
A4.12 Architectures......Page __sk_0321.djvu
A4.13 Optimization and Search......Page __sk_0327.djvu
A4.14 Clustering and Density Estimation......Page __sk_0331.djvu
A4.15 Filtering and State Estimation......Page __sk_0335.djvu
A4.16 Linear and Nonlinear Time Series......Page __sk_0337.djvu
Bibliography......Page __sk_0342.djvu
Index......Page __sk_0352.djvu