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Figure 2.1 (page 106):
Dynamical regimes for the planar system dx/dt = Ax, A \in \mathbb {R}^{2 x2}.
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Figure 2.2 (page 107):
Dynamical behavior on the region boundaries for the planar system dx/dt = Ax, A \in \mathbb {R}^{2 x2}.
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Figure 2.4 (page 127):
Function f(x)=\exp \big (-8\big (\frac {x}{\pi }\big )^{2}\big ) and truncated trigonometric Fourier series approximations with K=2,5,10. The approximations with K=5 and K=10 are visually indistinguishable from the exact function.
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Figure 2.5 (page 128):
Truncated trigonometric Fourier series approximation to f(x)=x, using K=5,10, 50. The wiggles get finer as K increases.
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Figure 2.6 (page 130):
Function f(x)=\exp \big (-8x^{2}\big ) and truncated Legendre-Fourier series approximations with n=2,5,10.
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Figure 2.7 (page 131):
Function f(x)=H(x) and truncated Legendre-Fourier series approximations with n=10,50,100.
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Figure 2.12 (page 176):
Leading-order inner U_0, outer u_0, and composite solutions u_{0c}, for Example \ref {ex:rxequil} with \epsilon =0.2, K=1, and k_{2}=1.
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Figure 2.15 (page 187):
Contours of an energy function V(x_{1},x_{2}) or H(x_{1},x_{2}).
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Figure 2.16 (page 191):
Energy landscape for a pendulum; H = \frac {1}{2} p^2 -\kappa \cos q; \kappa =2.
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Figure 2.17 (page 192):
Landscape for H=\frac {1}{2}p^{2}+\frac {1}{4}q^{4}-\frac {1}{2}q^{2}.
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Figure 2.18 (page 194):
A limit cycle (thick dashed curve) and a trajectory (thin solid curve) approaching it.
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Figure 2.19 (page 196):
Periodic (left) and quasiperiodic (right) orbits on the surface of a torus. The orbit on the right eventually passes through every point in the domain.
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Figure 2.20 (page 197):
A limit cycle for the R\"ossler system, a=b=0.2, c=1.
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Figure 2.21 (page 197):
A strange attractor for the R\"ossler system, a=b=0.2, c=5.7.
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Figure 2.22 (page 199):
Bifurcation diagram for the saddle-node bifurcation.
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Figure 2.23 (page 201):
Bifurcation diagram for the transcritical bifurcation.
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Figure 2.24 (page 202):
Bifurcation diagrams for the pitchfork bifurcation.
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Figure 2.25 (page 207):
Approximate solutions to \dot {x}=-x using explicit and implicit Euler methods with \Delta t=2.1, along with the exact solution x(t)=e^{-t}.
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Figure 2.26 (page 210):
Stability regions for Adams-Bashforth methods; \dot {x}=\lambda x.
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Figure 2.27 (page 211):
Stability regions for Adams predictor-corrector methods; \dot {x}=\lambda x.
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Figure 2.28 (page 212):
Stability regions for Runge-Kutta methods; \dot {x}=\lambda x.
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Figure 2.29 (page 215):
Hat functions for N=2.
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Figure 2.30 (page 216):
Approximate solutions to \eqref {eq:MWRexample} using the finite element method with hat functions for N=6 and N=12. The exact solution also is shown.
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Figure 2.31 (page 220):
Dependence of \left | c(j) \right | on j for the Legendre-Galerkin approximation of \eqref {eq:MWRexample} with n=10.
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Figure 2.34 (page 242):
Stability regions for Adams predictor-corrector methods; \dot {x}= \lambda x; APCn' uses nth-order predictor and nth-order corrector.
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Figure 4.1 (page 363):
Normal distribution, with probability density p_\xi (x) = (1/\sqrt {2\pi \sigma ^2}) \exp (-(1/2) (x-m)^2/\sigma ^2).
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Figure 4.2 (page 368):
Multivariate normal for n=2. The contour lines show ellipses containing 95, 75, and 50 percent probability.
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Figure 4.5 (page 381):
A joint density function for the two uncorrelated random variables in Example~\ref {ex:jointunc}.
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Figure 4.6 (page 385):
A nearly singular normal density in two dimensions.
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Figure 4.8 (page 392):
Histogram of 10,000 samples of uniformly distributed x.
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Figure 4.9 (page 392):
Histogram of 10,000 samples of \displaystyle y=\DOTSB \sum@ \slimits@ _{i=1}^{10} x_i .
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Figure 4.10 (page 407):
The multivariate normal, marginals, marginal box, and bounding box.
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Figure 4.11 (page 431):
The sum of squares fitting error (top) and validation error (bottom) for PCR versus the number of principal components \ell ; cross validation indicates that four principal components are best.
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Figure 4.12 (page 432):
The sum of squares validation error for PCR and PLSR versus the number of principal components/latent variables \ell ; note that only two latent variables are required versus four principal components.
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Figure 4.13 (page 433):
Predicted versus measured outputs for the validation dataset. Top: PCR using \textit {four} principal components. Bottom: PLSR using \textit {two} latent variables. Left: first output. Right: second output.
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Figure 4.14 (page 434):
Effect of undermodeling. Top: PCR using \textit {three} principal components. Bottom: PLSR using \textit {one} latent variable.
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Figure 4.15 (page 437):
The indicator (step) function f_1(w;x) and its smooth approximation, f(w;x).
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Figure 4.16 (page 453):
Typical strain versus time data from a molecular dynamics simulation. The data are available on the website \jbrawweb .
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Figure 4.17 (page 454):
Plot of y versus x available on the website \jbrawweb .
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Figure 4.18 (page 460):
Smooth approximation to a unit step function, H(z-1).
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Figure 5.2 (page 472):
Sampling faster on the last plot in Figure \ref {fig:wiener}; the sample time is decreased to \Delta t = 10^{-9} and the roughness is restored on this time scale.
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Figure 5.3 (page 479):
A representative trajectory of the discretely sampled Brownian motion; D=2, V=0, n=500.
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Figure 5.4 (page 479):
The mean square displacement versus time; D=2, V=0, n=500.
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Figure 5.5 (page 487):
Two first-order reactions in series in a batch reactor, c_{A0}=1, c_{B0}=c_{C0}=0, k_1=2, k_2=1.
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Figure 5.6 (page 489):
A sample path of the unit Poisson process.
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Figure 5.7 (page 489):
A unit Poisson process with more events; sample path (top) and frequency distribution of event times \tau .
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Figure 5.9 (page 496):
Stochastic simulation of first-order series reaction A-> B-> C starting with 100 A molecules.
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Figure 5.11 (page 500):
Solution to master equation for A + B <-> C starting with 20 A molecules, 100 B molecules and 0 C molecules, k_1=1/20, k_{-1}=3.
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Figure 5.12 (page 501):
Solution to master equation for A + B <-> C starting with 200 A molecules, 1000 B molecules and 0 C molecules, k_1=1/200, k_{-1}=3.
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Figure 5.13 (page 503):
The equilibrium reaction extent's probability density for Reactions \ref {rxn:rxn2} at system volume \Omega =20 (top) and \Omega = 200 (bottom). Notice the decrease in variance in the reaction extent as system volume increases.
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Figure 5.14 (page 507):
Simulation of 2 A \mathrel {\mkern 4mu}\mathrel {\smash {\mathchar "392D}}\mathrel {\mkern -2.5mu}-> B for n_0=500, \Omega =500. Top: discrete simulation; bottom: SDE simulation.
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Figure 5.15 (page 508):
Cumulative distribution for 2 A \mathrel {\mkern 4mu}\mathrel {\smash {\mathchar "392D}}\mathrel {\mkern -2.5mu}-> B at t=1 with n_0=500, \Omega =500. Discrete master equation (steps) versus omega expansion (smooth).
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Figure 5.16 (page 522):
The change in 95% confidence intervals for \hat {x}(k\vert k) versus time for a stable, optimal estimator. We start at k=0 with a large initial variance P(0), which gives a large confidence interval.
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Figure 5.17 (page 532):
Deterministic simulation of reaction A + B <-> C compared to stochastic simulation.
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