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This chapter focuses on the error and convergence analysis of the technique based on the use of a discrete method for ODEs endowed with some continuous extension, which is called the standard approach. The technique is described for some of the main types of DDEs and neutral DDEs, ranging from constant or non-vanishing time dependent delays, to arbitrary time dependent delays, up to state dependent delays. It is shown that for time dependent and state dependent delays, the resulting method may become implicit even if the underlying ODE method is explicit and, even in this case, well-posedness is proven. The non-trivial problem of tracking the discontinuities is also considered. It is shown that despite the fact that any discrete method is, in principle, suitable for the standard approach, one-step methods (essentially Runge-Kutta methods) are preferable to multistep methods. A list of available codes is given.

This tutorial gives a practical introduction to the Hybrid Monte Carlo algorithm and the analysis of Monte Carlo data. The method is exemplified at the ϕ 4 theory, for which all steps from the derivation of the relevant formulae to the actual implementation in a computer program are discussed in detail. It concludes with the analysis of Monte Carlo data, in particular their auto-correlations.

This chapter concerns the statement and properties of the steady Navier–Stokes equations and the corresponding weak formulation. This includes discussion of stability theory, bifurcation and nonlinear iteration. This is followed by a description of finite element discretization and error analysis of discrete solutions.

The use of neural networks (NNs) to predict an outcome or the output results as a function of a set of input parameters has been gaining wider acceptance with the advance in computer technology as well as with an increased awareness of the potential of NNs. A neural network is first trained to learn the underlying functional relationship between the output and the input parameters by providing it with a large number of data points, where each data point corresponds to a set of output and input parameters. Sumpter and Noid demonstrated the use of NNs to map the vibrational motion derived from the vibrational spectra onto a PES with relatively high accuracy. In another application, Sumpter et al. trained an NN to learn the relation between the phase-space points along a trajectory and the mode energies for stretching, torsion, and bending vibrations of H2O2. Likewise, Nami et al. demonstrated the use of NNs to determine the TiO2 deposition rates in a chemical vapor deposition (CVD) process from the knowledge of a range of deposition conditions. In view of the success achieved in obtaining interpolated values of the PESs for multi-atomic systems using an NN trained by the ab initio energy values for a large number of configurations, it is reasonable to ask whether we can successfully compute the results of an MD trajectory for a chemical reaction using an NN trained by the data obtained by previous MD simulations. If this can be done successfully, it becomes possible to execute a small number of trajectories, M, and then utilize the results of these trajectories as a database to train an NN to predict the final results of a very large number of trajectories N, where N >> M, that can be used to increase the statistical accuracy of the MD calculations and to further explore the dependence of the trajectory results upon a wide variety of variables without actually having to perform any further numerical integrations. In effect, the NN replaces the computationally laborious numerical integrations.

We consider solving a problem that is closely related to one that we have previously solved or whose parts have previously been solved. We study backward error analysis for sparse problems, obtaining entries of the inverse of a sparse matrix, sparsity in nonlinear problems, solution methods based on orthogonal transformations, and hybrid methods that combine iterative and direct methods.

In 2012, a Fairleigh Dickinson University (FDU) survey made head­lines. The headlines questioned Fox News viewers’ intelligence. The Nation’s headline read: “It’s Official: Watching Fox Makes You Stupider.” It claimed that “[a] ccording to a new study by Farleigh Dickinson University, Fox viewers are the least knowledgeable audience of any outlet, and they know even less about politics and current events than people who watch no news at all.” It concluded that Fox News “fails the fundamental test of journalism: Are you informing your audience?” The Huffington Post (2012) claimed that “people who only watch Fox News are less informed than all other news consumers.” The New York Times’ Timothy Egan (2014) repeated the assertion. Conservative-leaning publications interpreted FDU’s findings differently. The Examiner’s headline read “Democrats Use Biased ‘Study’ to Smear Fox News.” It claimed that the pollsters “abandoned all integrity to vindictively trash Fox News and peddle the partisan smear that anyone who watches ‘right-wing propaganda’ (anything that includes multiple sides of the story) is stupid.” FDU’s report on its Public Mind Poll (2012) focused not on how respondents answered individual recall questions, but on an aggregate PK scale that FDU manufactured. Like nearly all published PK scales, FDU’s scale was formed by adding the number of correct answers respondents gave to a small set of recall questions. Such scales typically range in value from zero-to-five or zero-to-seven, with the high number representing the total number of recall questions included in the scale. If a respondent answers no questions correctly, they get a score of zero. If they answer all questions correctly, they get the highest possible score. PK scales are regularly used to represent “the range of factual information about politics that is stored in long-term memory.” FDU’s report and the subsequent media reports are based on the finding that Fox News viewers scored lower on FDU’s PK scale than did viewers of other networks. In this chapter, I examine this case and other claims that are based on PK scales.