Computational Exercises

Problems at the end of the chapter consist of three different types (a) Basic Concepts (True False), which seek to test the reader s comprehension of the key concepts in the chapter (b) Short Exercises, which seek to test the reader s ability to compute the required parameters for a simple data set using simple or no technological aids, and this section also includes proofs of theorems and (c) Computational Exercises, which require not only a solid comprehension of the basic material but also the use of appropriate software to easily manipulate the given data sets. 

Determine if the following statements are true or false and state why this is the case. 

The median absolute difference is a robust measure of dispersion. 

A left-skewed data set has many values in the left tail. 

Outliers are data points whose values are abnormal. 

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A similar computational exercise was performed by Guirao et al. (1979). They used a code based on the Eulerean FCT approach. Blasts produced by four different, but energetically equivalent, sources ... 

In principle, one may combine equilibrium and critical data in one database for the parameter estimation. From a numerical implementation point of view this can easily be done with the proposed estimation methods. However, it was not done because it puts a tremendous demand in the correlational ability of the EoS to describe all the data and it will be simply a computational exercise. 

The chapters close with a few problems whose purpose is to encourage a deeper investigation of the chapter material, often with suggestions for computational exercises. 

The number of data points to generate in order to get an optimum value for the statistic is not obvious. Intuition indicates that the value of the statistic may very well be a function of the number of data points used in its calculation. At first glance, this would also seem to be a showstopper in the use of this statistic for the purpose of quantifying nonlinearity. However, intuition also indicates that even so, use of sufficiently many data points will give a stable value, since sufficiently many eventually becomes an approximation to infinity , and therefore even in such a case will at least tend toward an asymptotic value, as more and more data points are used. Since we have already extracted the necessary information from the actual data itself, computations from this point onward are simply a computer exercise, needing no further input from the original data set. 

So there are three equations, (625), (632), and (633), in two unknowns A and . These are enough to solve for the components of A and for for any boundary condition. For any physical boundary condition, there will be longitudinal as well as transverse components of A in the vacuum, and will in general be phase-dependent and structured. This computational exercise shows that the Lorenz condition is arbitrary and, if it is discarded, the values of A and from Eqs. (625), ( 632), and (633) change. 

Linear stability theory results match quite well with controlled laboratory experiment for thermal and centrifugal instabilities. But, instabilities dictated by shear force do not match so well, e.g. linear stability theory applied to plane Poiseuille flow gives a critical Reynolds number of 5772, while experimentally such flows have been observed to become turbulent even at Re = 1000- as shown in Davies and White (1928). Couette and pipe flows are also found to be linearly stable for all Reynolds numbers, the former was found to suffer transition in a computational exercise at Re = 350 (Lundbladh Johansson, 1991) and the latter found to be unstable in experiments for Re > 1950. Interestingly, according to Trefethen et al. (1993) the other example for which linear analysis fails include to a lesser degree, Blasius boundary layer flow. This is the flow which many cite as the success story of linear stability theory. 

Identification task. The first source of data was the computer exercise that followed the initial instructional session. (The format of this exercise was shown in Figure 5.1.) The items in this task were selected randomly for each student from a pool of 100 items, and they are similar to those of Table 7.1. During the exercise, one item at a time was displayed, and the student responded to it by selecting the name of the situation depicted in the item from a menu containing all five names Change, Group, Compare, Restate, and Vary. The student received immediate feedback about the accuracy of the answer, and if the student responded incorrectly, the correct situation was identified by SPS. 

At this point, each student was then given an abbreviated refresher lesson in SPS. This lesson lasted for about 5 min and merely described the situations again, using the original examples from the first SPS session. No computer exercises were presented. Finally, students were given a second set of 20 problems to sort, with explicit instructions to use their situational knowledge to form the groups. The entire session lasted about 1 hour. 

For each student, the model simulated the student s performance on the set of items presented to the student in the computer exercise. The items appeared in the same order that the student saw them. These items were new to the model they were not ones that had been presented in Experiment I. As before, the model made one response to each item. 

Validation without an independent test set. Each application of the adaptive wavelet algorithm has been applied to a training set and validated using an independent test set. If there are too few observations to allow for an independent testing and training data set, then cross validation could be used to assess the prediction performance of the statistical method. Should this be the situation, it is necessary to mention that it would be an extremely computational exercise to implement a full cross-validation routine for the AWA. That is. it would be too time consuming to leave out one observation, build the AWA model, predict the deleted observation, and then repeat this leave-one-out procedure separately. In the absence of an independent test set, a more realistic approach would be to perform cross-validation using the wavelet produced at termination of the AWA, but it is important to mention that this would not be a full validation. 

Before turning to specific applications at our institutions, we present a brief theoretical section on each of the three areas of quantum chemistry, molecular mechanics, and molecular dynamics. Our plan is that these theory sections will present the context in which our applications take place. This is to emphasize our contention that computational exercises should not take place without adequate background into the theory behind the exercise. Otherwise the user will treat the computer as a blackbox. At the least the user needs to appreciate what kinds of chemical questions can be answered by particular computational methodologies. 

Koopmans theorem and the concept of frozen orbital are readily introduced at this level. We have found it extremely helpful to employ flow charts, such as Figure 8, to assist the student to keep track of all the features of their computational exercise. 

At the second level is the use of the computer in a laboratory-like setting to accompany and support the quantum chemistry lectures. In the 1970s an effort was begun to develop a series of computer exercises that could serve as the laboratory component for theoretical chemistry. Students find quantum chemistry to be an abstract, highly mathematical subject, and unless its concepts are translated into action in some way it is unlikely that they will master its principles or discover its applications in other disciplines. Hands on in quantum chemistry means hands on the keyboard of a computer. Therefore, the goal was to create a repertoire of computer exercises that juxtaposes the theoretical framework of quantum chemistry and its computational methodology. 

Some computer exercises will be discussed briefly in this section. They serve as laboratory exercises for the second semester of physical chemistry sequence when quantum chemistry is taught. These computer exercises are also used in the lecture demonstration format. Sometimes they are given as extended problem assignments. Four software packages are used to various degrees in these exercises BASIC programs, spreadsheet templates, Mathematica, and Mathcad. 

Physical chemistry texts introduce molecular orbital theory with the example of the hydrogen-molecule ion. Lecture material on this subject is supported with a computer exercise based on a paper by Robiette.i fi This calculation has been done with BASIC, Mathematica, and Mathcad. Figure 20 and 21 show a Mathcad document for this calculation. 

A quick perusal of a typical daily schedule (Figure 1) shows that the students are kept busy from early morning until sometimes quite late at night. Also the schedule reveals the variety in the program. Lectures with demonstrations, extensive hands-on laboratory activities, computer exercises and science films are interspersed with talks by guest lecturers from other fields, tours of university facilities, visits to points of interest, and social events. 

The Parade of Trades exercise allows participants to develop a better intuitive understanding of various basic production concepts, such as variability and throughput. A single-line processing system is used to illustrate the impact that variability has on the performance of construction trades and their snccessors. Using simulation and computer exercises for this system shows the possibihty of reducing waste of resources and time by reducing the variability in work flow between trades (Tommelein et al. 1999). 

Au(CN)2] bridging ligands and substituted terpyridine (terpy) ancillary ligands ([Pb(DCA)2] (1), [Pb(terpy)(DCA)2] (2), pb(terpy)[Au(CN)2]2] (3), Pb(4 -chloro-terpy)[Au(CN)2]2] (4) and pb(4 -bromo-terpy)-(p-OH2)0.5[Au(CN)2]2] (5)) were spectroscopically examined by solid-state Pb MAS NMR spectroscopy to characterise the structural and electronic changes associated with Pb(II) lone-pair activity. In this computational exercise, the " Sn and Pb NMR spectra (data provided to the students)... 

FIGURE 2.5 Statistical structure of peptides. Upper and lower panels show and Fx< r)> respectively, corresponding to the uniform distribution. The open squares mark results of the computer exercise described in text. The results of the chemist s intuition have been omitted in the upper panel as they simply follow the horizontal line at height approximately 10 moles per gram. 

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