Illustrative Examples

We introduce a hypothetical two-sector economy from Miller and Blair (2009) to illustrate the concepts in the previous section. In addition, the corresponding LINGO code for the example is given in Appendix A. The shaded region in Table 8.3 represents the transactions (Z) matrix. The third column represents the monetary value of final goods consumed or the final demand vector (c). The fourth column represents the total output of each sector (x). The third row represents the amount of value added for each sector. The fourth row represents the total inputs required by each sector. It can be noted that the total inputs of each sector is equal to the total outputs of each sector. This ensures that we have a balanced economy. 

Taking the first row, it can be observed that Sector 1 produces 150 worth of output for its own intermediate consumption, 500 worth of output which will be used as inputs for Sector 2, and 350 worth of output is purchased as final goods by end users. The sum of the intermediate demand of each sector for Sector I s output and final demand for Sector I s output will yield the total output for the said sector. In addition, the first column shows that Sector 1 uses 150 worth of input from its own output and 200 worth of input from Sector 2 and 650 from value added. Value added can be broken down into employee 

The technical coefficients matrix. A, is derived by dividing the elements of transactions matrix (z j) by the respective column sum (x) such that 

For this example, we can compute for the Leontief inverse matrix, L, as  

In particular. Matrix L allows us to conduct a scenario analysis comparing the impact of a 1-unit worth of monetary increase in the final demand of each sector. Suppose that there is 1 increase in the final demand for Sector 1. This scenario will require the output of Sector 1 to increase by 1.25 and the output of Sector 2 to increase by 0.26. This increases the total output of the economy by 1.51. On the other hand, a 1 increase in the final demand of Sector 2 will require the output of Sector 1 to increase by 0.33 and the output of Sector 2 to increase by 1.12, which increases total output of the economy by 1.45. Comparing the impact between a dollar increase in the final demand for output of Sector 1 and Sector 2, the economic benefit is higher if the increase was on Sector 1. Evaluating 

The capabilities of the flexible model are illustrated by solving two SC designplanning problems. The first problem is an illustrative example, which intends to show some of the special characteristics of the flexible model. The last one constitutes a more sophisticated case study based on a real problem. The resulting MILP models have been solved to optimality in GAMS using CPLEX (11.0) on a computer with an Intel Core 2 Duo 2.0 GHz and 2 GB RAM. 

An SC design-planning problem comprising four potential locations for processing sites and distribution centers is presented. A planning horizon of five annual periods is considered. The STN representation of the production process is depicted in Fig. 5.4. Two final products (54 and 55) can be sold in two markets (Ml and M2). 51 and 52 are raw materials. A set of three equipment technologies (E -E1 ) is assumed to be available for the processing sites. It is assumed that task i can be performed in equipment E, task i2 in equipment E2, and task /3 in equipment E2 . The discount rate is equal to 35 %. Input data associated with this example can be found in Appendix C.l. 

5 Flexible Design—Planning of Supply Chain Networks 

This flexible model example consists of 4,391 equations, 935 continuous variables, and 64 binary variables. The total CPU time is 0.34 seconds and the optimal solution is found after 1,434 iterations. The LP-relaxed solution gives a value of 1,762,204m.u. for the objective function. 

Let us consider a data set containing Near-Infrared (NIR) spectra of 30 gasoline samples and five dependent variables (Y = [yl y2 y3 y4 ySj) [31], The original spectra and the so-called std spectrum are presented in Fig. 2. The std, i.e. standard deviation, spectrum it is a vector, the elements of which 

The goal is to construct calibration models (y = f(X)), which allow prediction of dependent variables for new samples, based on their NIR spectra. 

The original data set X (30, 256) was divided into two subsets the model set (20, 256) used to construct the model and the test set (10, 256) to evaluate predictive ability of the model. The splitting of the model and test sets was performed according to the Kennard and Stone algorithm [32]. This algorithm allows the selection of objects (samples) which are uniformly distributed over the experimental space and represent all sources of data variance. 

Evaluation of the constructed models and their predictive ability was based on the RMSCV and RMSEP, respectively. These parameters are defined 

A cross-validation procedure and a randomization test [33] were used to evaluate the complexity of the full-spectra models. 

Two cases are considered for the available dynamic data that correspond to locally identifiable and unidentifiable cases. For the first case, the modal data T consists of the identified modal frequencies for both modes of the frame. Rather than performing modal identification on simulated time histories, noisy versions of the modal frequencies are generated and the measurements are = 5.5 Hz and = 14.9 Hz. 

The prior PDF p(0 C) is taken to be independent log-normal PDFs with means of 0.9 and 1.2 and unit variance. Using the modal data V, the updated PDF for the stiffness parameter vector 0 is formulated as  

If it is assumed that only one sensor (at either the first or second floor) was used during the modal testing, only the modal frequencies can be identified. In this case, the two stiffness parameters are locally identifiable and the normalized modal goodness-of-fit function is given 

Bayesian Methods for Structural Dynamics and Civil Engineering 

In this equation, ax2 and are the standard deviations of the stationary displacement and velocity response of the top floor, respectively, obtained by the Lyapunov equation [249]  

Consider the X and y matrices in eq. (17.16), where variables have been centred to give a mean of zero. 

Clearly the first two rows show a large variation in x with no change in y, i.e. these variables are not related to the response. The last two rows display correlation and anti-correlation, respectively, with the y data in equal amounts. Solving with MLR equation (17.14) gives the solution vector a in eq. (17.17), which shows that both x columns are equally important in describing the y variation. The difference between the actual and predicted y indicates that there is a residual y variation that cannot be modelled by the x variables. 

So G°[N, J°] is defined. The simple figure would allow one to solve the problem by mere inspection. But our goal is to illustrate the formal procedure as outlined in Section 3.3. 

The decomposition of G° starts from some node, say (environment), giving the connected component 

Let us further select node a we have successively, according to Section A.4 

From the remaining nodes, let us take node/ we find 

We have K =3 and let us merge the nodes in each N,. The unmeasured arcs are deleted. Let be the node obtained by merging N. Then the incident nondeleted arcs are 

This process has an approximate settling time of 15 sec. Therefore, the lead element in the control signal trajectory specification for stable processes, or, is chosen to be = 3, which is approximately equal to the dominant process time constant. Then we let r = and tune the parameter a to determine the closed-loop response speed. The transfer function from the setpoint to the desired control signal is given by 

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Desulfurization will become mandatory when oxidizing catalysts are installed on the exhaust systems of diesel engines. At high temperatures this catalyst accelerates the oxidation of SO2 to SO3 and causes an increase in the weight of particulate emissions if the diesel fuel has not been desulfurized. As an illustrative example, Figure 5.22 shows that starting from a catalyst temperature of 400°C, the quantity of particulates increases very rapidly with the sulfur content. 

We consider first some experimental observations. In general, the initial heats of adsorption on metals tend to follow a common pattern, similar for such common adsorbates as hydrogen, nitrogen, ammonia, carbon monoxide, and ethylene. The usual order of decreasing Q values is Ta > W > Cr > Fe > Ni > Rh > Cu > Au a traditional illustration may be found in Refs. 81, 84, and 165. It appears, first, that transition metals are the most active ones in chemisorption and, second, that the activity correlates with the percent of d character in the metallic bond. What appears to be involved is the ability of a metal to use d orbitals in forming an adsorption bond. An old but still illustrative example is shown in Fig. XVIII-17, for the case of ethylene hydrogenation. 

A few illustrative examples are the following. Photohydrogenation of acetylene and ethylene occurs on irradiation of Ti02 exposed to the gases, but only if TiOH surface groups are present as a source of hydrogen [319]. The pho-toinduced conversion of CO2 to CH4 in the presence of Ru and Os colloids has been reported [320]. Platinized Ti02 powder shows, in the presence of water, photochemical oxidation of hydrocarbons [321,322]. Some of the postulated reactions are ... 

An illustrative example is provided by investigating the possible momenta for a single particle travelling in the v-direction, p First, one writes the equation that defines the eigenvalue condition... 

A marvellous and rigorous treatment of non-relativistic quantum mechanics. Although best suited for readers with a fair degree of mathematical sophistication and a desire to understand the subject in great depth, the book contains all of the important ideas of the subject and many of the subtle details that are often missing from less advanced treatments. Unusual for a book of its type, highly detailed solutions are given for many illustrative example problems. 

The field of gas phase reaction dynamics has been extensively reviewed elsewhere [1, 2 and 3] in considerably greater detail than is appropriate for this chapter. Here, we begin by simnnarizing the key theoretical concepts and experimental teclmiques used in reaction dynamics, followed by a case study , the reaction F + H2 HF + H, which serves as an illustrative example of these ideas. 

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

As an illustrative example, consider the vibrational energy relaxation of the cyanide ion in water [45], The mechanisms for relaxation are particularly difficult to assess when the solute is strongly coupled to the solvent, and the solvent itself is an associating liquid. Therefore, precise experimental measurements are extremely usefiil. By using a diatomic solute molecule, this system is free from complications due to coupling... 

In this section, I present a few illustrative examples of applications of NMR relaxation studies within different branches of chemistry. The three subsections cover one story each, in order of increasing molecular size and complexity of the questions asked. 

G) ILLUSTRATIVE EXAMPLES OF THE ELECTRONIC AND OPTICAL PROPERTIES OF MODERN MATERIALS... 

In other words, (n VQ ) is imaginary, making real. As an illustrative example, n) may assumed to be given by Eq. (24), in which case... 

Essential Dynamics In most applications details of individual MD trajectories are of only minor interest. An illustrative example due to Grubmuller [10] is documented in Figure 3. It describes the dynamics of a polymer chain of 100 CH2 groups. Possible stepsizes for numerical integration are confined... 

Having found a place (the sp -sp bon d t to establish the boundary between classical atom s and quantum atoms, the next cpiesiion is how to cap the quantum atoms. Let s first of all look at an illustrative example of the problem. ... 

A simple illustrative example of reciprocal space is that of a 2D square lattice where the vectors a and b are orthogonal and of length equal to the lattice spacing, a. Here a and b are directed along the same directions as a and b respectively and have a length 1/a... 

As an illustrative example we consider the Galerkin finite element solution of the following differential equation in domain Q, as shown in Figure 2.20. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

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