Representation of the Problem

Fig. 12.8 Graphical representation of the problem specification, with x set to 0.1 kg salt/kg water... 

Another question of interest is that of the transition between stepwise and concerted pathways. A schematic representation of the problem is given in Fig. 6 this is schematic in the sense that the system is represented as depending on a single reaction coordinate, viz., the distance between the atoms forming the bond to be broken during the reaction. This is obviously the main reaction coordinate as far as the concerted pathway (48) and the decomposition of RX- (47) are concerned. It is also an important reaction... 

Figure 6 Schematic representation of the problems that can be encountered during the synthesis of a zwitterionic diblock copolymer by aqueous ATRP. I represents the initiator fragment and represents the block junction...

Appropriate Representation of the Problem (Implicit Assumptions and Requirements)... 

Substituting these nondimensional numbers into eqs 11-18, and after some rearrangement, the general dimensionless representation of the problem is obtained as depicted in Table 1. These equations are valid not only for spherical pellet geometry, but also for the infinite cylinder and the infinite flat plate (slab). The dimensionless numbers x, , Bim and Bib must then be calculated on the basis of the respective characteristic length, i.e. the cylinder radius or the plate thickness. Moreover, the parameter b in eqs 32 and 33 is a factor depending on the pellet geometry. It is 2 for the sphere, 1 for the cylinder, and 0 for the flat plate. 

The derivatives in eq 188 are approximated by central differences. This leads to a discrete representation of the problem in the form of linear equations. For j = 1 and j = N we have slightly modified equations, since here the boundary conditions have to be considered. Because the concentration of three isomers has to be calculated for each of the N volume elements as a whole, we have a system of N x 3 linear equations. This can be expressed in matrix notation ... 

Fig. 4. Schematic representation of the problem geometry corresponding to a pattern involving a step distribution of the active-area density. The domain Q can be viewed as a two-dimensional, symmetric section of a larger, three-dimensional configuration, as indicated. The relative dimensions are shown, and the boundary segments of the corresponding boundary-value problem are indicated. (Reprinted by permission of the publisher, The Electrochemical Society, Inc. [25]).

From the viewpoint of scientific methodology there are three main tasks in CAPE representation of the problem, generation of several alternative solutions, and selection of the best one. These tasks correspond to the activities realized in four phases of any scientific method analysis (description of the problem and identification of the objectives), hypothesis (generation of solutions), synthesis (comparing the solutions), and validation (formulation of conclusions). The activities realized in the last two phases correspond to the selection task in CAPE. 

In the implementation of the method one may thus either assemble information obtained on an element by element basis into a global representation of the problem or assemble the information from the complete system form instead. Some assembling techniques are outlined by Jiang [84], chap 15. Finally, standard numerical techniques may be used to solve the global system matrix equation for the unknown field variables. 

This simplification allows us to use a convenient Lagrangian representation of the problem (6.2). Without the diffusion term the solution can be expressed by the method of characteristics through a set of ordinary differential equations... 

Though these properties have not been very rigorously proved, intuitive arguments show that they are satisfactory. Moreover, Balian and Toulouse12 showed that in the continuous case the properties of a chain can be studied precisely by starting from a Lagrangian representation of the problem (see Chapter 11) and by using a transfer matrix method. In particular, these authors verified that the critical exponents are v = 1 and y = 1 [see also the article by Thouless (1975)]. 3... 

Figure 17.1 A schematic representation of the problem of lack of data for chemicals in the environment and the need for predictive methods.

The use of successive approximations can be avoided altogether by means of the POINTER function, which will locate all real roots of any equation within a selected interval. The use of this function has the added advantage of providing a global overlay on the graphical representation of the problem under consideration. 

As the learner s semantic knowledge about the command device is limited, the consequence of these two characteristics of the learning of a command device is that the learner is unable to build a representation of the problem in terms of the command device operators. In other words, the learner could have built a problem space, but it is quite probable that this problem space cannot be searched through using the device operators. Two cases may happen (i) the learner has been given a worked out example, and this example is similar to the problem he (she) has to solve (ii) there is no available worked out example, but the source domain in which the problem is defined is very familiar to the learner. 

If no similar worked out example is available, and if the problem to be solved using the device is an easy problem in the source domain, the only way for the beginner learner to understand (i.e. to build a representation of) the problem is to rely on the knowledge of the source domain, hence to think about the problem solution in terms of the means (i.e. the operators) that are available in the source domain. 

There exist two ways to combine the numerical accuracy requirement with the desire of a simple nearly diabatic representation of the problem. One solution consists of performing a unitary transformation on the set of n eigenvectors i/ of... 

Conitzer Sandholm [31] propose automated mechanism design, in which a computational method is used to design mechanisms with respect to highly-enumerative description of the function space and agent type space. The challenge in this automated MD approach is to develop structured representations of the problem to constrain the input size to the optimization. However, automated MD cannot solve the wider issues presented by the second two problems because it is only applicable to direct revelation mechanisms and because it continues to ignore computational considerations in the formulation of the problem. 

In this case it is not necessary to do a graphical representation of the problem statement, but we need a mathematical formulation. The objective should be to maximize company profits, where we can define the profits (U) as follows Profits = input — output. Therefore, we can write U = PQ — CQ. 

One of the main objectives of using conceptual modeling is providing a relatively easily tmderstandable representation of a problem. Several conceptual and information modeling techniques are usually applied to obtain a comprehensive representation of the problem. The choice of techniques and modeling concepts depends upon objectives of the information modeling application. In the framework of supply chain configuration, several objectives can be identified ... 

In addition to the multi-criteria representation of the problem, relationships between projects, companies, and products should be taken into account. In such assessments, premium cost and maximal insured value can be found using Multi-Objective Decision-Making (MODM) methods and solving as a multi-criteria optimization problem (Figure 5.2) the same criteria can be reused for insurance portfolio optimization, and in the case of discrete alternatives (premium cost or insured value), the Multi-Attribute Decision-Making (MADM) approach can be used. Different risk assessment approaches can be adapted for MCDM for example, product lifecycle can be presented in detail and/or insured accidents can be presented implicitly along with business opportunities and other benefits. 

To build a model, important factors that act on the system must be included and unimportant factors that only make the model harder to build, understand, and solve should be omitted. For a continuous model, a set of equations that describe the behavior of a system as a continuous function of time t are written. Models use statistical approximations for systems that cannot be modeled using precise mathematical equations. While building a model it must be taken to ensure that it remains a valid representation of the problem. In order to get this purpose, a scientific model necessarily embodies elements of two conflicting attributes-realism and simplicity [24]. 

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