Expression of the model

The expression of the models in different modes of representation occurred during the whole process and, as has been previously commented upon, exerted an essential role in the development of students knowledge. This was particularly relevant for those students who could understand the relevance of the choice of a given code and level of representation in order to better express the mental model previously produced. In several activities, students were asked to propose a concrete model for a specific system. This was shown to be essential for the development of students ideas because, from the concrete models, they could produce simulations of the chemical process and think about details related to the mechaiusm of the chemical reactions (such as the directions of the necessary collisions between the molecules, something that they had not studied before). 

The role of the teacher has been shown to be vital. As students had to express their models for the sub-micro level in a concrete mode of representation, the teacher could identify aspects of their concrete representation (related to the codes of representation or to the ideas they wished to represent) that students did not express verbally. She questioned them, checking what they had really thought at different levels. The concrete expression of the models also allowed students to think about... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

Fitting of model potentials to supermolecular interactions like in Eq. (3-1) has its disadvantages the calculations have to be repeated many times and a predefined analytical expression of the model potentials in terms of atomic parameters is required. These parameters are not easily transferred to other situations. The internal state of, e.g., system A depends on the presence of another system X, and this will be different near system Y. Furthermore, the chosen analytical form of the potential may give rise to problems and errors. For example, Hartree-Fock (HF) calculations fitted to a power series in 1/r suggest that the 1/r6 terms have to do with dispersion, which is not part of the HF energy. Finally, such empirical potentials are best for describing situations close to those to which the parameters are fitted. Whenever the situation is very different from that, the results will be doubtful. 

In the example above, a maximum point was found within the explored domain. This is, however, not often encountered. Most frequently, the response surface is either monotonous in the explored domain or describes saddle-shaped surfaces or ridge systems. In such cases, it is not easy to comprehend the shape of the response surface from the algebraic expression of the model. A transformation to... 

Section VIII). In this case, with = 1, Eq. (7-27) is reduced to the simplest expression of the model ... 

A second-order model with square terms can describe a variety of differently shaped response surfaces. It is rather difficult to comprehend how the surface is shaped by mere inspection of the algebraic expression of the model. A canonical analysis constitutes a mathematical transformation of the original model into a form which only contains quadratic terms ... 

A description of how the rotation of the coordinate system is effected is given below. For the moment it is sufficient to realize that the algebraic expression of the model is highly simplified in the zi..z system. As the squared variables z- in the model cannot be negative we shall see that the shape of the surface is determined by the sign and the magnitude of the coefficients. .. 

The following conclusions can be drawn directly form the mathematical expressions of the models An increase in the yield of enamine, y, can be expected by increasing the settings of all three variables. A decrease in the yield of byproduct, 72, can be expected by increasing Xj and Xj, and decreasing Xj. Fig. 12.21 shows the contour plots projected in the planes spanned by the experimental variable axes. These projections lead to the same conclusions. 

More particularly, Fig. 21 shows the selectivity lines of membranes consisting of an anion exchanger and different diluents. As the dielectric constant of the diluent rises, the selectivity coefficient drops. The same trend is seen in Fig. 22. This phenomenon may well be an expression of the model of Eisenmann [20], in which he showed that the potential of a liquid membrane is composed of two factors ... 

Anyone offering or considering a different qualitative model for donor-acceptor reactivity should express it using an empirical equation, as was done with Equation 2 for the covalent-electrostatic model. The 500 enthalpies compiled (9) should be fit to the proposed equation. If the quantitative expression of the model leads to a poor fit but reproduces the correct trends with parameters consistent with the model imposed then it can be concluded that the qualitative model is acceptable. If the quantitative expression does not reproduce the trends, then the qualitative model is not acceptable. If the new model requires... 

In appendix, a formal description of the model can be found, using the Express language, based on the STEP (STandard for the Exchange of Product model data) standardized approach (ISO 10303). 

In this section, we illustrate the applicability of the model to some important special cases, and summarize the relationship between aromaticity and chemical reactivity, expressed in the properties of transition states. 

The relationship between fiber and matrix moduH and fiber volume fraction for a unidirectional lamina loaded in the direction transverse to the fibers is not simple. A lower bound (1) is given by the expression of the series spring model. 

This situation would fit the thermal model as seen in Figure A-7b. It does not matter in this ease not all the elements of the model are known sinee all the elements above the lead temperature node are known for this first step. The thermal expression for the temperature rise above the measured lead temperature is... 

Local Thermodynamic Equilibrium (LTE). This LTE model is of historical importance only. The idea was that under ion bombardment a near-surface plasma is generated, in which the sputtered atoms are ionized [3.48]. The plasma should be under local equilibrium, so that the Saha-Eggert equation for determination of the ionization probability can be used. The important condition was the plasma temperature, and this could be determined from a knowledge of the concentration of one of the elements present. The theoretical background of the model is not applicable. The reason why it gives semi-quantitative results is that the exponential term of the Saha-Eggert equation also fits quantum-mechanical expressions. 

Assumption 2 Tlie form of the model is appropriate. Statistieally, tliis is expressed as E(e) = 0 for all data, where is tlie random eiTor. 

In the bundle-like nucleation, there are two possibilities one is the attachment of a whole molecule to the crystal substrate and the other is the attachment of part of a molecule to the substrate. For long molecules, the former situation is unfavorable because of the high free energy created on forming a thick nucleus. It would be reasonable to consider that each molecule is partially incorporated into the crystal substrate. This bundle-like nucleation can be expressed by the model shown in Fig. 14. 

The minimum number of postulates of the model of a desorption process with no explicit analytical expression of the heating schedule are required if the primary output data are treated according to Eqs. (10) and (12), viz. by numerical or graphical derivations and integrations of the recorded pressure data. After an adaptation of the analyzer, these operations can be performed by means of electrical circuits. The known temperature-time relationship (either in the form of an analytical function or established... 

The tissue-specific patterns of expression of the PPAR. isotypes suggested that these proteins have distinct physiological roles, and this was further supported when each was specifically disrupted in mouse gene knockout models. 

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