Multiplicity modelling solutions

The multiplicity of solutions at the continuum level can be viewed as arising from a constitutive deficiency in the theory, reflecting the need to specify additional pieces of constitutive information through some kind of phenomenological modeling (see, for instance, Truskinovsky, 1987 Abeyaratne and Knowles, 1991). Here we take a different point of view and interpret the nonuniqueness as an indicator of essential interaction between macro and micro scales. 

In the models discussed here we have considered primarily as bifurcation parameters the affinity of reaction as measured by the parameter B in model (1) or the length l as in Section VI. The results have illustrated that when B or l increase, the multiplicity of solutions increases. This is not astonishing, as a variation in length is a simple way through which the interactions of the reaction cell with its environment can be increased or decreased. 

Another approach is the modify the MILP transshipment model P2 so as to have preferences among multiple global solutions of model P2 according to their potential of vertical heat transfer between the composite curves. Such an approach was proposed by Gundersen and Grossmann (1990) as a good heuristic and will be discussed in section 8.3.3. 

The basic idea in discriminating among multiple global solutions of the MILP transshipment model of Papoulias and Grossmann (1983) is to use as criterion the verticality of heat transfer,... 

Remark 3 Model P4 will provide good results only when the heat transfer coefficients are equal or close in values, since in this case the vertical heat transfer results in minimum heat transfer area. If however, the heat transfer coefficients are different, then nonvertical heat transfer can result in less heat transfer area. Therefore, for such cases the vertical MILP model P4 is not applicable since it will discriminate among multiple global solutions of P2 with the wrong criterion. 

After performing pharmacophore analysis on a set of compounds, typically the user will have to select the model(s) with biological and/or statistical relevance, often from multiple possible solutions and use for further research purposes. 

In the steady state case example, because the data were sparse and there were more unknowns than constraining equations, an additional restriction to the linear programming method was needed. This was done by utilizing a mixed integer-programming model such as in a stepwise multiple regression solution. In the multiple regression method, the identification problem was formulated as follows ... 

Another consequence of this spatial isomorphism is that spatial indeterminacy due to multiple models can be dramatically reduced. For example, the solution to the following problem is indeterminate because there are two spatial models consistent with the text (from Johnson-Laird, 1996) ... 

As an example of how this may be used, we return to the group contribution model for the octanol-water partition coefficient discussed above. As already shown, this model was quite good for monofunctional (i.e., only one nonalkyl group) solutes when applied to multiple functional solutes, the large deviations shown in Fig. 4a were found. The failure of the GCSKOW model is the result of strong proximity effects in multifunctional compounds. 

In this section, we attempt to extend our theory of thermoreversible gelation from pairwise association to the more general multiple association. As a model solution, we consider a mixture of associative molecules R A/ in a solvent. Molecules are distinguished by the number / of associative groups they bear, each group being capable... 

Marek M, Hlavacek V. Modelling of chemical reactors. VI. Heat and mass transfer in a porous catalyst particle on the multiplicity of solutions for the case of an exothermic zeroth-order reaction. Collection of Czechoslovak Chemical Communications 1968 33 506-517. 

When So = 2, this model reduces to the variable multiplicity model in which junctions of arbitrary multiplicity can coexist at the probability determined by the thermodynamic balance. In the case of micro-crystalline junctions, for instance, it is natural to assume that a minimum number Sq greater than 2 of the crystalline chains is required for a junction formation. This is because, the surface energy terms will prevent small-k units from being stable, leading to the existence of the critical multiplicity for the nucleation of the crystallites. Similarly, a minimum aggregation number is required for the stability of micelles formed by hydrophobes on water-soluble polymers. As we will see later, surfactants added to the solution cause complex interaction with hydrophobically modified polymers due to the existence of this minimum multiplicity. 

Almost every study in the multi-objective network design models uses an interactive approach to solve the problem. Multiple efficient solutions are generated and presented to the DM. The DM guides the procedure until a best compromise solution is reached. 

Multiple models often arise as possibilities for representing the same physical situation when the solution approach for solving an engineering... 

Schaefer et al. ( ) theoretically examined the stability of a coimtercurrent shaft furnace by treating a simple step change in heat generation rate and reported the multiplicity of solution depends on model parameters and boundary conditions. Mori and Muchi (83) treated the case of a first order reaction occurring in a catalytic moving bed and examined the reactor stability. 

It is necessary to explore the multiplicity of solutions of the model and the validity of extrapolation for predicting the maximum boiling point, although this last fact could not be a critical issue. 

The minimutn in the Ai singlet Hessian eigenvalue at 3.9ao arises because of an avoided crossing with a second Hartree-Fock state as indicated by the dotted line terminated by an arrow in Figure 10.3. This second RHF solution represents an excited (albeit unphysical) state of /4 symmetry. (For distances shorter than 4.0oo, the calculation of the excited RHF state becomes difficult and no plot has been attempted in this region.) Multiple RHF solutions close in energy are often found in regions where the Hartree-Fock wave function provides an inadequate description of the electronic system and where it is necessary to go beyond the Hartree-Fock model for a proper description of the electronic system. 

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