Solution representation

We begin with some simple problems to illustrate the basic ideas. The simplest is the flow induced by rotation of a sphere of radius a with angular velocity f2 in an unbounded, quiescent fluid. Although we could formulate this problem in dimensionless terms by using ila as a characteristic velocity (where il = f2 ) and the sphere radius as a characteristic length scale, it is more convenient because the solution representation was carried out in terms of dimensional variables to simply solve it in dimensional form. 

When the mole fraction of species A is small compared to 1, we obtain the dilute solution representation for the diffusive flux... 

Generate initial individual population Each solution individual is usually generated randomly based on pie-specified solution representation and population size. 

At least the graphical solution display requires a cheap solution representation not only at the integration points and the computational grid. Our global solution representation and evaluation is done by means of two quite different Hermite interpolation variants. We found that doing first the interpolation in time then the interpolation in space gives better results than vice versa. 

Fig. 9. Mutarotation in solution representation of the different species in equilibrium for D-glucose (see Table 2).

Percent-by-weight solution—representation in which the concentration of the solute is expressed as a percentage of the total weight of the solution. 

Ruas A, Moisy P, Simonin JP, Bernard O, Dufieche JF, Turq P (2005) Lanthanide salts solutions representation of osmotic coefficients within the binding mean spherical approximation. J Phys Chem B 109 5243-5248... 

Previous works on GVRP-MDMPDR are reported by Sombuntham and Kachitvichyanukul (2012) and Sombuntham (2010). All these works are completed using the algorithms based on GLNPSO, a variant of PSO with multiple social learning terms. They proposed three solution representations and decoding procedures, namely SDl, SD2, and SD3. 

According to the literatures, they found that on average, the computational results from SDl provide the best solution quality but take much longer computational time than those of SD2 and SD3. Considering the solution quality of these three solution representation and decoding procedures, they found that solution quality obtained from SDl and SD3 is comparable. Consequently, SD3 is used for decoding the data into the suitable format in this work. 

The solution representation SD3 is described here. The solution vector consists of dimension 2r + 3m, where r and m represent the number of requests and the number of vehicles, respectively. For each request, there is a corresponding pair of origin-destination which is used for generating pickup-delivery priority list. 

Figure 5 shows the isothermal data of Edwards (1962) for n-hexane and nitroethane. This system also exhibits positive deviations from Raoult s law however, these deviations are much larger than those shown in Figure 4. At 45°C the mixture shown in Figure 5 is only 15° above its critical solution temperature. Again, representation with the UNIQUAC equation is excellent.

The continuous line in Figure 16 shows results from fitting a single tie line in addition to the binary data. Only slight improvement is obtained in prediction of the two-phase region more important, however, prediction of solute distribution is improved. Incorporation of the single ternary tie line into the method of data reduction produces only a small loss of accuracy in the representation of VLE for the two binary systems. 

In many cases, the methods used to solve identification problems are based on an iterative minimization of some performance criterion measuring the dissimilarity between the experimental and the synthetic data (generated by the current estimate of the direct model). In our case, direct quantitative comparison of two Bscan images at the pixels level is a very difficult task and involves the solution of a very difficult optimization problem, which can be also ill-behaved. Moreover, it would lead to a tremendous amount of computational burden. Segmented Bscan images may be used as concentrated representations of the useful... 

Finally, we consider the complete molecular Hamiltonian which contains not only temis depending on the electron spin, but also temis depending on the nuclear spin / (see chapter 7 of [1]). This Hamiltonian conmiutes with the components of Pgiven in (equation Al.4,1). The diagonalization of the matrix representation of the complete molecular Hamiltonian proceeds as described in section Al.4,1.1. The theory of rotational synnnetry is an extensive subject and we have only scratched the surface here. A relatively new book, which is concemed with molecules, is by Zare [6] (see [7] for the solutions to all the problems in [6] and a list of the errors). This book describes, for example, the method for obtaining the fimctioiis ... 

FigureBl.7.2. Schematic representations of alternative ionization methods to El and PI (a) fast-atom bombardment in which a beam of keV atoms desorbs solute from a matrix (b) matrix-assisted laser desorption ionization and (c) electrospray ionization.

Figure Bl.20.9. Schematic representation of DLVO-type forces measured between two mica surfaces in aqueous solutions of KNO3 or KCl at various concentrations. The inset reveals the existence of oscillatory and monotonic structural forces, of which the latter clearly depend on the salt concentration. Reproduced with pennission from [94].

Solution of the scattering amplitude may then be detemrined from the asymptotic fomi of 4 (r jdirectly or from the integral representation... 

This decomposition into a longitudinal and a hansverse part, as will be discussed in Section III, plays a crucial role in going to a diabatic representation in which this singularity is completely removed. In addition, the presence of the first derivative gradient term W l Rx) Vr x (Rx) in Eq. (15), even for a nonsingular Wi i (Rx) (e.g., for avoided intersections), introduces numerical inefficiencies in the solution of that equation. 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Reactive atomic and molecular encounters at collision energies ranging from thermal to several kiloelectron volts (keV) are, at the fundamental level, described by the dynamics of the participating electrons and nuclei moving under the influence of their mutual interactions. Solutions of the time-dependent Schrodinger equation describe the details of such dynamics. The representation of such solutions provide the pictures that aid our understanding of atomic and molecular processes. 

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