Scalar case

The transformation to this LP program is graphically depicted in Fig. 6 for the case when c is a scalar. For each data pair (x, y,) the term y, -represents two im + l)-dimensional hyperplanes, z =yi Lc dkiXi) and z = -y, + LckOkOti)- For the scalar case, these correspond... 

Property (ii) is also controlled by the behavior of Sg(0)Cg(0). In general, the diffusion matrix should have the property that it does not allow movement in the direction normal to the surface of the allowable region.100 Defining the surface unit normal vector by n(0 ), property (ii) will be satisfied if Sg(0 )Cg(0 )n(0 ) = 0, where 0 lies on the surface of the allowable region. This condition implies that (e 10 )n(0 ) = 0, which Girimaji (1992) has shown to be true for the single-scalar case. Thus, the FP model satisfies property (ii), but the user must provide the unknown conditional joint scalar dissipation rates that satisfy (e 0 )n(0+) = 0. 

Phenomenological quasiparticle model. Taking into account only the dominant contributions in (7), namely the quasiparticle contributions of the transverse gluons as well as the quark particle-excitations for Nj / 0, we arrive at the quasiparticle model [8], The dispersion relations can be even further simplified by their form at hard momenta, u2 h2 -rnf, where m.t gT are the asymptotic masses. With this approximation of the self-energies, the pressure reads in analogy to the scalar case... 

The general features of the Newton method are very well known. Nevertheless, it is perhaps worthwhile to offer a very brief review for the scalar case, which is finding a solution to F(y) = 0. The algorithm is stated as... 

The BEM methodology listed in Algorithm 11 will be applied again with the exception of the matrices dimensions. The BEM matrices assembly will be performed in a similar way as in the scalar case. Algorithms 14 and 15 show the new assembly methodology for the matrices and the integral calculation of the components, including Telles transformation for matrix G. 

In the scalar case (i.e., N = 1), wave solutions are easily constructed with the equilibrium diagram y(x) or Y(X). According to the above considerations, typical scalar problems are a binary nonreactive distillation process, a ternary reactive distillation process with a single chemical reaction, a reactive distillation process with Nc components and Nc - 2 chemical reactions, or a chromatographic reactor with Ns solutes... 

Fig. 5.5. Construction of wave solutions in the scalar case, (a) Constant pattern wave (b) spreading wave (c) combined wave solution.

Similarly to the scalar case (equation (13.65)), an arbitrary source field in the vector equation can be represented as the sum of point pulse sources ... 

As in the scalar case (equation (13.66)), we can conclude that, using Green s tensor G of the vector wave equation, one can find the solution to this equation with an arbitrary right-hand side F (r, t), as the convolution of the Green s tensor G with the function F (r, t), i.e.,... 

In the scalar case we obtained these conditions by analyzing (for the sake of simplicity) the asymptotic behavior of spherical waves. At the same time, as it has been demonstrated above, the same result could be obtained by analyzing the conditions required to ensure that the corresponding Kirchhoff integral goes to zero over a large sphere expanding to infinity. 

In full analogy with the electromagnetic case, we can apply the quasi-linear (QL) approximations introduced in Chapter 14 for acoustic and vector wavefield inversion. We begin our discussion of the basic principles of QL inversion with the simpler scalar case of acoustic waves. 

This equation is weakly singular if 0 < p < 1 and regular for p > 1. In the former case, we must give explicit proofs for existence and uniqueness of the solution. Hence, results that are very similar to the corresponding classical theorems of existence and uniqueness, known in the scalar case of first-order equations, are discussed subsequently [49]. 

It will ease our discussion considerably to deal with the scalar case first, so that we denote the particle state by z. Further, we neglect any dependence on the continuous phase variables and consider this in Section 7.1.3. 

Before closing this section, we briefly consider the case in which there are several environmental variables forming a vector with realizations described by the vector y. The relevant equations for the scalar case are readily modified by replacing y with y, and Y with Y, and replacing the partial derivatives with respect to y in Eqs. (7.3.12), (7.3.14), (7.3.15), and (7.3.17) with the partial divergence... 

The matrix exponential function is herein defined in terms of an infinite series as in the scalar case... 

For notational simplicity we restrict the presentation to the scalar case. 

Figure 4. Quantization and probabilities of compositions, scalar case. In a) the composition profile c(x) (solid solution composition as a function of space at a given time) is represented. There is a sharp fi ont (corresponding compositions have zero probability) and a continuous evolution, wherein the spatial spreading of a specific composition 2, lying between compositions I and 3 is represented its probability p is proportional to f (c), i.e. the difference of the neighbouring velocities. In b) and c) the application of this rule is given for a continuous isotherm the envelope f between the extreme points is shown. The probability distribution is given in c). In d) and e) the same method is applied for a discontinuous isotherm (isotherm is given in d) and probability distribution in e)) Guy, 1993, wifii permission from Eur. J. Mineral.

In our problem the probability aspects are related to the preceding spatial quantization (refer again to Fig. 4 and 5, and to Fig. 6). The probability to find a specific rock composition Co on the field is proportional to the ratio of the surface covered by Co to the total sur ce of outcrop of transformed rocks (Fig. 6). The spatial spreading of a composition (Le. the proportion of the composition with respect to the whole of the transformed rocks) is proportional to the difference of the velocities of the compositions before and after it. This statement allows to compute the probability density p of co in the scalar case p is proportional to f "(cq). In the case of systems, p is proportional to VXk-Hc (scalar product with eigen vector it is the co-ordinate of VXk along rk). 

As in the scalar case, integral and series representations for the translation coefficients can be obtained by using the integral representations for the vector spherical wave functions. First we consider the case of regular vector spherical wave functions. Using the integral representation (B.26), the relation r = 0 + T" ) and the vector spherical wave expansion... 

We note that the coefficients ai(-) and fei( ) can be expressed in terms of the coefficients a( ) by making use of the recurrence relations for the associated Legendre functions. As in the scalar case, the integration with respect to the azimuthal angle a gives... 

J. G. Fikioris, P.C. Waterman, Multiple scattering of waves, II, Hole corrections in the scalar case, J. Math. Phys. 5, 1413 (1964)... 

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