### Table 1. Mean Field Theories

"... In PAGE 5: ...solated atoms. We rst present the results of the Iben et al. model. Table1 gives the calculated ground state ionization potentials, , and the probability densities, 2, at the nucleus for a screened Coulomb potential with Z taking on values from 1 to 6 and Debye radius RD = 0:45 , which is the solar value at R=R = 0:06. For Z = 1, Debye-H... In PAGE 6: ...the rate reduction factors, FIKS, by which the bound state capture rate is reduced due to screening, FIKS = 2e = 2 0e 0; (3) where the subscript 0 indicates unscreened values. Thus, we see from Table1 that bound state screening reduces the total capture rate by a factor R = (wc + FIKSwb1)=(wc + wb) = 0:85; (4) or by 15% . Screening e ects on continuum electrons were studied by Bahcall amp; Moeller (1969), who integrated numerically the Schroedinger equation for continuum electrons.... In PAGE 6: ... For 7Be under solar conditions, screening corrections are small but larger than our calculational accuracy. Let the screening corrections for continuum electrons be represented by FBM = lt; 2 gt; = lt; 2 0 gt; : (5) Table1 gives values of FIKS and FBM for di erent nuclear charges Z; solar values at R=R = 0:06 were used for and RD. The total electron capture rate should be calculated using a density enhancement factor wIKSBM = FBMwc + FIKSwb1; (6) where we make the excellent approximation that screened excited bound states give a negligible contribution.... In PAGE 7: ... The rst order expansion of the potential gives = Zr e?r=RD Zr ? Z RD : (7) Thus the potential near the nucleus is a Coulomb potential plus an approximately constant correction. In statistical equilibrium, the constant change in the potential reduces the electron density at the nucleus by a Boltzmann factor, FS = exp(? Z=RD), and the density enhancement factor is given by wS = FS(wc + wb): (8) Table1 compares, in the last two rows, our numerical values obtained from the detailed quantum mechanical calculations summarized by Eq. (6), and the simple Salpeter-like formula, Eq.... ..."

### Table 1: Comparison of algorithms with linear and quadratic ap- proximations.

"... In PAGE 10: ... Each optimization used the same initial point. This led to the results shown in Table1 . Clearly, very simi- lar optimization results are obtained for these two designs in terms of the coding gain.... ..."

### Table 3.1 Number of linear-quadratic problems to be solved to evaluate the derivatives of j(p) and i(p).

### TABLE 6 Integration Dimension amp; Problem Solving Effectiveness: Linear and Quadratic Effects

### Table 4: Iteration count and CPU time for the de ation scheme constant linear quadratic

1997

"... In PAGE 23: ... All CPU times (in seconds) are for an SGI Onyx with su cient RAM to ensure that disk swapping is not required. Table4 shows the number of iterations and CPU time required to reduce the pres- sure residual for the rst time step by ve orders of magnitude for the de ation scheme using piecewise (discontinuous) constant, bilinear, and biquadratic coarse grid spaces. Our parallel production code [16] has been based upon the piecewise constant prolongation op- erator.... In PAGE 23: ... The higher-order coarse grid spaces were studied in [18] in an attempt to improve this scheme. Although the results of Table4 show a two-fold reduction in CPU time for the rst time step, these extensions yielded only a thirty percent reduction in subsequent steps, and would be even less e ective in lR3 due to the rapid increase in the dimension of the coarse grid problem. These two considerations motivated the present study.... ..."

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### Table 4: Iteration count and CPU time for the deflation scheme constant linear quadratic

in of

"... In PAGE 23: ... All CPU times (in seconds) are for an SGI Onyx with sufficient RAM to ensure that disk swapping is not required. Table4 shows the number of iterations and CPU time required to reduce the pres- sure residual for the first time step by five orders of magnitude for the deflation scheme using piecewise (discontinuous) constant, bilinear, and biquadratic coarse grid spaces. Our parallel production code [16] has been based upon the piecewise constant prolongation op- erator.... In PAGE 23: ... The higher-order coarse grid spaces were studied in [18] in an attempt to improve this scheme. Although the results of Table4 show a two-fold reduction in CPU time for the first time step, these extensions yielded only a thirty percent reduction in subsequent steps, and would be even less effective in lR3 due to the rapid increase in the dimension of the coarse grid problem. These two considerations motivated the present study.... ..."

### lable there exists a state-feedback controller , u(t) = ?Kx(t), such that the poles (eigenvalues) of the closed-loop system can be located arbitrarily. State-space theory for feedback design was introduced by Kalman in the early sixties [10].Many text books are now available on this approach, see for example [9]. One state-space design theory, which is especially well suited for multivariable feedback systems, is the so-called linear-quadratic (LQ) theory. In the LQ theory the problem is to nd a state- feedback control law which minimizes an integral quadratic per- formance measure of the form

### Table 2. Number of Iterations to Decrease Residual by 10?6, 2D, Square Domain AFIF Linear Quadratic J

1997

"... In PAGE 19: ... 4.2 2D tests In this section, we report on similar performance results for the model problem ? u + u = f in ; @u @n = 0 on @ : Table2 summarizes the number of iterations required to reduce the initial residual by a factor of 10?6 using the -algorithm in the case = [0; 1]2. For this particular problem, the number of iterations approaches a value of approximately 10, 20, and 30, for the linear, quadratic, and AFIF elements, respectively.... ..."

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### Table 8: kFk2 and cond(X) for the stabilization method. A comparison of Table 8 and Table 5 shows that the conditioning of the closed-loop system via the stabilization method is much better. It should be pointed out that in the case n = 20, the resulting closed-loop eigenvalues via the pole placement algorithm had no correct digits but those via the stabilization method had 7 valid digits. For further results in this direction, in particular for large sytems of several hundred states, see [12]. In this section we have demonstrated that for the problem of stabiliza- tion, the pole placement problem should not be considered as a substitute problem. A much better substitute (though not perfect) is the stabilization via the solution of a linear-quadratic control problem. In a similar way, one

1995

"... In PAGE 27: ... To demonstrate the superiority of this approach consider the stabilization algorithm proposed in [12] applied to one of the previous examples. Example 8 For Example 6 in the case of = 1, the result obtained from the stabilization algorithm proposed in [12] is given in Table8 . Observe that the stabilization method and the pole placement method are comparable in this case, since the eigenvalues of the closed-loop systems are both ?n; ?(n?... ..."

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