### 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 2. Symbols introduced in mean-field model.

"... In PAGE 9: ... Mean-field approximation We relegate details of the mean-field analysis to Appendix B and present the results here. Table2... ..."

### Table 1. Marginal probabilities of MFT compared with the exact results. The first two columns show that any single mean-field solution on its own results in a very poor approximation of the exact marginals.

1998

### Table 3: Percentage of classes scheduled using the different methods. The averages and highest and lowest values were obtained using 10 independent runs for simulated annealing (SA) and mean-field annealing (MFA). The expert system (ES) is deterministic so the results are from a single run. No preprocessor was used with the three methods.

1998

Cited by 35

### Table 3: Percentage of classes scheduled using the different methods. The averages and highest and lowest values were obtained using 10 independent runs for simulated annealing (SA) and mean-field annealing (MFA). The expert system (ES) is deterministic so the results are from a single run. No preprocessor was used with the three methods.

1998

### Table 1: C4BD approximation error of single node marginals for the fully connected graph C3BL and the 4 nearest neighbour grid with 9 nodes, with varying potential and coupling strengths B4CSpotBN CScoupB5. Three different variational methods are compared: MF/Tree derives a lower bound with mean field approximation for A8BV and tree-reweighted belief propagation for A8; MF/SDP derives a lower bound with the SDP relaxation used for A8; Tree/MF derives an upper bound using tree- reweighted belief propagation for A8BV and mean field for A8. SDP denotes the heuristic use of the dual parameters in the SDP relaxation, with no provable upper or lower bounds.

2004

"... In PAGE 7: ... To assess the accuracy of each approximation, we use the C4BD error, defined as BD D2 D2 CG D7BPBD CYD4AIB4CG BE BVB5 A0 CQ D4AIB4CG BE BVB5CY (36) where CQ D4AI denotes the estimated marginal. The results are shown in Table1 for the single node case, and in Table 2... ..."

Cited by 6

### Table 1: Marginal probabilities of MFT as compared to the exact results. The first two columns show that any single mean field solution on its own results in a very poor approximation of the exact marginals.

1999

Cited by 12

### Table 3. Results of a typical run of MFT with a random sample of evidence for the observable nodes in our demonstration network of figure 2. We have listed some switches with high failure probability. The mean-field results are compared with exact results obtained by using HUGIN (right-hand column). Switches 3 and 5 are probably already down at t D 0 or at least at t D 1. Switch 6 is probably slow at t D 2 and may be even down at t D 3. Mean-field Exact

1998

"... In PAGE 11: ...0067 the iteration at 30 different points typically resulted in about 10 different solutions, whereby according to the mixture weights typically only two or three of all solutions are dominant. Table3 shows a typical result, which is quite useful for root cause analysis. Occasionally we found less accurate results like those in table 4.... ..."

### Table 1: Our results for the inverse transition temperature c at di erent values of the parameter are listed. These CVM results are compared with Mean Field and Monte Carlo previous results.

"... In PAGE 6: ...n Fig. 2 we have plotted c as a function of . At su ciently low values of , that is lt; tr = 0:87 0:01, the nature of the transition changes over to a rst order behaviour which is strengthened as is lowered. In Table1 our results for c are compared with Monte Carlo and mean eld approxi- mation results obtained in [10]; Monte Carlo and CVM predictions are in good agreement. Table 1... ..."