### Table 15. Summary: Delay Cost Impacts and Expansion

2007

"... In PAGE 48: ... Most of this accrues on Reach 4, followed by Reach 2 and 1. The effect of the expansion on the change in equilibrium between the base case without expansion in 2020 and that with an expansion was also evaluated ( Table15 ). The expansion results in reduced delay costs of $61 million (about $1.... ..."

### Table 4. Reductions and Expansions for WDC i

"... In PAGE 9: ... With these two-sided circuits (nets), cut-elimination is replaced by normalization, as given by reductions and expansions, as well as by some rewrites that may best be thought of as \permuting reductions quot;, as they are not \directed quot; in any natural way. The reduc- tions and expansions for the multiplicative fragment of our logic are given in (Blute et al 1992) and are shown in Table4 . The reductions and expansions for the exponentials ! and ? are given in Tables 5, 6, and 7.... ..."

### Table 1 and Figure 8). This is because the interfaces are implemented as C language macros which cannot take variable arguments. Macros are used for inline expansion of the interfaces. The inline expansion is used to reduce the overhead for event recording.

"... In PAGE 7: ... Four kinds of interfaces (macros) for thread creation are provided. Table1 lists interfaces for thread creation. Some of them will be explained.... ..."

### Table 2: Time to reduce relative residual norm to 10?5. The degree of multipole expansion is fixed at 7. All times are in seconds.

1997

"... In PAGE 10: ... The degree of multipole expansion is fixed at 7 and the parallel runtimes to reduce the residual norm by a factor of 10?5 are noted. These times are presented in Table2 . (The overall time was capped at 3600 seconds and therefore the one missing entry in the table.... ..."

Cited by 10

### Table 5: The e ects of full predicating on code expansion rate and speedup

"... In PAGE 24: ...4. Table5 shows the e ects of the threshold value on the... In PAGE 25: ...69 with the threshold value of 16. The performance doesn apos;t vary much with the threshold values used in Table5 . This result shows that we can reduce the code expansion rate to a reasonable range without harming the performance.... ..."

### Table 2. Comparison on the Reduced Model Sizes

2003

"... In PAGE 4: ... The LU factorization at one expansion point might be reused at another point not far apart as an approxi- mated Jacobian in the iteration. Finally, for the state-equation form of (7), we compare the worst-case reduced model sizes generated by three methods in Table2 , where each method is used to match the moments of or amp; up to kth order. In the table, the op- timal strategies outlined in previous section are used for the meth- od of [6][8], and NORM(mp) is the equivalent zeroth-order Input: 1.... ..."

Cited by 13

### Table 1. Performance of the baseline and experimental query expansion runs (retrospective evaluation)

in proximity

2005

"... In PAGE 4: ...f 0.2426, and MI has the AveP of 0.3872. First, the effect of using the exponential distance factor proposed by Gao et al. on ranking query expan- sion terms was tested (run MID_Exp in Table1 ). The average precision of MID_Exp (0.... In PAGE 5: ... Two runs were conducted: MID_lgd, using Equation 6 as the distance factor, and MID_lgd2, using Equation 7. Performance measures of these two runs are presented in Table1 . The average precision, and preci- sion at 10, 15, 20, 30 and 100 documents are signifi- cantly higher in MID_lgd2 than in MID_lgd.... In PAGE 5: ... df(x, y) = log2(2 + fr(x, y)/D(x, y)) (8) where fr(x, y) is the joint frequency in the relevant document set. The distance factor in Equation 8 (run MID_lgd3 in Table1 ) puts more emphasis on frequent words, thereby reducing the problem of overweighting low- frequency words. The performance of MID_lgd3 is somewhat higher than that of MID_lgd2.... In PAGE 5: ... The performance of MID_lgd3 is somewhat higher than that of MID_lgd2. To test the effect of the constant in the logarithm function, Equation 9 (run MID_lgd4 in Table1 ) was further proposed, setting the constant as 3. The performance of MID_lgd4 is however worse than that of MID_lgd3.... In PAGE 5: ... df(x, y) = log2(3 + fr(x, y)/D(x, y)) (9) Next, we evaluated the distance function without the logarithm format (Equation 10). df(x, y) = fr(x, y)/D(x, y) (10) The run MID_d5 was conducted, and its performance results are presented in Table1 . After continuous improvement of the distance factor, the best perform- ance was obtained by using Equation 10.... In PAGE 5: ... Therefore, we evaluated separately the effect on performance of joint frequency (Equation 11, Run MID_d6) and of distance as a linear function (Equation 12, Run MID_d7). df(x, y) = fr(x, y) (11) df(x, y) = 1/D(x, y) (12) As seen from Table1 , MID_d6 performs worse than MID_d5, suggesting that term proximity information has some positive effect on query expansion term selection. On the other hand, linear reduction of the MI score with the increase in the average distance between two terms (Equation 12, Run MID_d7) leads to worse performance than a combination of joint frequency and inverse average distance.... In PAGE 5: ... The 11-point precision- recall graphs of runs using different distance factors are presented in Figure 1. For comparison we also show in Table1 the results obtained by using Robert- son selection value (RSV) [27], which is a well-known QE term selection method demonstrating consistently high performance in TREC experiments. MID_d5 method does not perform better than RSV in these experimental settings.... ..."

### Table 1. Performance of the baseline and experimental query expansion runs (retrospective evaluation)

2005

"... In PAGE 5: ...f 0.2426, and MI has the AveP of 0.3872. First, the effect of using the exponential distance factor proposed by Gao et al. on ranking query expan- sion terms was tested (run MID_Exp in Table1 ). The average precision of MID_Exp (0.... In PAGE 6: ... Two runs were conducted: MID_lgd, using Equation 6 as the distance factor, and MID_lgd2, using Equation 7. Performance measures of these two runs are presented in Table1 . The average precision, and preci- sion at 10, 15, 20, 30 and 100 documents are signifi- cantly higher in MID_lgd2 than in MID_lgd.... In PAGE 6: ... df(x, y) = log2(2 + fr(x, y)/D(x, y)) (8) where fr(x, y) is the joint frequency in the relevant document set. The distance factor in Equation 8 (run MID_lgd3 in Table1 ) puts more emphasis on frequent words, thereby reducing the problem of overweighting low- frequency words. The performance of MID_lgd3 is somewhat higher than that of MID_lgd2.... In PAGE 6: ... The performance of MID_lgd3 is somewhat higher than that of MID_lgd2. To test the effect of the constant in the logarithm function, Equation 9 (run MID_lgd4 in Table1 ) was further proposed, setting the constant as 3. The performance of MID_lgd4 is however worse than that of MID_lgd3.... In PAGE 6: ... df(x, y) = log2(3 + fr(x, y)/D(x, y)) (9) Next, we evaluated the distance function without the logarithm format (Equation 10). df(x, y) = fr(x, y)/D(x, y) (10) The run MID_d5 was conducted, and its performance results are presented in Table1 . After continuous improvement of the distance factor, the best perform- ance was obtained by using Equation 10.... In PAGE 6: ... Therefore, we evaluated separately the effect on performance of joint frequency (Equation 11, Run MID_d6) and of distance as a linear function (Equation 12, Run MID_d7). df(x, y) = fr(x, y) (11) df(x, y) = 1/D(x, y) (12) As seen from Table1 , MID_d6 performs worse than MID_d5, suggesting that term proximity information has some positive effect on query expansion term selection. On the other hand, linear reduction of the MI score with the increase in the average distance between two terms (Equation 12, Run MID_d7) leads to worse performance than a combination of joint frequency and inverse average distance.... In PAGE 6: ... The 11-point precision- recall graphs of runs using different distance factors are presented in Figure 1. For comparison we also show in Table1 the results obtained by using Robert- son selection value (RSV) [27], which is a well-known QE term selection method demonstrating consistently high performance in TREC experiments. MID_d5 method does not perform better than RSV in these experimental settings.... ..."

### Table 2.2: Iteration values using eqn.(2.12), c = 2:8281 and = 10?3. for a required precision is therefore reduced, yielding a shorter execution time for the formation of the multipole expansion and all other p-dependent parts of the algorithm. (Note, however, that the multipole expansion is not actually evaluated as such; the multipole coe cients are used to calculate a p-term local expansion.)

1994

Cited by 10

### Table 2 displays the numbers m; c m; apos;m and apos;c m and it it clear that the overall number of trees in the expansion is reduced a great deal by opting for the canonical expansion (3.3) in place of (2.4), whether with explicit integrals or with quadrature formulae. The canonical expansion (3.3) has another important and most welcome charac- teristics: as we are about to prove, it is time-symmetric. A dynamical system

1998

"... In PAGE 12: ...Table2 : The cardinality of di erent sets of binary trees. m m apos;m c m apos;c m 0 1 1 1 1 1 1 0 1 0 2 2 1 1 1 3 5 2 2 1 4 14 4 3 1 5 42 8 6 1 6 132 21 11 3 7 429 52 23 4 8 1430 138 46 7 9 4862 362 98 11 10 16796 980 207 20 completeness, we sketch the proof of this classical result.... In PAGE 15: ...3) to obtain order 6. According to Table2 , this results in eight trees, except that (adding an extra root to emphasize that we are dealing with R t 0 H ( )d , rather than with... ..."