### Table 5: PRIME performance measures as a function of the type of calibration model. Perfor- mance of the normalized vs. unnormalized forms was essentially identical, normalized models were used herein.

1996

"... In PAGE 32: ... Note that a di erent gray scale mapping is used for each image. As seen in Table5 , the mean acquisition time did not vary signi cantly between model types. This is primarily due to the compute burden associated with managing the datacube hardware and with nding the nominal position of the laser pro le.... In PAGE 38: ... The size of processed imagery was adjusted to maximize the amount of range data, while maintaining frame-rate throughput for each calibration model. The results of these performance benchmarks is given in Table5 . These benchmarks were also used in the model selection process.... ..."

Cited by 7

### Tables like [3] list several hundred such identities. Since binomial coe cients satisfy many relations, the expression on the left-hand side may appear under numerous disguises, which makes it di cult to locate it in such tables (or to implement table lookup in a computer algebra system). However, a sort of normal form follows from the observation that in many identities with left-hand side Pk f(k), the function f(k) satis es

1994

### Tables like [3] list several hundred such identities. Since binomial coe cients satisfy many relations, the expression on the left-hand side may appear under numerous disguises, which makes it di cult to locate it in such tables (or to implement table lookup in a computer algebra system). However, a sort of normal form follows from the observation that in many identities with left-hand side Pk f(k), the function f(k) satis es

### Table 3. Identities involving functions and transformations. Identity Example 3.1:

"... In PAGE 7: ...f objects. For example, moments can be expressed as cumulants. These identities are illustrated here with examples. (X)(Y Z) = (XY Z) multiplication is associative (X)(Y Z) = (X)(Y )(Z) multiplication is associative X = X Expectation is a linear operator XY = X;Y + X Y de nition of of cumulants X;Y = XY ? X Y de nition of of cumulants X Y = g X; Y + n?1 f X e Y de nition of disjoint averages g X; Y = X Y ? n?1X Y de nition of disjoint averages General de nitions of these identities in terms of operations on lists are pre- sented in Table3 . The functions and transformations involved are composed of the basic elements of Tables 1 and 2.... In PAGE 12: ... Table 8. Standard Objects T ype Function fType fType fffW; Xg; fY gg; ffZggg A Q1 A2 Q2 Q3 W XY Z DA DA1 Q2 Q3 g W XY; Z M Q1 2 Q2 Q3 WXY Z K Q1 2 Q3 WX;Y Z A[M] Q1 A2 Q2 3 Q3 WX Y Z DA[M] DA1 2 Q2 Q3 g WXY ; Z A[K] Q1 A2 2 Q3 WX;Y Z DA[K] DA1 2 Q3 g WX;Y ; Z Objects of one form may be transformed into objects of another form using the identities of Table3 . The elementary transformations are shown in Figure 1 with lines.... ..."

### Table 1: Some multiple sum identities found byPSLQ

"... In PAGE 4: ...[13] some base-3 formulas were obtained, including the identity 2 = 2 27 1 X k=0 1 729 k quot; 243 (12k +1) 2 ; 405 (12k +2) 2 ; 81 (12k +4) 2 ; 27 (12k +5) 2 ; 72 (12k +6) 2 ; 9 (12k +7) 2 ; 9 (12k +8) 2 ; 5 (12k + 10) 2 + 1 (12k + 11) 2 # 5. Identi cation of Multiple Sum Constants A large number of results were recently found using PSLQ in the course of research on multiple sums, such as those shown in Table1 . After computing the numerical values of these constants, a PSLQ program was used to determine if a given constant satis ed an identity of a conjectured form.... In PAGE 4: ... Eventually,elegant proofs were found for many of these speci c and general results [6, 7]. Three examples of PSLQ results that were subsequently proven are given in Table1 . In the table, (t)= P 1 j=1 j ;t is the Riemann zeta function, and Li n (x)= P 1 j=1 x j j ;n denotes the polylogarithm function.... ..."

### Table 1: Synthesis of an example function.

2006

"... In PAGE 7: ... Consider the 3-variable reversible function specified by the permutation [1;0;3;2;5;7;4;6] in Boolean domain. The spectra for this function are shown in tabular form in the column labelled RM in Table1 . We want to select Toffoli gates to transform this specification into that of the identity (Table 1, column Id).... In PAGE 7: ... The spectra for this function are shown in tabular form in the column labelled RM in Table 1. We want to select Toffoli gates to transform this specification into that of the identity ( Table1 , column Id). The first row of the function specification does not match the first row of the RM spectra of the identity function.... In PAGE 7: ... This can be fixed by applying the NOT gate TOF(a). Application of this NOT gate from the output side transforms the specification into the one shown in Table1 , column S1. The first 5 rows in specification S1 match the first 5 rows of the RM spectra of the identity.... ..."

Cited by 2

### Table 1: Some multiple sum identities found by PSLQ In another application to mathematical number theory, PSLQ has been used to inves- tigate sums of the form

"... In PAGE 4: ...[13] some base-3 formulas were obtained, including the identity 2 = 2 27 1 X k=0 1 729k quot; 243 (12k + 1)2 ? 405 (12k + 2)2 ? 81 (12k + 4)2 ? 27 (12k + 5)2 ? 72 (12k + 6)2 ? 9 (12k + 7)2 ? 9 (12k + 8)2 ? 5 (12k + 10)2 + 1 (12k + 11)2 # 5. Identi cation of Multiple Sum Constants A large number of results were recently found using PSLQ in the course of research on multiple sums, such as those shown in Table1 . After computing the numerical values of these constants, a PSLQ program was used to determine if a given constant satis ed an identity of a conjectured form.... In PAGE 4: ... Eventually, elegant proofs were found for many of these speci c and general results [6, 7]. Three examples of PSLQ results that were subsequently proven are given in Table1 . In the table, (t) = P1 j=1 j?t is the Riemann zeta function, and Lin(x) = P1 j=1 xjj?n denotes the polylogarithm function.... ..."

### Table 1: Romanized form of the symbolic operators

"... In PAGE 4: ... This allows us to di erentiate between the function and, which is logical conjunction, and the function and which is the function which folds the conjunction over a list. The full list of conversions can be seen in Table1 . The parse scheme for identi ers is thus: PE[id] = new ParsedIdent(R(i)) where R romanizes the operator symbols (and leaves other functions alone).... ..."

### Table 8. Exponential Function Coefficients, 0-400 Meter Interval Removed Study Area Functional Form R-squared

2007

"... In PAGE 8: ...able 7. Exponential Function Coefficients ..................................................................... 24 Table8 .... In PAGE 35: ...8991 0% 10% 20% 30% 40% 50% 60% 70% 80% 400-800 800-1200 1200-1600 1600-2000 2000-2400 2400-3200 3200-4000 gt; 4000 Network Distance from Residence to Trail Access Sample 1 Sample 2 Sample 3 Sample 1 Trend Sample 2 Trend Sample 3 Trend Figure 8. Geographic Sub-samples, 0-400 Meter Interval Removed Removing the 0-400 meter interval from the graph results in more consistent functional forms across the three study areas (See Figure 8; Table8 . In this variation, the slopes for samples one and three are virtually identical, while the curve for sample two drops off more steeply.... ..."

### Table 6: polarisation values in bins of cos for the combination of all analyses. The errors are statistical only. The nal column shows the 2 for each bin of the combination. The combined data were tted to the functional form of the polarisation given in Eq. 3. In contrast to the case for hP i, PZ and its associated systematic uncertainty are dependent on the correlations introduced between di erent bins in cos . By combining the simulated data distributions for bins of cos re ected in cos = 0, which had identical acceptance e ects, the uncertainty in PZ due to simulation statistics was greatly reduced. Other systematic uncertainties in the polarisation for the di erent channels a ect PZ if

1995

"... In PAGE 25: ... The results of the di erent analyses were combined taking into account the correlations in each of the six bins. The results are shown in Table6 and in Fig. 15.... In PAGE 27: ... The polar angle dependence is displayed in Fig. 15 and in Table6 . The data have been combined with the published 1990 results [2].... ..."