### Table 4 Lower bound L, upper bound U, approximation A, and tree size T for random permutations on n elements.

"... In PAGE 25: ... In the third set of experiments we studied the quality of the approximation algo- rithm and the upper bound algorithm on random permutations. Results are given in Table4 , where dev denotes the standard deviation. Note that the maximum di erence between the upper and lower bound, which is what limits the range of optimal solu- tion, was at most 2 reversals for n up to 30, while the average di erence between the bounds was around 2.... ..."

### Table 2: Lower Bound for Approximation Error

"... In PAGE 23: ... However, this is in a large part due to the poor performance of the actual frequency weighted reduction algorithm rather than a de ciency of the proposed approaches. By studying the lower bound on the approximation error ( Table2 ) the advantage of the approaches we apos;ve presented are self-evident. They indicate the possible existence of a seventh order (P; )-admissible controller and give an initial starting point for parameter optimisation to nd such a controller.... ..."

### Table 21: Non-approximability of various levels of the Function Bounded NP Query Hierarchy. y Applies only to complete problems.

"... In PAGE 89: ...P query hierarchy. Though the correspondence is not exact ([Kre88, p. 492]; [CP91, p. 243]), there is a pattern of approximability and non-approximability (see Table21 ). This pattern may assume greater signi cance in the light of future discoveries of lower limits on approximability.... ..."

### Table 21: Non-approximability of various levels of the Function Bounded NP Query Hierarchy. y Applies only to complete problems.

"... In PAGE 82: ...P query hierarchy. Though the correspondence is not exact ([Kre88, p. 492]; [CP91, p. 243]), there is a pattern of approximability and non-approximability (see Table21 ). This pattern may assume greater signi cance in the light of future discoveries of lower limits on approximability.... ..."

### Table 3: Lower bound

"... In PAGE 9: ...7, an optimal drawing of a K9 has 36 crossings. Referring to Table3 , an optimal drawing has at least 27 non-internal edge crossings (Lemmas 4.... ..."

### Table 1: Lower Bounds, Heuristics and Optimal Values

1997

"... In PAGE 18: ... Then, arcs are dropped or added following a dichotomic scheme which, based on the sorting criterion, attempts to identify the most interesting arcs. Using a set of ten representative instances, Table1 illustrates the di#0Eculty of standard methods and state-of-the-art software packages to address these problems. All instances belong to the class of single O-D networks, since we are going to show results of the tabu search heuristic of Crainic, Gendreau and Farvolden #5B21#5D, which is tailored for these problems.... ..."

### Table 3b. Solution Statistics for Model 2 (Minimization)

1999

"... In PAGE 4: ...6 Table 2. Problem Statistics Model 1 Model 2 Pt Rows Cols 0/1 Vars Rows Cols 0/1 Vars 1 4398 4568 4568 4398 4568 170 2 4546 4738 4738 4546 4738 192 3 3030 3128 3128 3030 3128 98 4 2774 2921 2921 2774 2921 147 5 5732 5957 5957 5732 5957 225 6 5728 5978 5978 5728 5978 250 7 2538 2658 2658 2538 2658 120 8 3506 3695 3695 3506 3695 189 9 2616 2777 2777 2616 2777 161 10 1680 1758 1758 1680 1758 78 11 5628 5848 5848 5628 5848 220 12 3484 3644 3644 3484 3644 160 13 3700 3833 3833 3700 3833 133 14 4220 4436 4436 4220 4436 216 15 2234 2330 2330 2234 2330 96 16 3823 3949 3949 3823 3949 126 17 4222 4362 4362 4222 4362 140 18 2612 2747 2747 2612 2747 135 19 2400 2484 2484 2400 2484 84 20 2298 2406 2406 2298 2406 108 Table3 a. Solution Statistics for Model 1 (Maximization) Pt Initial First Heuristic Best Best LP Obj.... In PAGE 5: ...) list the elapsed time when the heuristic procedure is first called and the objective value corresponding to the feasible integer solution returned by the heuristic. For Table3 a, the columns Best LP Obj. and Best IP Obj.... In PAGE 5: ... report, respectively, the LP objective bound corresponding to the best node in the remaining branch-and-bound tree and the incumbent objective value corresponding to the best integer feasible solution upon termination of the solution process (10,000 CPU seconds). In Table3 b, the columns Optimal IP Obj., bb nodes, and Elapsed Time report, respectively, the optimal IP objective value, the total number of branch-and-bound tree nodes solved, and the total elapsed time for the solution process.... ..."

### Table 1 Variance Accounted for by Seven Sources of Variance for Religiousness

"... In PAGE 4: ... We intended that the results gen- eralize beyond the raters, participants, and years of measurement included in this study, so we treated the sources of variance as random effects. Approximately 95% of the variance in our measures could be attributed to substantive sources (see Table1 ). Roughly 66% of the variance could be attributed to consistent differences among the 148 participants (i.... ..."

### Table 5 Kolmogorov-Smirnov with Lilliefors significance correction

### Table 2: Objective values of the relaxations. A higher value means a better lower bound for the (unknown) optimal value.

2007

"... In PAGE 42: ... We used the Matlab built-in function fmincon to search for the minima from some different starting points. Table2 shows the results. Note that the third point local min point cost function value (0, -0.... In PAGE 42: ...0018 (0,0,0,0) 65.1556 Table2 : Local minima found by the Matlab function fmincon. is the global minimum.... In PAGE 42: ... To get an idea of the shape of the function, Figure 3 plots the function-values along a line from one local minimum to the global minimum. To the left is the values on the line from the first point in Table2 and on the right is the second point. Iterations Figure 4 shows the performance of the algorithm in two cases.... In PAGE 91: ...349 .0418 Table2 : Average execution times (s) for 100 random instances of each problem. For the large scale test we used the known rotation problem.... In PAGE 113: ... By adding the (ho- mogenization) variable xn+1, the problem can be transformed to the same form as in (6). Table 1 shows the execution times and Table2 displays the obtained estimates for different n. For the subgradient method, 10 iterations were run and in each iteration, the 15 smallest eigenvectors were computed for the approximation set S in (18).... ..."