### TABLE 2: EDCF PARAMETERS USED FOR THREE ACS. High Medium Low AIFS [us] 34 52 63

2002

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### Table 2 indicates the overall worst-case complexities1 to establish arc consistency with algorithms AC3, AC3rm, AC2001 and AC3.2. It is also interesting to look at worst- case cumulated time complexities to seek successive supports for a given cn-value (C,X,a). Even if it has not been intro- duced earlier, it is easy to show that optimal algorithms admit a cumulated complexity in O(d). By observing Table 3, we do learn that AC3 and AC3rm are optimal when the tightness is low (i.e. c is O(1)), and that, unlike AC3, AC3rm is also optimal when the tightness is high (i.e. s is O(1)).

"... In PAGE 4: ...Time AC3 O(e + nd) O(d P C;X;a c(C;X;a) + P C;X;a s(C;X;a)) AC3rm O(ed) O(ed2 + P C;X;a c(C;X;a) s(C;X;a)) AC2001 O(ed) O(ed2) AC3.2 O(ed) O(ed2) Table2 : Worst-case complexities to establish AC. Tightness Any Low Medium High AC3 O(cd + s) O(d) O(d2) O(d2) AC3rm O(cs + d) O(d) O(d2) O(d) AC2001 O(d) O(d) O(d) O(d) AC3.... In PAGE 4: ..., 2004]. By taking into account Proposition 3 and Table2 , we ob- tain the results given in Table 4. It appears that, for the longest branch, when gt; d2, MAC3 and MAC3rm have a better worst-case time complexity than other MAC algo- rithms based on optimal AC algorithms since we know that, for any branch, due to incrementality, MAC3 and MAC3rm are O(ed3).... ..."

### Table 5: Medium distillates prices [$/m3]. Sub-regions Pr (Apr/2001) Scenario 1 Scenario 2 PA/AP 426.30 1,771.84 1,415.02 AM/RR/RO/AC 415.80 1,746.66 1,395.16 MA/PI

in A SPATIAL PRICE OLIGOPOLY MODEL FOR REFINED PETROLEUM PRODUCTS: AN APPLICATION TO A BRAZILIAN CASE.

### Table 1. Comparison of Threshold Parameter a with Theoretical Lower Bound Ncr H33360 1.86225 and Upper Bound GH20851H20851H20851fH20852H20852 [Eq. (8)] for Various Initial Profiles CA0H20849H20849rH20862r0H20850H20850H20850a

2000

"... In PAGE 2: ... The theoretical lower and upper bounds for the critical power can be written in terms of a as Ncr H33527 alb #a#aub H33527 GH20851fH20852 . (8) Inasmuch as, at present, there is no known analytic technique with which to calculate the critical power for these profiles, in Table1 we give numerical values of a as defined in Eq. (1) for various profiles in a bulk medium as determined from numerically integrating the NLSE.... In PAGE 3: ... (6). As shown in Table1 , from our numerical calculations using the criteria discussed above we find that the parameter a for the critical power for catastrophic collapse for various initial profiles is equal to the Townesian value. The reason for this behavior is that, unlike in bulk media, the walls prevent the shedding of energy and keep the energy localized in the transverse domain.... ..."

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