### Table 23: E ect of using constrained space interval extensions instead of natural interval extensions to determine ranges of mole fraction weighted averages in using the UNIQUAC equation with reference temperature.

"... In PAGE 13: ... For this comparison we used the reference temperature case for each of the three UNIQUAC examples. Results of this comparison are summarized in Table23 , which gives the total time required for each problem and interval extension, as well as the number of leaves that occur in the binary tree generated in the bisection process. The results clearly indicate that use of the constrained space interval extensions for mole fraction weighted averages results in a dramatic increase in computational e ciency.... In PAGE 14: ... (1997). The use of a constrained space interval extension for nding exact (within roundout) bounds on mole fraction weighted averages greatly improves the computational e ciency of the method, as shown is on Problems 6-8 ( Table23 ). We also considered two approaches for handling the temperature dependence of the activity coe cients, one in which the activity coe cient is eval- uated at a reference temperature and then treated as independent of temperature, and the other in which a fully temperature dependent activity coe cient model is used.... ..."

### Table 1: Problem 1: SRK, hydrogen sul de(1) and methane(2) at P = 40.53 bar and T = 190 K. Comparison of using natural interval extensions (F ), constrained space interval extensions (F CS), and constrained space plus monotonic interval extensions (F CSM).

1998

"... In PAGE 17: ...Newton uniqueness test described above. Thus, for example, in Table1 , for the z1 = 0.0115 feed, the x1 value found for the third root was actually x1 = [0.... In PAGE 18: ...and Gani, 1996), a code that in general we have found to be extremely reliable, incorrectly predicts that this mixture is stable. As indicated in Table1 , several other feed compositions were tested using the IN/GB approach, with correct results obtained in each case. It should be noted that the presence of multiple real volume roots in this problem does not present any di culty, since the solver simply nds enclosures of all roots for the given system.... In PAGE 18: ... Thus, nothing needs to be done to select the right volume roots (or compressibility factors). Also included in Table1 are the number of root inclusion tests performed in the computation and the total CPU time on a Sun Ultra 1/170 workstation. This is done for the case (F ) in which just the natural interval extensions are used, the case (F CS) in which the constrained space interval extension is used, and the case (F CSM) in which both the constrained space and monotonic interval extensions are used.... In PAGE 29: ... Table1 . Problem 1: SRK, hydrogen sul de(1) and methane(2) at P = 40.... ..."

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### Table 1: Problem 1: SRK, hydrogen sul de(1) and methane(2) at P = 40.53 bar and T = 190 K. Comparison of using natural interval extensions (F ), constrained space interval extensions (F CS), and constrained space plus monotonic interval extensions (F CSM).

"... In PAGE 9: ...c2 = 190.6 K, Pc2 = 46.0 bar, !2 = 0.008, and a binary interaction parameter k12 = 0.08. Several feed compositions were considered, as shown in Table1 , which also shows the roots (stationary points) found, and the value of the tangent plane distance D at each root. For the z1 = 0.... In PAGE 10: ... Michelsen apos;s algorithm, as implemented in LNGFLASH from the IVC-SEP package (Hytoft and Gani, 1996), a code that in general we have found to be extremely reliable, incorrectly predicts that this mixture is stable. As indicated in Table1 , several other feed compositions were tested using the interval Newton/generalized bisection approach, with correct results obtained in each case. It should be noted that the presence of multiple real volume roots in this problem does not present any di culty, since the solver simply nds all roots for the given system.... In PAGE 10: ... It should be noted that the presence of multiple real volume roots in this problem does not present any di culty, since the solver simply nds all roots for the given system. Also included in Table1 are the number of root inclusion tests performed in the computation and the total CPU time on a Sun Ultra 1/170 workstation. This is done for the case (F ) in which just the natural interval extensions are used, the case (F CS) in which the constrained space interval extension is used, and the case (F CSM) in which both the constrained space and monotonic interval extensions are used.... ..."

### Table 1: Problem 1: SRK, hydrogen sul de(1) and methane(2) at P = 40.53 bar and T = 190 K. Comparison of using natural interval extensions (F ), constrained space interval extensions (F CS), and constrained space plus monotonic interval extensions (F CSM). Molar volume roots v are given in cm3/mol.

"... In PAGE 9: ...c2 = 190.6 K, Pc2 = 46.0 bar, !2 = 0.008, and a binary interaction parameter k12 = 0.08. Several feed compositions were considered, as shown in Table1 , which also shows the roots (stationary points) found, and the value of the tangent plane distance D at each root. For the z1 = 0.... In PAGE 10: ... Michelsen apos;s algorithm, as implemented in LNGFLASH from the IVC-SEP package (Hytoft and Gani, 1996), a code that in general we have found to be extremely reliable, incorrectly predicts that this mixture is stable. As indicated in Table1 , several other feed compositions were tested using the interval Newton/generalized bisection approach, with correct results obtained in each case. It should be noted that the presence of multiple real volume roots in this problem does not present any di culty, since the solver simply nds all roots for the given system.... In PAGE 10: ... It should be noted that the presence of multiple real volume roots in this problem does not present any di culty, since the solver simply nds all roots for the given system. Also included in Table1 are the number of root inclusion tests performed in the computation and the total CPU time on a Sun Ultra 1/170 workstation. This is done for the case (F ) in which just the natural interval extensions are used, the case (F CS) in which the constrained space interval extension is used, and the case (F CSM) in which both the constrained space and monotonic interval extensions are used.... ..."

### Table 5: Newton on the Mor e-Cosnard nonlinear integral Equation with initial intervals in [?108; 0] (and the Hansen-Segupta apos;s operator) is e ective when near a solution. It is also interesting to stress the importance of box-consistency on the natural extension in this example to reduce the growth factor. Without it, the algorithm takes about 48 and 440 seconds instead of 27 and 61 for Newton for n = 80 and n = 160, since the distributed interval extension loses precision due to the dependency problem. Finally, it is interesting to compare Newton with traditional interval methods. HRB takes 0.34 seconds on n = 5 with 18 branchings, about 18 seconds for n = 10 with about 300 branchings, and does not return after more than an hour on n = 20. 6.2 Discretization of a Nonlinear Integral Equation This example comes from [23] and is also a standard benchmark for nonlinear equation solving . It consists in nding the root of the functions fk(x1; : : :; xm) (1 k m) de ned as xk +

1997

"... In PAGE 23: ... Once again, we observe that box-consistency over the natural extension is helpful when far from a solution while box-consistency on the Taylor extension is useful to terminate the search quickly. Table5 gives the result for the initial intervals of size [?108; 0], which shows that the algorithm continues to perform well in this case. Finally, Table 6 gives the results for the HRB algorithm on this problem.... ..."

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### Table 6: Effect of Low Temperature on the Growth and Nature of Original Experimental Plants

"... In PAGE 127: ... During summer the plant Duranta repens goldiana showed drying and burning of the plants. But in winter ( Table6 ) the plant started developing new shoots and leaves. All these results indicated that the best selected plant for the arid condition is Rhagodia spinescens followed by Furcraea gigantea.... In PAGE 130: ... Increase in size of leaves. Table6 : The effect of Low Temperature on the Growth and Nature of the Experimental Plant Name of plant Effects of High Temperature Rhagodia spinescens Excellent growth. Good, extensive growth of foliage gives the plant a bushy appearance.... ..."

### Table 4.6: Stability Margin 1= kTzwk1 The resulting behaviour of the control system is characterised in terms of the closed- loop norms in Table 4.6 and Table 4.7 on the next page.15 Even though the approach to evaluate the stability and the performance of a gain-scheduled control scheme in xed operating points is simplistic, it is naturally the next step before extensive simulation studies. Although the reserve in the stability margin is perhaps slightly too small at low and slightly too large at high manifold pressure, the stability properties are 15The notation in the captions of the tables is slightly abused, in that the transfer function Ted also includes the channel from the rst component of w1 to the error e. Furthermore, the components of the signals z and w pertaining to the EGR feedback control loop are excluded from the transfer function Tzw to be consistent with other norm calculations.

### Table 2. Procrustes matching parameters Slices Rotation X-Centroid Y-centroid X-Variance Y-Variance

1997

"... In PAGE 10: ... The method is not restricted to merely aligning adjacent pairwise sections. The results shown in Figure 5 and summarized in Table2 were obtained with an eye towards averaging of forms. Incorporating correspondence and outlier rejection in Procrustes averaging is a natural extension of our approach.... ..."

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### Table 4. Aggregate analysis of the project

2003

"... In PAGE 10: ...2% A natural extension of the subsystem results would be to aggregate them into corresponding project results. Table4 presents two worst-case scenarios and one average-case scenario. For the worst-case scenarios the minimum average efforts and maximum average efforts are totaled.... ..."

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