### Table 2: Effect of analytical terminal propagation (ATP) on placement of PEKO benchmarks. In the default mode, during global placement all the cells are assigned to the center of the blocks they belong to at each level of partitioning. For analytical terminal propagation, at each level of partitioning, a global quadratic minimum solution is computed. This solution satisfies the block boundary constraints and center of gravity constraints. This illegal locations of cells is used for terminal propagation. This effects the placement quality significantly.

2003

"... In PAGE 3: ... This may affect the terminal propagation and hence the final placement quality. In our experiments, we were able to significantly improve the per- formance of Capo on those benchmarks (see Table2 ) by combining Capo with a simple SOR-based analytical placer along the lines of [25]. This placer is called after every round of min-cut partitioning in Capo and finds locations of cells that minimize a quadratic wire- length objective, subject to region constraints (a cell must be within the top-down block where it is assigned by recursive partitioning) and center-of-gravity constraints (weighted locations of cells in a given block must average to the geometric center of the block).... In PAGE 3: ... A sample analytical placement is shown in Figure 3. As the results in Table2 indicate, this hybrid placer achieves much better wirelengths than Capo. 4.... ..."

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### Table 2: Scattering of an incident plane wave on a circular cylinder of radius a

"... In PAGE 6: ... 7 Numerical results A matrix free iterative solver has been implemented by utilizing the two-dimensional version of the high-frequency integrator described in the preceding sections in conjunction with the GMRES algo- rithm [15]. Table2 shows results produced by means of this two-dimensional solver on a 1:5 GHz PC, applied to a circular cylinder of radius a. Errors were computed by comparison with an exact solution for the integral equation, and de ned as Z @ j exact slow (r) slow(r)j2 ds(r) 1=2 .... ..."

### Table 1. This table shows the calculation time and precision with which our algorithm performs. This table was generated using a 2.4 GHz Intel r XeonTMprocessor. The error percentages are based on comparisons of results obtained in the range where 0:8 gapless using the island method [4]. The exception is for n = 1, where we have an analytical solution (16), we calculate the error based on the results obtained through the known equation. It should be noted that the percent error inherent in the island method for the simple scoring system described by Eq. (1) is 0.5%.

"... In PAGE 10: ... This process iterates until converges to the solution for . 5 Results for More Complex Scoring Systems Table1 summarizes the performance of the computer program outlined in sec- tion 4. For each value of the integer n, a combination of and is chosen that leads to a gapped of approximately 0:8 gapless.... ..."

### Table 5: Initial data (pressure p, density , velocity v) for two common relativistic blast wave test problems. The decay of the initial discontinuity leads to a shock wave (velocity vshock, compression ratio shock) and the formation of a dense shell (velocity vshell, time{dependent width wshell) both propagating to the right. The gas is assumed to be ideal with an adiabatic index = 5=3.

"... In PAGE 31: ...Table5 ) has been considered in 1D, 2D and 3D, respectively. Methods are described in Sects.... In PAGE 32: ...Table5 ) has been considered in 1D, 2D and 3D, respectively. Methods are described in Sects.... In PAGE 34: ...density distribution for the relativistic blast wave Problem 1 de ned in Table5 . The nal frame of the movie also shows the analytical solution (blue lines).... In PAGE 35: ...Table5 ) has been considered. Methods are described in Sects.... In PAGE 37: ...density distribution for the relativistic blast wave Problem 2 de ned in Table5 . The nal frame of the movie also shows the analytical solution (blue lines).... ..."

### Table 2 Results for the variant GLS-FLS-S4 using the circular list strategy.

"... In PAGE 6: ... GLS-FLS-S4 produced the best results when combined with the strategy while GLS-FLS-S5 and GLS-FLS-S3 also improved their performance. Results for GLS-FLS-S4 using using the circular list strategy are reported in Table2 . As we can see in this table, GLS-FLS-S4 achieves a stunning average excess of only 1.... In PAGE 6: ...4. Another result worth noting in Table2 is that the best known solutions were found in each and every run of GLS-FLS-S4 for eleven of the instances (see rows with mean excess equal to 0). The average user time for the variant was a reasonable 4.... In PAGE 7: ... Generally speaking, if the average behaviour of the CALMA methods was well documented then further conclusions could be drawn. As mentioned above, GLS-FLS-S4 has a very small variance with regard to solution quality (see Table2 ) and that could be a significant advantage over other heuristic methods which frequently, they are very sensitive to the random starting point chosen in each run. Finally, Tabu Searches (TS) developed by the CALMA project members are not very competitive, the main reason being the use of best improvement local search which given the large number of variables is a very expensive option for these problems.... ..."

### Table 1 Estimates of the values of mk for which the circular self-similar solution bifurcates into a noncircular solution k Estimate

"... In PAGE 3: ...re nonradial self-similar focusing solutions possessing k-fold symmetry, i.e., the symmetry of cos.k /. Numerical studies show that for each k 3 the bifurcation point m D mk is unique so that the Graveleau solutions are linearly stable with respect to perturbations with k-fold symmetry for m gt;mk and unstable for mk gt;m gt;1. Moreover, the mk are ordered 1 gt;m3 gt;m4 gt; gt;mk gt; amp;1: Crude estimates for the first four mk are given in Table1 (see also Table 4). The results in [3] give no information about stability with respect to perturbations with wave number k D 2.... In PAGE 11: ...nd 29 (c). This flow is not self-similar. The dashes indicate that for this values of k and m the solution tends to the circular case (Graveleau solution). This result serves as another validation, since the solution becomes unstable for the values of m reported in Appendix B (see also Table1 ). Note that for m D 1:3 and k D 4, the aspect ratio I becomes closer to the circular case (D 1) because we are near the bifurcation.... ..."

### Table 1 Results for the coastally-trapped waves in a circular basin

"... In PAGE 14: ...efinement for both a constant and an adaptive resolution are illustrated in Fig. 6. The vorticity adaption cri- terion (Popinet, 2003) was used to follow the waves as they travelled around the coastline. Table1 summarises the results for Gerris and reproduces the results obtained using FVCOM (Chen et al., 2003) and ROMS (Shchepetkin and McWilliams, 2005).... In PAGE 15: ...75 km). The results are given at the end of Table1 and are close to those obtained on a mesh with a constant resolution equal to that of the coarser half of the domain (18.75 km), which confirms that no significant spurious reflections occur at refinement boundaries.... ..."

### Table 5: Average field-free propagation velocities v0 of the planar wave front in different solution mixtures. The number in parenthesis is the number of experiments done.

2004

"... In PAGE 64: ...Data given in Table5 show that the propagation velocity for a given excitability determined by [H2SO4] = 0.15 M and [NaBrO3] = 0.... In PAGE 67: ...alculated from Eq. 7 at 25 oC and extreme dilution in aqueous solution, i.e. 0.0007041 cm2 V- 1s-1, and using v0 given in Table5 we obtain theoretical EA ranges for solution mixtures, they are I: 1.... ..."

### Table 1. Refraction indices of alcaline{earth uoride crystals and Schott IR glasses(a)

1995

"... In PAGE 1: ...aterials and in Sect. 4 we draw our conclusions. 2. The selection of achromatic pairs The refraction indices of IR glasses and alkaline{earth uoride crystals at standard laser wavelengths are listed in Table1 . A convenient method to preselect a com- bination of materials which is achromatic over a given 1 lt; 2 lt; 3 wavelength range is that described by Herzberger amp; Salzberg (1962, cf.... In PAGE 3: ... 2. The wavelength dependent `Abbe number apos; = [n( i) ? 1]=[n( i?2) ? n( i+2)] is computed at the wavelengths i (i=3,5) of Table1 . A pair is achromatic if the ratio between the `Abbe numbers apos; of the crown ( C) and the int ( F) does not vary with , and the best combinations are those with the largest C and C= F .... ..."

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### Table 1: Relative errors in L2 norm of the re ected and transmitted waves for di erent rational approximants.

2002

"... In PAGE 7: ...=3. For the propagator Pj, we use the [(q 1)=q] Pad e approximant [14]. For simplicity, we use a xed reference refractive index n = 3:25. In Table1 , we list the results for m =... ..."

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