"... In PAGE 68: ...f the kidney-shaped region displayed in Figure 3.21. Using the new, spectral method5 and a nite di erence approach6, we compare the area lost over a time t = 0:0125 with the exact answer, 0:0125 = 0:0785398 (see [51]). From Table1 , we see that the new, spectral method is adequate for nding solutions to within a 1% error. As we shall see in the next chapter, even more accurate results are practical using the transform methods 5A direct evaluation of the Fourier summations was carried out.... In PAGE 96: ...e consider the motion by mean curvature of a collapsing sphere with initial radius 0.4. Using the new, spectral discretization of the MBO-method the volume lost over a time t = 0:03 was compared to the exact answer, 0:16 , for several . The results for a number of experiments are given in Table1 1, below. Error Conv.... In PAGE 105: ...he exact answer, 0.044508, which was found by integration of (5.34). The results for a number of experiments are given in Table1 2, below. Error Conv.... ..."

"... In PAGE 6: ...idney-shaped region displayed in Fig. 9. Using the new, spectral method and a nite di erence implementation3, we compare the area lost over a time t = 0:0125 with the exact answer, 0:0125 = 0:0785398 (see [10]). From Table1 , we see that the new, spectral implementation is adequate for nd- ing solutions to within a 1% error. The nite di erence approach, however, becomes impractical when accurate solutions are sought (see Table 2).... ..."

"... In PAGE 7: ... Basically, Table 1 is reconsidered under the form of Table2 where, for completeness, the unity column has been added 1 .... In PAGE 10: ... This special feature of S-SAR will be illustrated later, in the application section. It is now clear that once expanded, observing its first column, the determinant (17) generates the searched full solution of the basic SAR problem of Table2 with minimization of errors included and independent of the orthogonalization order. Remarkably, apart from being conceptually new through considering the spectral (orthogonal) expansion of the input data space (of both activity and descriptors) through the system (16), the present method also has the computational advantage of being simpler than the classical standard way of treating SAR problem previously exposed.... ..."

"... In PAGE 18: ...8a) is almost as good as the CFE multigrid.In Table3 , the spectral radii are presented for solving (4.2).... In PAGE 19: ... Surprisingly, one smoothing is enough to ensure the convergence of nonnested multigrids when our new intergrid transfer operator eIk is in use. Columns 2 and 5 of Table3 show that Brenner apos;s method is better (consistent with Table 2). But Columns 3 and 6 of Table 3 imply that our method is better.... In PAGE 19: ... Columns 2 and 5 of Table 3 show that Brenner apos;s method is better (consistent with Table 2). But Columns 3 and 6 of Table3 imply that our method is better. It tells us that eIk is more stable than Ik k?1.... ..."

"... In PAGE 14: ...1 The 8 Bands For computing IR uxes due to water vapor, carbon dioxide, and ozone, we divide the spectrum into eight bands. Table1 shows the spectral ranges of these bands, together with the absorbers involved in each band and the parameterization methods used to compute the transmittance in each band. The transmittance parameterizations are discussed in Section 4.... In PAGE 19: ... The transmittances due to water vapor continuum absorption in Bands 3 through 6 are computed using a one-parameter scaling approach. The applications of these parameterizations to the di erent spectral bands and absorbers are summarized in Table1 . The code allows \HIGH quot; and \LOW quot; options to be speci ed depending on the desired accuracy in the middle atmosphere.... In PAGE 23: ...Table1 , in Bands 1, 2, and 7 water vapor line absorption can be computed using either this method or the transmittance tables described in the next subsection; in all other bands we use only the k-distribution method. In Band 3, the 15 m band, we use a somewhat more complicated procedure described in Section 4.... In PAGE 24: ... 4.3 One-Parameter Scaling For Water Vapor Continuum Absorption As shown in Table1 , water vapor continuum absorption is included in Bands 3 through 6. The water vapor continuum absorption coe cient, kc , depends on the water vapor partial... In PAGE 27: ... T = 1 2 + R B 0 1( ) 0 2( )d R B d : (49) If the overall shapes of the absorption curves due to both absorbers are uncorrelated with each other and with B , the second term on the right-hand side of (49) can be neglected and the total transmittance becomes T = 1 2: (50) Overlapping of absorption in individual bands are shown in Table1 . As shown in Chou et al.... In PAGE 27: ... Second, the CO2 absorption coe cients di er by several orders of magnitude between the band center and the wings (see the top panel of Figure 3). Rather than trying to parameterize the correlation e ect or the variations in CO2 absorption, we simply divide the band into three sub-bands (see Figure 3 and Table1 ) and then combine the parameterized transmittances of the sub-bands into a single band transmittance. This transmittance is then used in the usual way to solve the transfer equations for the entire band.... In PAGE 49: ...Table1 0: Same as Table 9, except for the sub-arctic winter atmosphere. Spectral Band (cm?1) F# sfc F quot; top 0 { 340line-by-line 40.... ..."

"... In PAGE 9: ... This has resulted in 114 new spectra and 90 new redshift measurements which we present here. The journal of obser- vations and the new redshifts are given in Table4 and we present the spectra in Appendix C. Notes on some individ- ual spectra are given in Section 5.... In PAGE 9: ...3 below. Note that three sources are presented (\EXTRAS quot; in Table4 ) that are not in our nal sample. These were part of an earlier version of the sample and are included here to provide a published reference to their redshifts.... ..."