CLOSED-CURVE

It is sometimes very usefiil to look at a trajectory such as the synnnetric or antisynnnetric stretch of figure Al.2.5 and figure A1.2.6 not in the physical spatial coordinates (r. . r y), but in the phase space of Hamiltonian mechanics [16, 29], which in addition to the coordinates (r. . r ) also has as additional coordinates the set of conjugate momenta. . pj. ). In phase space, a one-diniensional trajectory such as the aiitisymmetric stretch again appears as a one-diniensional curve, but now the curve closes on itself Such a trajectory is referred to in nonlinear dynamics as a periodic orbit [29]. One says that the aihiamionic nonnal modes of Moser and Weinstein are stable periodic orbits. 

We will now look at how different types of wave functions behave when the O-H bond is stretched. The basis set used in all cases is the aug-cc-pVTZ, and the reference curve is taken as the [8, 8J-CASSCF result, which is slightly larger than a full-valence Cl. As mentioned in Section 4.6, this allows a correct dissociation, and since all the valence electrons are correlated, it will generate a curve close to the full Cl limit. The bond dissociation energy calculated at this level is 122.1 kcaPmol, which is comparable to the experimental value of 125.9 kcal/mol. 

The 1000 A column did not show any resolution between 312 nm and 57 nm particle sizes. Shown in Fig.2 are the calibration curves for the 2000 A and 3000 A columns and for their combination. The 57 nm particle standard appears to have been erroneously characterized by the supplier. This was subsequently confirmed by electron microscopy. The 2000 X column exhibited a sharp upturn in its calibration curve close to the exclusion limit. It is to be noted that while data points corresponding to 312 and 275 nm diameter particles appear on individual column calibration curves, they are not indicated for the calibration curve of the combination. This is because these larger diameter particles were completely retained in the packed colimms, generating no detector response. The percentage recovery for these particles from individual columns was considerably less than 100 resulting in their complete retention when the columns were combined in series. 

It follows that from the slope of the linear section in the polarization curve close to the equilibrium potential, we can determine the exchange CD of the overall reaction. 

Figure 5A shows experimentally derived profiles of pH vs rate for reactions in H2O and D2O [30, 50, 71]. The magnitude of the apparent isotope effect (ratio of rate constants in H2O and D2O) is 4.4 and the profiles appear to support the possibility that a proton is transferred from (Mg -bound) water molecules. However, careful analysis led us to conclude that a metal ion binds directly to the 5 -oxygen. Since the concentration of the deproto-nated 2 -oxygen in H2O should be higher than that in D2O at a fixed pH, we must take into account this difference in pKa, namely ApKa (=pKa °-pKa ), when we analyze the solvent isotope effect of D2O [30, 50, 68, 71]. We can estimate the pKa in D2O from the pKa in H2O using the linear relationship shown in Fig. 5B [30, 68, 73-75]. If the pKa for a Mg -bound water molecule in H2O is 11.4, the ApKa is calculated to be 0.65 (solid line in Fig. 5B). Then, the pKa in D2O should be 12.0. Demonstrating the absence of an intrinsic isotope effect (kH2o/kD20=l)> the resultant theoretical curves closely fit the experimental data, with an approximate 4-fold difference in...

The third oxide used for physical decolorizing is neodymium oxide. Its absorption curve closely compliments an average mixture of ferrous and ferric oxides especially with the strong absorption band at 589 nm. Neodymium oxide is also stable against any state of oxidation change in the furnace. Neodymium is exceptionally good as a decolorizer for potassium silicate and lead glasses. If the redox balance is not quite correct for the... 

Data in Figure 6 show the effect of varying the pH and sodium chloride concentration on emulsion capacity of peanut protein isolate. Shifting the pH to levels above or below the isoelectric point improved emulsion capacity of peanut protein isolate in O.IM or 0.2M NaCl. Similar trends were noted when distilled water was used as the continuous phase (data not.shown). At the 0.5M NaCl concentration, however, little difference was noted in emulsion capacity at pH 3, 4, or 5 appreciable increases occurred when the pH was raised to 6 and above. At the highest salt concentration (1.OM NaCl), a gradual increase in emulsion capacity occurred when the pH was increased from 3 to 10. An overall suppression in emulsion capacity occurred as salt concentration increased except at pH 5 and 6. These emulsion-capacity curves closely resemble the protein-solubility curves of peanut protein shown in Figure 7... 

Figure 10 shows the performance of the large Du Pont still for the full year 1959. The production curve follows the solar radiation curve closely. The efficiency, although erratic, tends to increase with increased solar radiation. The production curve for bay 14 is also shown in Figure 10 and includes estimated values for the earlier months of the year, based on leakage rates derived from Figure 9. 

Thalifasine (516), C40H46N2O9, [a] [67.9° (c 0.80, MeOH), is the last of the six 12 —8 ether-linked aporphine-benzylisoquinoline dimers (the others being 302,304,473, 514, and 515) isolated from Thalictrum faberi. The UV base shift of the alkaloid and a NMR study of its O, O-diacetate suggested the indicated location of the hydroxy substituents. The CD curve, closely resembling that of thalifaberine (302), indicated the same configuration (545). 

The argument / varies in the interval -n to +n, the parameters xb x2,..., x5 are the values of the different features of one object, so that one object vector is represented by /(/). Curves close to each other represent similar objects. 

As a result of this enhancement, when the spectral absorption characteristics of the four Rhodonines are plotted on one graph, the resulting family of curves closely resembles [Figure 5.5.9-1] from Mees James. 

The BOVB method has been successfully tested for its ability to reproduce dissociation energies and/or dissociation energy curves, close to the results (or estimated ones) of full Cl or to other highly accurate calculations performed with the same basis sets. A variety of two-electron and odd-electron bonds, including difficult test cases as F2, FH, and F2 (38,42), and the H3M-C1 series (M = C, Si, Ge, Sn, Pb) (39,43,44) were investigated. 

To obtain geometries, 10-orbital 10-electron complete active space self-consistent field (CASSCF) [82-84] calculations were performed with the GAMESS-UK program [6], The occupied orbital order in an SCF for flat benzene is n,2c,2n. In the bent molecule, there is no clear distinction between a- and tt-orbitals and we want to include all the tt-orbitals in the CAS-space. Thus, 10 orbitals in the active space are required. Obviously, the 5 structure VB wavefunction would have been a preferable choice to use in the geometry optimisation. However, at that time, the VB gradients were not yet available. The energies of the VBSCF at the CASSCF geometries followed the CASSCF curve closely. 

Figure 7.4. Total, elastic, and viscous stress-strain curves for collagen fibers from rat tail tendon. The total stress-strain curve (open boxes) was obtained by collecting all the initial, instantaneous, force measurements at increasing time intervals and then dividing by the initial cross-sectional area. The elastic stress-strain curve (closed diamonds) was obtained by collecting all the force measurements at equilibrium and then dividing by the initial cross-sectional area. The viscous component curve (closed squares) was obtained as the difference between the total and the elastic stresses. Error bars represent one standard deviation of the mean.

Figure 7.7. Total, elastic, and viscous stress-strain curves for uncrosslinked self-assembled type I collagen fibers.Total (open squares), elastic (filled diamonds), and viscous (filled squares) stress-strain curves for self-assembled uncrosslinked collagen fibers obtained from incremental stress-strain measurements at a strain rate of 10%/min. The fibers were tested immediately after manufacture and were not aged at room temperature. Error bars represent one standard deviation of the mean value for total and viscous stress components. Standard deviations for the elastic stress components are similar to those shown for the total stress but are omitted to present a clearer plot. The straight line for the elastic stress-strain curve closely overlaps the line for the viscous stress-strain curve. Note that the viscous stress-strain curve is above the elastic curve suggesting that viscous sliding is the predominant energy absorbing mechanism for uncrosslinked collagen fibers.

Hydrogen-Absorption Isotherms. The isotherms for the 25 weight % uranium alloy constitute a family of curves closely resembling each other. Seven of the 13 isotherms which were measured are plotted in Figure 3. Isotherms intermediate between each adjacent pair were omitted to reduce the complexity of the plot. The isotherms at 572° C. (not shown) and at 601° C. cross only two phase boundaries, because they are below the eutectic temperature. 

Figure 27illustrates some electrostatic field change measurements which follow the theoretical curve closely. 

So that an azeotrope with acetone does not form, the alcohol used must have a high enough boiling point. This requirement is reliably established only if vapor-liquid equilibrium data for at least two, preferably three, of the members of the series with acetone are known. The Pierotti-Deal-Derr method (4) (discussed later) or the Tassios-Van Winkle method (5) can be used in this case. In the latter method a log-log plot of y°i vs. P°i should yield a straight line. Figure 1 presents results for n-alco-hols and benzene from the isobaric (760 mm Hg) data of Wehe and Coates (6). Reliable infinite dilution activity coefficients are established for the other n-alcohols from data for at least two, and preferably three, of them. These y° values are used with equations like those of Van Laar or Wilson (7) to generate activity coefficients at intermediate compositions and to check for an existing azeotrope or a difficult separation (x-y curve close to the 45° line). 

Both males and females carry horns, which are long, slightly curved and elliptical. The inside of the curve, close to the base, is ridged. They are usually an almost even, dark brown to black in colour, though the horns of a young animal can be paler and show more colour variation (Fig. 7.2). The horns are mostly hollow with a solid tip, but are thicker than ox horn. 

Nichols pointed out that the degree of overlap of two distribution curves can be determined by locating the tube corresponding to the intersection of the distribution curves (arrows in Figure 23-8). The area under the intersected portions of the distribution curves is a quantitative estimate of the degree of nonseparation. The area under any portion of a distribution curve can be determined from statistical tables, since the shapes of the curves closely approximate the gaussian distribution. 

Figure 13 Summary of efforts to calibrate 6 B variations in calcite as a function of the pH of precipitation. Results are replotted from Sanyal et al. (1995, 1996, 2000, 2001). The shape of the dashed curves is that predicted by the pH control on the of B(OH)J. The position of these curves was adjusted to fit the O. universa and G. sacculifer data, highlighting the nearly constant boron isotopic offset between these two species of planktonic foraminifera. The upper curve closely approximates the calculated S B of B(OH)jT (see Figure 12). Note that the O. universa and G. sacculifer data plotted include both cultured foraminifera and core-top samples.

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