Case equality

Figure Bl.13.8. Schematic illustration of (a) an antiphase doublet, (b) an in-phase doublet and (c) a differentially broadened doublet. The splitting between the two lines is in each case equal to J, the indirect spin-spin coupling constant.

In this case equals about half the wavelength used for the... 

In an ideal gas, the only forces are those due to interactions between the gas and the container and it can be shown that the virial in this case equals —3PV. This result can also be obtained directly from PV = NkeT. 

In the author s group, much lower-melting benzenesulfonate, tosylate, or octyl-sulfate ionic liquids have recently been obtained in combination with imidazolium ions. These systems have been successfully applied as catalyst media for the biphasic, Rh-catalyzed hydroformylation of 1-octene [14]. The catalyst activities obtained with these systems were in all cases equal to or even higher than those found with the commonly used [BMIM][PF6]. Taking into account the much lower costs of the ionic medium, the better hydrolysis stability, and the wider disposal options relating to, for example, an octylsulfate ionic liquid in comparison to [BMIM][PF6], there is no real reason to center future hydroformylation research around hexafluorophosphate ionic liquids. 

The value for DR is converted to Log (DR-1) value, which in this case equals 0.48. The pA2 is calculated with the equation... 

Fig. 70.—Weight fractions of various finite species and of gel in a simple trifunctional condensation as a function of a, which in this case equals the extent of reaction p. Curves have been calculated from Eqs. (36) and (45). ...

Therefore, the surface integral in Equation (1.78), as in the previous case, equals zero and correspondingly inside the volume we have... 

Dissolve 0-01 mol of the phenohc ether in 10 ml. of warm chloroform, and also (separately) 0-01 mol of picric acid plus 5 per cent, excess (0-241 g.) in 10 ml. of chloroform. Stir the picric acid solution and pour in the solution of the phenohc ether. Set the mixture aside in a 100 mh beaker and allow it to crystallise. Recrystallise the picrate from the minimum volume of chloroform. In most cases equally satisfactory results may be obtained by conducting the preparation in rectified spirit (95 per cent. CjHgOH). The m.p. should be determined immediately after recrystallisation. It must be pointed out, however, that the picrates of aromatic ethers suffer from the disadvantage of being comparatively unstable and may undergo decomposition during recrystaUisation. 

The singularity of A can also be ascertained by inspection of the determinant lAI which in this case equals zero. As a result, the characteristic equation has the form of a degenerated quadratic ... 

The last term of the characteristic equation is always equal to the determinant of A, which in this case equals zero. Hence we obtain ... 

The ratio of the time constants, Xr/Xm, which for this case equals (k Xm) will determine whether C is significantly different from Cm. When this ratio is less than 1.0 the measurement lag will be important. If Xj/Xm >10, then Cm = Cr and the measurement dynamics become unimportant. 

Equations (2.7) and (2.8) can be added directly because the number of electrons produced equals the number of electrons consumed. If this is not the case, equalization must be done as the preliminary step. After multiplication with four and realizing that H+ + OH- —> H20, the final equation showing the stoichiometry of the total redox reaction is as follows ... 

PrPres Prion protein that is partially resistant against proteinase K digest (in most cases equal to PrPSc or PrP27-30)... 

So in some cases equal might be defined to mean identical, but this is by no means general. Suppose you re dining at a restaurant. When your waiter comes up to take your order, you point at the next table and say, I ll have what she is having. If the waiter interprets your request in terms of object identity rather than your intended equality or similarity, he need not expect a tip from either of you 1... 

This has been particularly evident with the me thyIcarhamate and phosphoramidothioate insecticides where in almost all cases derivatization has resulted in improved mammalian toxicity. Further, in many cases, equal or greater insecticidal activity has also been observed although in some cases insecticidal activity may be substantially reduced. Improvement in mammalian toxicity has been attributed to the delayed factor provided by the derivatizing group, giving the animal the opportunity to metabolize the compound to non-toxic products. 

As the late Carl Sagan often stated, Extraordinary claims require extraordinary evidence. Was this particular Artist of the Century —or any other one— worth all this labored thought and scholarly travail In this case, equally extraordinary are the claims that 1) Marcel Duchamp did systematically employ hermetic references and alchemical iconography in creating his artworks and/or that 2) Marcel Duchamp was never influenced by Alchemy. Both positions are, therefore, obligated to produce their extraordinary evidence. And here you have just been presented with concrete proof, heaps of it, repeatedly substantiating the first of these two extraordinary art-historical arguments. 

We further address the influence of the pulse shape on the Sq S2 S process dynamics. For this purpose, simple pulse profiles are used in Ref. [42], shown in Figure 9.18, which can be modeled by analytical functions. More specifically, the profiles shown in the left panel of Figure 9.18 correspond to pulses composed from one, as pulse A, or three, as pulses B, C, and D, monochromatic pulses, as defined in Eqs (9.69) and (9.70). However, in all cases, the intensity and phase of each monochromatic pulse used in Eq. (9.69) is equal to 4TW/cm and zero, respectively. Moreover, the carrier frequencies of the monochromatic pulses are also the same in all cases, equal to 4.81 eV different carrier frequencies considered for profiles shown in the right part of Figure 9.18, are discussed below. The other parameters in Eqs. (9.69) and (9.70) regarding the pulses A, B, C, and D are as follows for pulse (A) with = 1, we have = 70 fs and = 24 fs for... 

To decay to about one-sixteenth the original amount would take 5 half-lives, which in this case equals 120 years. 

Simultaneous Optimization of Density and Temperature. Although near-baseline resolution was achieved for all eight sample components via the optimization of a single variable (density), as illustrated in Figure 1, a better (or in rare cases, equal) result will always be obtained if all variables of interest are optimized. The window diagram method is now considered for the simultaneous optimization of density and temperature for the separation of the eight component sample of Table VI, to provide a comparison with the SFC separation obtained with the density-only optimization (Figure 6). 

Scenario 1 For the symmetrical case (equal flow rates and viscosities), almost no penetration of the liquid layers into each other is achieved [68], Symmetrical fluid trajectories along the two symmetry planes of the mixing channel result (see Figure 1.55). 

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