Double-valued function

Thus Uk is a double-valued function of e, as depicted in Fig. 1. Strictly speaking, then, the eigenvalues of tr2A2 do not determine those of Ai, and consequently Ai cannot be determined from A2 alone. 

Figure 1. An eigenvalue of Ai as a (double-valued) function of the corresponding eigenvalue of tT2A2.

As we pointed out above, metastability is manifest as hysteresis in a sorption isotherm (like the one plotted in Fig. 5.9). Metastabihty involves a range of finite width As around the true transition point over which for the same T and fj, p (sz) is a double-valued function. To distinguish the metastable from the thermodynamically (i.e., globally) stable phase one needs to compare w for the two states pertaining to different brandies of the sorption isotherm at the same p, and [see Eq. (1.68)]. The branch having lowest uj is the globally stable phase the other one is only metastable. In Fig. 5.9 we plot only data for thermodynamically stable phases identified according to this... 

Using the replacement x = /exp(—/ ) the total wave-function can be rewritten in terms of double-valued functions as TJ = -h In terms of x ... 

The time dependence of the adiabatic electronic populations can be calculated either by using the S matrix [Eq. (7)] or by defining suitable adiabatic projection operators. The S matrix is a double-valued function of the coordinates and possesses a branch point at the conical intersection. This problem is circumvented by using adiabatic projectors in the diabatic electronic representation... 

The amount of the ester sulfonates, besides the mono- and disalt of the a-sulfo fatty acid, can be calculated by two titrations, one in the acid and one in the basic range. In the basic range both sulfonates and carbocylate functionalities are negatively charged and titrated with the cationic surfactant hyamine. In acid medium the RCOOH group is protonated and no longer available for the titration. Since hyamine-methylene blue (acid conditions) titrates only sulfonate and hyamine-phenol red (basic conditions) determines both sulfonates and carbo-cylates, substraction of the titration value with phenol red from the double value of the titration with methylene blue yields only the a-sulfo fatty acid ester. This is the only species of the three which has merely the sulfonate function [106]. 

For small asymmetries, the superconducting state is homogeneous and the order parameter preserves the space symmetries. For most of the systems of interest the number conservation should be implemented by solving equations for the gap function and the densities of species self-consistently. In such a scheme the physical quantities are single valued functions of the asymmetry and temperature, contrary to the double valued results obtained in the non-conserving schemes. 

Equation (10) is an empirical relationship, which undoubtedly can be improved upon as more data become available. [A theoretical relationship between p and pp has also recently been derived (Walukiewicz et al., 1982)]. The solution for the true electron mobility p as a function of the apparent electron mobility R0/p0 is shown in Fig. 2. It is seen that some of the solutions are double valued. Usually, the correct choice can be decided by... 

Figure 1.8. The top left sphere shows the positive (shaded) and negative (unshaded) regions for the real-valued function 2 - The top right sphere shows the pure real (solid) and pure imaginary (dashed) meridian for the function 72,2- The bottom picture shows the zero points (double-dashed) as well as the pure real (soUd) and pure imaginary (dashed) meridians of 12,1 There are colored versions of these pictures available on the internet. See, for instance, [Re].

Fig. 9.5 Computed average velocities (proportional to volumetric flow rate) as a function of pressure drop. Length, 50 in Vi,z 5.66 in/s. Curve A, steady state curve B, adiabatic high inlet temperature curve C, adiabatic low inlet temperature. Note the double-valued flow rates at a given pressure rise in the adiabatic operation and the maximum pressure rise at finite flow-rate values. [Reprinted by permission from R. E. Colwell and K. R. Nicholls, The Screw Extroder, Ind. Eng. Chem., 51, 841-843 (1959).]...

For a van der Waals gas determine the Joule Thomson coefficient and inversion temperature Ti in terms of a, b, P, T, and Cp. Make a sketch of versus T note and comment on the double-valued nature of the function. 

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