Double-valued

Type B thermocouples (Table 11.56) offer distinct advantages of improved stability, increased mechanical strength, and higher possible operating temperatures. They have the unique advantage that the reference junction potential is almost immaterial, as long as it is between 0°C and 40°C. Type B is virtually useless below 50°C because it exhibits a double-value ambiguity from 0°C to 42°C. 

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 example, the ir-eiectron energy change in the dimerization of benzyl is taken as a twofold difference in the rr-electron energies of benzene and benzyl. With the SCF data, a double value of the valence state ionization potential of carbon [I in eq. (25)] has to be added to this difference. The entries of Table XII show that in all equilibria considered, a dimer is favored. 

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. 

This algorithm works in most circnmstances but it is slow, and the end condition is not unique. The 1- field narrows sharply near the optimum position, so that at certain intensity levels the intensity contonr is double-valued in that is, it is shaped like the Greek letter rather than a simple parabola. With some starting conditions the peak will be found on one of the two left-hand points of the rather than the correct right-hand point. An algorithm been developed by Loxley et al. which avoids this problem. Fewster s condition is... 

The presence of the pores adds two parameters - the pore volume fraction and the pore radius. The predicted Rp Increases as the pore radius decreases suggesting a preference tor small pore packings. However, for a small pore radius of 1.0 pm a single value of the separation factor corresponds to two values of the particle diameter (13). Such double-valued behavior Is of course undesirable In an analytic technique. 

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.

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... 

Even though this Nernst equation does not look like the one in the preceding example, Box 14-2 shows that the numerical value of E is unchanged. The squared term in the reaction quotient cancels the doubled value of n in front of the log term. 

Group theory predicts that time-odd interactions for double-valued representations must transform as the symmetrized direct square. For E and E" representations one has ... 

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).]...

Monodisperse melts appear to exhibit a plateau region in the stress vs shear rate flow curve [51,62,65]. The capillary flow behavior actually closely resembles the oscillatory shear behavior in the sense that the flow curve essentially overlaps on the absolute value of complex modulus G vs the oscillation frequency (0 [62]. Thus.it appears that the transition-like capillary flow behavior of highly entangled monodisperse melts reflects constitutive bulk properties of the melts and is not interfacial in origin. It remains to be explored whether this plateau indeed manifests a real constitutive instability, i.e., whether it is double-valued. 

All data are for the pseudocells and compositions staled. The true cells have doubled values of b and can be in varying degrees disordered a and c must often be differently defined. The structure of C-S-H(I) has little more than two-dimensional order, and that of C-S H(II) has less than full three-dimensional order. 

It may be concluded that, when the external and internal motions may be considered separely, the local full NRG is isomorphic to the direct product of the restricted NRG by the symmetry point group of the molecule in its most symmetric configuration (20). This conclusion holds in the case of synunetric linear molecules, in which the reference frame is an axis, except that double valued groups have to be used, but it does not hold in the case of centro-symmetric molecules in whidi the reference frame is a single atom. 

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. 

To our knowledge, the first PES suitable for dynamical studies of the title reaction was proposed in 1976 by Murrell and Sorbie [58] from spectroscopic data, but the first dynamical studies have been carried out in 1980 by Schinke and Lester in a PES fitted from ab initio data [32]. To better reproduce the experimental rate constant, those authors propose two different potential energy surfaces named SLl and SL3. Subsequently, Murrell, Carter, Mills and Guest have published a double-valued PES [57] to correctly account for the different crossings between the diabatic surfaces and Murrell and Carter proposed a simpler single-valued PES that approximates it [59] this last one is known as MC PES. 

Recently, mainly based on the very accurate ab initio results of Partridge and Schwenke [66] complemented with other data [68,69] and on a careful description of the long range interactions between the different dissociation channels [70], Brandao and Rio presented another double-valued PES for this system [43]. This BR PES displays a small van der Waals minimum and a small saddle point under the dissociation limit for the C2v approach of the O ( D) atom to the H2 molecule and a very small barrier (< 0.4 kJmoP ) to collinear addition ( 27+ surface) in agreement with the findings of Walch and Harding [68]. 

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