Single valuedness

It was shown by Baer and Alijah that the single-valuedness of W in the region surrounded by ris guaranteed if D(T) is diagonal [26]. Next, since D(T). just like... 

Eqs. (22) ascribe to each point T a certain direction, in which it will be traversed by the 6-points. If the external field of force does not depend explicitly on t (and this is the case we are considering), then these directions of propagation in the T-space are the same for all times. Therefore they can bo used to make up (lrA -1) fixed 6-paths. Because of the single-valuedness of the dynamical equations (disregarding singular points) each T-point is traversed by only one 6-path. 

The conditions of normalizability and single-valuedness require that the spectrum of (5.4.4) is discrete. It is given explicitly by... 

Mead and Truhlar [52] introduced an elegant way of incorporating the geometric phase effect, namely the vector potential approach. In this method, the real electronic wave function 4>(a), where a is any internal angular coordinate describing the motion around the Cl, is multiplied by a complex phase factor c(a) to ensure the single-valuedness of the new complex electronic wave function ... 

Inasmuch as we are going to interpret the square of the absolute value of a wave function as having the physical significance of a probability distribution function, it is not unreasonable that the wave function be required to possess certain properties, such as single-valuedness, necessary in order that this interpretation be possible and unambiguous. It has been found that a satisfactory wave mechanics can be constructed on the basis of the following auxiliary postulates regarding the nature of wave functions ... 

C. Conical Intersections and the Non-Single-Valuedness of Real Adiabatic Electronic Wave Functions... 

In the absence of such an intersection, the no-geometric-phase (NGP) boundary condition associated with the single-valuedness of xad(R) is instead... 

The quantitative definition of temperature is completed by establishing a partially arbitrary functional relationship between the temperature t and a suitably selected property P of the thermometer. The thermometric property P and those other properties of the thermometer which are held constant in the operation of measurement must determine the thermodynamic state of the thermometer. The care necessary in choosing the variables to be held constant is illustrated by the fact that p and v are not always sufficient to determine the thermodynamic states of a pure system. Thus, for example, in water at temperatures between 0°C and 8°C, the dielectric constant may have two different values at the same values of p and v. The relationship between t and P, t = t P), though partially arbitrary, must be such that t P) is a continuous, monotonic, and single-valued function of P. Single-valuedness is required so that a set of systems with the same value of P remain in equilibrium on contact with each other, i.e., there is only one value of t for each P monotonicity... 

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