Single-valued function

Figure B3.6.2. Local mterface position in a binary polymer blend. After averaging the interfacial profile over small lateral patches, the interface can be described by a single-valued function u r. (Monge representation). Thennal fluctuations of the local interface position are clearly visible. From Wemer et al [49].

Whereas a linear relation between flow stress and lattice-parameter change is obeyed for any single solute element in nickel, the change in yield stress for various solutes in nickel is not a single-valued function of the lattice parameter, but depends directly on the position of the solute in the Periodic Table... 

See Benderskii et al. [1993] for exceptions to this statement the period may be not a good parameter to use for parametrizing this family because the action may be not a single-valued function of the period however this difficulty is easily circumvented by parametrizing the solutions, for example, by their energy. 

A relatively simple approach suggests itself if the interfaces are known to be almost flat. In that case, the interface position can be described by a single-valued function z(x,y), where (A,y) are cartesian coordinates on a flat parallel reference plane. The functional (21) can be approximated by... 

Since the energies are single-valued functions of the volumes and entropies of the substance in the two states, and since the equations (9), (10), (11) give three relations between the four quantities vi, Si, v2, S2, it is evident that if one is fixed the others are determined, so that the two points lie on two definite curves on the model ... 

This notation admits of generalisation (Gibbs, 1876). The total energy of a homogeneous fluid is a continuous and single-valued function of the masses mi, m2, m3,. . mh of its constituents, of the total volume Y, and the total entropy S ... 

In general, for polymerization reactions, the heat generation rate is not a single-valued function of temperature, g(t), but also a function of monomer and catalyst concentrations, f(c). This is particularly important in high conversion reactions where a certain amount of peaking can be tolerated. 

When the relation y = f x) is such that there is only one value of y for each acceptable value of x, f(x) is said to be a single-valued function of x. Thus, if the function is defined for, say, x = jc , the vertical line x = x intercepts the curve at one and only one point, as shown in Fig. S. However, in many cases a given value of x determines two or more distinct values of y. 

In addition, Turner and Trimble defined a slip equation of state combination as the specification of mass flux, momentum flux, energy density, and energy flux as single-valued functions of the geometric parameters (area, equivalent diameter, roughness, etc.) at any z location, and of mass flux, pressure, and enthalpy,... 

For some polymers, like polystyrene or poly(methyl methacrylate), narrow standards of known molar mass and small polydispersity are commercially available, which can be used for calibration. Unfortunately, such standards are not available for all polymers and then the obtained true molar masses of a specific polymer might differ by a factor of two from the value obtained by calibration with, e.g., polystyrene [30] (see Section 9.1). This problem can be resolved by the so-called universal calibration, which is based on the finding that the retention volume of a polymer is a single-valued function of the hydrodynamic volume of the polymer, irrespective of its chemical nature and... 

When random variable X can be split into intervals over which the function has this desirable property. We can use this method to find the distribution of Y = X2 where X is a normal random variable with zero mean and unit variance... 

Maximum likelihood estimators also have another desirable property invariance. Let us denote the maximum likelihood estimator of the parameter 6 by maximum likelihood estimator of f(0) is /([Pg.904]

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. 

Although it has been shown that thermodynamic models which imply phase separations can create difficulties with uniqueness in solving the reaction equilibrium equations (10, 25, 24), there proved to be only one solution to equation (26) under the conditions studied. It is conceivable that more than one critical point could be found for some reacting mixtures at certain reaction extents (two critical points are indeed indicated in Figure 1 for some C02 - CO mixtures), in which case F(e) will not be a single-valued function. This possibility was not explored. 

Calibration of size-exclusion chromatography columns based on the finding that the retention volume of a molecular or particulate species is usually a single-valued function of an appropriate size parameter of this molecule or particle, irrespective of its chemical nature and structure. 

The brain is a thermodynamic system. The mind is a thermodynamic single valued function because it depends only on the states of the system. It means that mind is a function of anger, suspicion, greed. Mathematically... 

Separable solutions to (8.26) that are single-valued functions of are of the form... 

In the turbulent core it has been conventional to employ the velocity deficiency (B2) as a single-valued function of the relative position in the channel in order to correlate the velocity distribution outside the boundary flow. The velocity deficiency is defined by... 

The early application of volumetric data for hydrocarbons made use of the perfect gas laws. They were not sufficiently descriptive of the actual behavior to permit their widespread use at pressures in excess of several hundred pounds per square inch. The need for accurate metering aroused interest in the volumetric behavior of petroleum and its products at elevated pressures. Table II reviews references relating to the volumetric behavior of a number of components of petroleum and their mixtures. For many purposes the ratio of the actual volume to the volume of a perfect gas at the same pressure and temperature has been considered to be a single-valued function of the reduced pressure and temperature or of the pseudo-reduced (38) pressure and temperature. The proposals of Dodge (15), Lewis (12), and Brown (8) with their coworkers serve as examples of the nature of these correlations. The Beattie-Bridgeman (2) and Benedict (4) equations of state describe the volumetric behavior of many pure substances and their mixtures with an accuracy adequate (31) for most purposes. However, at pressures above 3000 pounds per square inch the accuracy of representation with existing constants leaves something to be desired. 

Figure 5-11 illustrates the results of an oxide interdiffusion experiment. Clearly, the transport coefficients are not single valued functions of composition. From the data, one concludes that for a given composition, the chemical diffusion coefficients depend both on time and location in the sample [G. Kutsche, H. Schmalzried (1990)]. Let us analyze this interdiffusion process in the ternary solid solution Co. O-Nq. O, which contains all the elements necessary for a phenomenological treatment of chemical transport in crystals. The large oxygen ions are almost immobile and so interdiffusion occurs only in the cation sublattice of the fee crystal. When we consider the following set ( ) of structure elements... 

There exist systems, namely systems which undergo processes involving hysteresis (plastic deformation or ferromagnetism, lor example) for which no equation of slate can he indicated. Although Ihe laws of thermodynamics may apply to such systems, the rigorous results of classical thermodynamics arc not applicable because the science of thermodynamics is developed on the assumption of the existence of the single-valued function. 

The requirement that a function be single valued, for a given input value for the independent variable, will hold for the majority of associations between one number and another. However, the association between any real number and its square root always yields both a positive and a negative result. For example, the two square roots of 9 are +3, and so we say that 9 is associated with both -3 and 3. Thus, if we write this association as y = x1/2, then we cannot define the function y =J x) = xl/2. However, if we explicitly limit the values of y to the positive (or negative) roots only, then we can redefine the association as a single-valued function. Alternatively, if we square both sides to yield y2 = x, we can take the association between x now as a dependent variable and y as an independent variable and define the function x=g(x)=y2, for which there is only one value of x for any value of y. 

---