Linear parametric equations

The linear parametric equation of state proposed by Schofield [3], is maybe the most versatile model based on the scaling law. The equation of state is in terms of parametric variables r and where r is essentially the radial distance to the critical point, and 0 describes an angular position in the p,T plane (Figure 4). All anomalies represented by the power laws are incorporated in the r-dependence, whereas the -dependence is strictly analytic. The transformation proposed by Schofield is ... 

One of the most extensively studied applications of this method focuses on the prediction of C NMR spectra (see Quantitative Structure-Property Relationships (QSPR)). The objective is the development of linear parametric equations that relate a set of descriptors to C NMR chemical shifts. The equations take the form ... 

During the moving phase, the differential equation (1) has to be solved, which is a system of nine linear differential equations with six additional constraints due to the orthogonality of the DCM. However, it is well-known that the group of proper rotations of the three-dimensional Euclidean space defined by the orthogonal transformation matrices R t) with det = 1 is itself a three-dimensional manifold. Some parametrizations of the rotation group with less than nine parameters and some of their attributes are listed below ... 

Equations (i2.iil and ri2.i2l are parametric equations of a linear relationship between He and XeA If we let w vary (by varying the relative amounts of solutions and B) and plot He against Xc, we obtain a straight line that passes from points (xa, Ha) and (xb, Hb) (see Figure i2-c l. This leads to a simple graphical procedure for calculating the temperature of the final state ... 

The linear parametric model is valid in a limited region around the critical point, located within approximately 40% of the critical density and 1% of the critical temperature. Outside this region it fails severely. Thus in order to carry out the necessary shock calculations we need a classical far-field equation of state. 

To define an equation of state one needs to introduce an approximation for the function M 6) in the expression eq 10.8 for the order parameter cpi. The most common choices are the linear model in which" M(0) = k6 and the cubic model in which M(9) = k9(l + c0 ). In these equations and c are universal constants, while a and k represent the two system-dependent coefficients that are related to the critical amplitudes. In this chapter we specify the relevant equations only for the simplest parametric equation which is the linear model with order parameter... 

The linear-model expressions for the critical amplitudes are presented in Table 10.5. The values of the universal critical-amplitude ratios implied by the restricted linear model are included in Table 10.3. For a corresponding set of expressions for the cubic model the reader is referred to the literature.More sophisticated parametric equations have also been considered in the literature, that are not discussed here. 

We note that the asymmetry coefficients a, b, and C3 affect the critical amplitudes but not the amplitude ratios which continue to have the same universal values listed in Table 10.3. Substitution of the parametric expressions for the scaling fields, sealing densities, and scaling susceptibilities from Table 10.4 into eqs 10.39 to 10.44 yields a linear-model equation of state consistent with complete scaling. 

To obtain a simple extended linear-model equation of state consistent with eq (10.53) in terms of the parametric variables r and 6, defined by eq (10.9), one has proposed to extend eq 10.10 for the order parameter to... 

A somewhat less fundamental but more practical approach has been developed by Kiselev who has formulated a phenomenological extension of the linear-model parametric equation described in Section 10.2.2 with variables... 

In these equations k, a, ct, d ( = Pu in eq 10.29), and g are system-dependent coefficients with g being related to the inverse of the Ginzburg number AIq. Slightly different versions for the crossover function R q) have also been used. In the critical limit 0 one recovers the linear-model parametric equation in Section 10.2.2 with coefficients a and k. In the classical limit q-rcc. Ad becomes an analytic function of AT and Ap. For a comparison of this phenomenological parametric crossover equation with the crossover Landau models the reader is referred to some previous publications. " " ... 

Non-linear models, such as described by the Michaelis-Menten equation, can sometimes be linearized by a suitable transformation of the variables. In that case they are called intrinsically linear (Section 11.2.1) and are amenable to ordinary linear regression. This way, the use of non-linear regression can be obviated. As we have pointed out, the price for this convenience may have to be paid in the form of a serious violation of the requirement for homoscedasticity, in which case one must resort to non-parametric methods of regression (Section 12.1.5). 

For simplicity, we will consider the case in which surface charge and potential are positive, and that only anions adsorb. Furthermore, the potential drop in the Gouy-Chapman layer will be assumed to be small enough that its charge/potential relation can be linearized. The V o/oo/pH relationship can then be derived parametrically, with the charge in the Gouy-Chapman layer cr4 as the parameter. The potential at the plane of anion adsorption can then be calculated and substituted in Equation 28 to give ... 

No one of the equations introduced here are defined as in the standard Bom-Oppenheimer approach. The reason is that electronic base functions that depend parametrically on the geometry of the sources of external potential are not used. The concept of a quantum state with parametric dependence is different. This latter is a linear superposition the other are objects gathered in column vectors. 

In this paper we present preliminary results of an ab-initio study of quantum diffusion in the crystalline a-AlMnSi phase. The number of atoms in the unit cell (138) is sufficiently small to permit computation with the ab-initio Linearized Muffin Tin Orbitals (LMTO) method and provides us a good starting model. Within the Density Functional Theory (DFT) [15,16], this approach has still limitations due to the Local Density Approximation (LDA) for the exchange-correlation potential treatment of electron correlations and due to the approximation in the solution of the Schrodinger equation as explained in next section. However, we believe that this starting point is much better than simplified parametrized tight-binding like s-band models. 

A curious feature of the space Ms of thermodynamic variables in an equilibrium state S is that its dimensionality varies with the number of phases, p, even though the values of the intensive variables (which might be used to parametrize the state S) do not. The intensive-type ket vectors R/ of (10.8) can actually be defined for all c + 2 intensities (T, —P, fjL, pi2, , pic) arising from the fundamental equation of a c-component system, U(S, V, n, ri2,. .., nc), even if only /of these remain linearly independent when p phases are present. 

Although the parametrization of coordinates in equation (3) is useful to indicate the separation into vibrations and rotations, it does not clearly separate the five nuclear modes into vibrational and rotational parts. However, it must always be possible to move to a set of rotating coordinates gA, formed from a linear combination of the general coordinates gA that do separate into rotational and vibrational coordinates, so that the Hamiltonian in equation (5) can be written as a sum of 7Yrot and Hvib over the rotational coordinates and vibrational coordinates, respectively. The vibrational problem in particular is then easier to solve as the number of variables is reduced. 

The problems of parametric estimation and model identification are among the most frequently encountered in experimental sciences and, thus, in chemical kinetics. Considerations about the statistical analysis of experimental results may be found in books on chemical kinetics and chemical reaction engineering [1—31], numerical methods [129—131, 133, 138], and pure and applied statistics [32, 33, 90, 91, 195—202]. The books by Kendall and Stuart [197] constitute a comprehensive treatise. A series of papers by Anderson [203] is of interest as an introductory survey to statistical methods in chemical engineering. Himmelblau et al. [204] have reviewed the methods for estimating the coefficients of ordinary differential equations which are linear in the... 

---