Undetermined parameters

Three undetermined parameters which appear in the LSZK equation of state are evaluated from the experimental data, and the remaining constants required are evaluated from the experimentally determined dependence of detonation velocity on density... 

The variation with temperature of n and for an undoped GaAs crystal grown by the horizontal Bridgman method is shown in Fig. 3. The parameters that give the best fit to Eq. (12) are also shown. The power of this method is illustrated by the small probable errors in the fitted parameters, i.e., less than 15% for Nd and less than 1% for ED0. Very few techniques can lay claim to such accuracy. Note that for maximum reliability it is necessary to know the Hall r factor [Eq. (A 17)], since n = r/eR. A variational calculation, with NAS as the only undetermined parameter, was used to fit the fi versus T data, and also obtain r versus T (Meyer and Bartoli, 1981 Look et al., 1982a). 

A different dependence of the parameters in kinetic equations was reported by Horiuti [11] who suggested a method for determining the number of independent parameters. The method consists of the numerical estimation of a rank for some Jacobian matrix. (It is known that this procedure can result in a considerable error.) Later, these problems were analyzed in detail by Spivak and Gorskii [52, 53] but they did not aim at the elucidation of the physico-chemical reasons for the appearance of dependent and undeterminable parameters. It is this aspect that we will discuss below. 

The as yet undetermined parameter p follows from an evaluation of this result for ft = AT... 

In the present calculations, the interaction parameter (x or x") is the only undetermined parameter. For the PI chains-toluene interactions, x is selected 0.35, which is close to the experimental values of the Floiy-Huggins interaction parameter45 OfFH=0.32—0.36 in the concentration range temperature range 25-55 °C). For the PS chains-toluene interactions, x is taken 0.2, which is in the lower range of the experimental Floiy-Huggins interaction parameters45 (a,fh=0.04-0.67 in the concentration... 

Equations 1-4 can easily be set up on a PC and used to simulate or fit temperature-dependent mobility data. In an n-type sample, the only undetermined parameter is the acceptor concentration Na, so usually Na is varied to give the best fit to the data. (For this fit, the approximate carrier concentration, nn = 1/Re, can be used when n is required in the various scattering formulas.) Then, the Hall factor r = can be calculated at... 

Equation (2) for the number of particles per unit volume of aqueous phase has one undetermined parameter Af which is the difierence between the surfactant charged and the surfactant adsorbed on the surface of the polymer particles. 

This formula indicates that the reconstruction can be achieved but it will always have one undetermined parameter as a free parameter. In this particular case is the free parameter an additive constant in the imaginary part iy(x,y) of holomorphic function W(z). After performing derivative of H (z), the free parameter vanishes. It is therefore obvious that from electric potential one can recover the current density without any free parameter. In the case when data are known at discrete points, reconstruction, in general, is non-unique and can have any finite number of free parameters, which is evident from the following. Let an exact solution f(z) = u(x,y)+iv(x,y) of the problem be found by any method. It is obvious, that the modulus must be real therefore it satisfies the following ... 

In the approach which has come to be known as the Connolly-Williams method (Connolly and Williams, 1983), a systematic inversion of the cluster expansion is effected with the result that the parameters in the effective Hamiltonian are determined explicitly. In this case, the number of energies in the database is the same as the number of undetermined parameters in the cluster expansion. Other strategies include the use of least-squares analysis (Lu et al. 1991) and linear programming methods (Garbulsky and Ceder, 1995) in order to obtain some best choice of parameters. For example, in the least-squares analyses, the number of energies determined in the database exceeds the number of undetermined parameters in the effective Hamiltonian, which are then determined by minimizing a cost function of the form... 

The last expression contains only a single undetermined parameter p which can be determined from the requirement of Q attaining a maximum, equivalent to / being a minimum... 

In fact we have already prepared the way in the earlier sections if we make a specific type of systematic approximation to that is, we choose a which is of known mathematical form containing some undetermined parameters, the expression for E becomes a function of these parameters and we know how to find the minimum in a function ... 

The basic constraint dynamics method employed There are two distinct methods of constraint dynamics the analytical method and the method of undetermined parameters. The reason for the names will become apparent. 

The analytical method deserves a detailed discussion for at least two major reasons. First, if used in conjunction with some constraint correction scheme," it is important in its own right as a practical method of solution of the constrained dynamics problem. Second, the method of undetermined parameters, central to the subject matter of this chapter, is an outgrowth of the analytical method hence a thorough understanding of the analytical method is an essential prerequisite for understanding the method of undetermined parameters. 

As mentioned above, Ryckaert et al. introduced the method of undetermined parameters, which is essentially the analytical method modified to ensure that the constraints are satisfied exactly at each time step. The method of undetermined parameters is discussed in detail, in its most general form, in a later section. Basically, the aforementioned modification requires that the highest derivatives with respect to time of the Lagrangian multipliers (i.e., of order be replaced by a set of undetermined parameters with values to be determined such that the constraints are satisfied exactly at each time step. As will be seen, the algorithm does not introduce into the trajectories additional numerical errors any worse than the error already present in the integration scheme itself. 

Again, the method of undetermined parameters deserves a detailed discussion in its most general form because most researchers are interested in its implementation with different integration algorithms (e.g., basic Verlet,- i velocity Verlet ), with holonomic constraints of various types (e.g., bond-stretch, angle-bend, and torsional constraints), and with particular techniques of solution (e.g., the ma-... 

The numerical procedure used to solve the final equations The analytical method leads to a system of equations linear in the unknowns (i.e., the Lagrangian multipliers and their time derivatives up to order Therefore standard numerical techniques for solving such systems can be employed. The method of undetermined parameters leads to an additional system of equations generally nonlinear in the unknowns (i.e., the derivatives of the Lagrange multipliers of order s ,3x)- The order of nonlinearity depends on the particular... 

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