Observation equations Gauss

The number of reflection intensities measured in a crystallographic experiment is large, and commonly exceeds the number of parameters to be determined. It was first realized by Hughes (1941) that such an overdetermination is ideally suited for the application of the least-squares methods of Gauss (see, e.g., Whittaker and Robinson 1967), in which an error function S, defined as the sum of the squares of discrepancies between observation and calculation, is minimized by adjustment of the parameters of the observational equations. As least-squares methods are computationally convenient, they have largely replaced Fourier techniques in crystal structure refinement. 

X2° = X30 = 0 assumed to be known exactly. The only observed variable is = x. Jennrich and Bright (ref. 31) used the indirect approach to parameter estimation and solved the equations (5.72) numerically in each iteration of a Gauss-Newton type procedure exploiting the linearity of (5.72) only in the sensitivity calculation. They used relative weighting. Although a similar procedure is too time consuming on most personal computers, this does not mean that we are not able to solve the problem. In fact, linear differential equations can be solved by analytical methods, and solutions of most important linear compartmental models are listed in pharmacokinetics textbooks (see e.g., ref. 33). For the three compartment model of Fig. 5.7 the solution is of the form... 

This chapter uses Gauss 1809 treatment of nonlinear least squares (submitted in 1806, but delayed by the publisher s demand that it be translated into Latin). Gauss weighted the observations according to their precision, as we do in Sections 6.1 and 6.2. He provided normal equations for parameter estimation, as we do in Section 6.3, with iteration for models nonlinear in the parameters. He gave efficient algorithms for the parameter... 

This equation shows the relationship between field values observed on various points of the surface S and can be interpreted from two points of view. If the charge e is known, eq. 1.34 can be considered to be an integral equation in an unknown variable the normal component of the field. In contrast, when the electric field is known, the use of the flux allows us to determine the sources of the field. If we wish to find the relationship between flux and source within an elementary volume, we can make use of Gauss s theorem ... 

The resulting numerical values of 204 gauss for a single-layer and 218 gauss for a doublelayer coil are much smaller than the observed maximum field strengths. It is suggested that equation (3), on which the models of Montgomery and Chandrasekhar and Hulm are based, cannot be used in the manner assumed p]. 

The liposomes formed under such conditions were mostly multilamellar vesicles (MLV) as can be seen in Fig. 2. They were characterized by the mean vesicle diameter and their standard deviation, determined by SEM and calculated according to the Gauss distribution equation. From results shown in Table 1 it can be observed that the increase in PRO concentration causes an increase in the mean vesicle diameter. It could be assumed that PRO was mostly located in the aqueous phase of the liposomes, between the lamellas, because of their distinctive polar character and the position of the hydroxyl groups seen in Fig. 1. All the dispersions examined have marked polydispersity of the liposomes. 

This observation can be used to extend the formula form of the Gauss-Jordan algorithm to give the inverse of matrix A every time it calculates the solution to the set of equations... 

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