Ellipse method

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

We can now draw the 95 percent confidence ellipse using the method outlined in Section 2.4. First, let us write the diagonal form of the matrix S as... 

The 11 points of the isochron diagram are drawn in Figure 5.13. Their respective error ellipse (1 a error) could be drawn using the method described in Section 2.4, but errors are too small for the ellipses to be clearly seen. 

Points with a constant Euclidean distance from a reference point (like the center) are located on a hypersphere (in two dimensions on a circle) points with a constant Mahalanobis distance to the center are located on a hyperellipsoid (in two dimensions on an ellipse) that envelops the cluster of object points (Figure 2.11). That means the Mahalanobis distance depends on the direction. Mahalanobis distances are used in classification methods, by measuring the distances of an unknown object to prototypes (centers, centroids) of object classes (Chapter 5). Problematic with the Mahalanobis distance is the need of the inverse of the covariance matrix which cannot be calculated with highly correlating variables. A similar approach without this drawback is the classification method SIMCA based on PC A (Section 5.3.1, Brereton 2006 Eriksson et al. 2006). 

This difference in j-coordinate can be explained by the shift of the centre in y-direction caused by the curvature of the Ewald sphere. The curvature of the Ewald sphere does not affect the position of the centre in the four-clicks method, but leads to a shift of the ellipse centre when calculating this centre from a single ellipse. Thus the four-clicks method is better for the centre calculation. 

Contrary to the classical mean and covariance matrix, a robust method yields a tolerance ellipse that captures the covariance structure of the majority of the data points. In Figure 6.1, this robust tolerance ellipse is obtained by applying the highly robust minimum covariance determinant (MCD) estimator of location and scatter [7] to the data, yielding pMCO and i MCD > and by plotting the points x whose robust distance... 

Unlike ellipses, parabolas do not lend themselves to simple mechanical drawing aids. The ones occasionally described in texts work crudely. Templates are hard to find. The two best methods for drawing parabolas both involve locating points on the parabola and connecting those points either by eye, or with the help of a draftsman s french curve. 

We believe J.W. Linnett, in his Methuen Monograph, Wave Mechanics and Valency Theory, 1956, was the first to use elliptical projections to display the phases of the spherical harmonics on the unit sphere. We adopt a projection in which both 0 and (f) coordinates are plotted on linear scales on the minor and major axes of a 30° ellipse of eccentricity a/3/2. This cartographic device is the one proposed by Apianus in 1524, and known as the Apianus II projection. In our early work on the Spherical Shell method we called this a modified Mollweide projection, reversing the historical sequence. 

Fig. 3-7 Location of Laue spots (a) on ellipses in transmission method and (b) on hyperbolas in back-reflection method. (C = crystal, F = film, Z.A. = zone axis.)...

In either Laue method, the diffraction spots on the film, due to the planes of a single zone in the crystal, always lie on a curve which is some kind of conic section. When the film is in the transmission position, this curve is a complete ellipse for sufficiently small values of 0, the angle between the zone axis and the transmitted beam (Fig. 8-12). For somewhat larger values of 0, the ellipse is incomplete because of the finite size of the film. When 0 = 45°, the curve becomes a parabola when 0 exceeds 45°, a hyperbola and when 0 = 90°, a straight line. In all cases, the curve passes through the central spot formed by the transmitted beam. 

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