Distance between points

Corresponding elements in the vector representing one particular sample and in the appropriate vector of means are worked up as in 2) to find the Euclidean distance between point i and its group mean (see lines marked with an asterisk ( ) in Table 4.16) this forms the second side of the appropriate triangle in Fig. 4.20. 

Here Lqp is the distance between points q and p. Note that G q, p) is called a Green s function. There are an infinite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point by a small spherical surface S and apply Equation (1.99) to the volume enclosed by surfaces S and S, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is. 

Here. s is a point located somewhere inside the sphere along the line Op, p and sq the distances between points q, p and q, s, respectively, and a. a constant. Now we demonstrate that for each observation point p it is possible to find such a point and coefficient a, that the Green s function is equal to zero at all points of the sphere. In order to prove it, we first choose the position of the point from the condition ... 

Taking into account the fact that the distance between points p and q is very small, the normal potential varies almost linearly and this gives ... 

Here N is the distance between points p and q measured along the perpendicular from the point q to the geoid. Fig. 2.9a. The linear behavior of the normal potential implies that the field y is constant between the geoid and the reference ellipsoid. The change of sign in Equation (2.260) is related to the fact that the field has a direction, which is opposite to the direction of differentiation. As follows from the first equation of the set (2.258 and 2.260) we have... 

Fig. 29.8. (a) Pattern of points in column-space S (left panel) and in row-space S" (right panel) before column-centering, (b) After column-centering, the pattern in 5 is translated such that the centroid coincides with the origin of space. Distances between points in S are conserved while those in S" are not. (c) After column-standardization, distances between points in S and 5" are changed. Points in 5" are located on a (hyper)sphere centered around the origin of space. 

Distances between points in and can also be defined formally as distances of chi-square ... 

Figure 3.13 Principal component analysis of repetitive GC/MS profiles of M. truncatula root (R), stem (S) and leaves (L). The first and second principal component of each GC/MS analysis were calculated and plotted. The relative distance between points is a measure of similarity or difference. The clustering shows good reproducibility within the independent tissues but clear differentiation of tissues. The results also show that roots and stems are more similar to each other than to leaves.

Figure 4.31 SEM image of MFI-type zeolite, (distance between points of size bar = 50 pm total length 500 pm.)...

Sampling Procedure. A set of twenty-five sampling points was marked out on each plot on a rectangular 5x5 grid. The distance between points was thus 40.5m on the long axis of the grid and 20.2m between rows of points. Each sampling point was permanently identified by number. 

Each spectrum Is regarded as a point In an N dimensional space. The coordinates of each point are the absorbance values at each wavenumber Interval. Several metrics are widely used.(19) The first Is the N dimensional cartesian distance between points 1 and j. 

The second point to note is that none of these distance measures necessarily have any physical meaning. They are simply measures of proximity that are interpreted as similarity. Therefore, in all of the pattern-recognition techniques it is the relative distance between points (i.e., is point A closer to B than it is to Q that is important. 

Prove that a tangent to the point of inflection of a Gaussian peak will intercept the baseline a distance 2a from center. (Therefore the distance between points where tangents on each side intercept the baseline is w = 4a, the effective zone width.)... 

This definition of the CSM implicitly implies invariance to rotation and translation. Normalization of the original shape prior to the transformation additionally allows invariance to scale (Figure 1). We normalize by scaling the shape so that the maximum distance between points on the contour and the centroid is a given constant (in this chapter all examples are given following normalization to 1 however, CSM values are multiplied by 100 for convenience of handling). The normalization presents an upper bound on the mean-squared distance moved by... 

To try to get a reasonable starting matrix D, one first builds a matrix L of lower distance bounds and a corresponding matrix U of upper bounds. Both matrices should contain any experimental distances as well as any covalently determined distances. In cases such as bond lengths, elements l,t may nearly equal ubut in the case of undetermined distances between points covalently far from each other, /i may be the sum of the van der Waal radii, whereas u will be some large number. 

The main parameters derived from ESR spectra are the g-values (ana-lagous to the chemical shift in NMR), the hyperfine coupling constants (like spin-spin coupling constants), line-widths and intensities. Because spectra are usually displayed as first derivatives (Fig. 3.2), line-widths are conveniently measured as the distances between points of maximum and minimum slope (a in Fig. 3.2). Intensities are often quoted as the total extent of each line (/ in Fig. 3.2), but this can be misleading, and it sometimes helps to integrate to give the normal absorption spectrum. 

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