Projections onto factors

In addition to expressing the spectral data as projections onto the spectral factors (basis vectors), we express the concentration data as projections onto the concentration factors (basis vectors). 

When we perform PLS on the data in Figure 74, we find that the difference between the PCA factors in Figures 70 through 72 and the PLS factors was so slight that there is no point in plotting the PLS factors in separate figures. Plots of the projections onto the PLS spectral factors vs. the projections onto the PLS concentration factors are shown in Figure 78. 

Efficiencies or efficiency factors Q are defined by dividing the cross-section by the cross-sectional area of the particle projected onto the plane perpendicular to the incident beam. For a spherical particle of radius a one writes... 

The Sturmian eigenfunctions in momentum space in spherical coordinates are, apart from a weight factor, a standard hyperspherical harmonic, as can be seen in the famous Fock treatment of the hydrogen atom in which the tridimensional space is projected onto the 3-sphere, i.e. a hypersphere embedded in a four dimensional space. The essentials of Fock analysis of relevance here are briefly sketched now. 

In Section 3.3 we have seen that the irradiance at the sensor array is proportional to the radiance given off by the corresponding object patch. In the following equation, we assume a scaling factor of one for simplicity, i.e. E(X,xj) = L(X, Xobj) where xobj is the position of the object patch that is projected onto x/ on the sensor array. Then we obtain the following expression for the intensity measured by the sensor. 

Effective one-electron equations for the channel orbital functions can be obtained either by evaluating orbital functional derivatives of the variational functional S or more directly by projecting Eq. (8.3) onto the individual target states p. With appropriate normalizing factors, ((")/ TV) = if/ps. Equations for the radial channel functions fps(r) are obtained by projecting onto spherical harmonics and elementary spin functions. The matrix operator acting on channel orbitals is... 

Table 1 displays an orthogonal array of strength three because it contains each of the 23 = 8 level combinations of any set of three factors exactly twice. In other words, its projection onto any set of three factors consists of two replicates of the complete 23 factorial design. 

Theorem 2. A necessary and sufficient conditionfor an OA(n, 2f, 2) with f >4 to have the property that the main effects and two-factor interactions can be estimated in all projections onto four factors is that it has no defining words of length three or four. 

Since we are considering a 2D system, we note that this is a monomolecular system and all of the centers of mass of the molecules are in the same plane (even if individual nuclei in the molecule are not). S Q), introduced earlier in this chapter, therefore only depends on the component of Q projected onto the scattering plane, Qparaiiei Just as Warren did, the case of lamellar systems like graphite and mica can now be considered, assuming that the scattering system has random orientations of the crystalhtes about an axis normal to the basal (or 2D) plane. Because the magnitude of the parallel component of the structure factor 5 (Q) is the relevant quantity... 

It is not always feasible to directly measure the ancilla independently from the information system in other words, it is sometimes impossible to perform a projection onto disentangled subspaces of H of the form 7T/0Span o ) in some cases, as for the example proposed in Sec. 3, one can only project onto entangled subspaces of the total Hilbert space H. In such a case the information initially stored in the vector ipi) = J2i=i r< Iu<) G Hi must be transferred into an entangled state of X and A of the form ip) = i r> H) where the I vectors 0) (i = 1.. .., I) which form an orthonormal basis of the information-carrying subspace C, are generally not factorized as earlier but entangled states. Nevertheless the same method as before can be used in that case to protect information, albeit in a different subspace C. 

The fluctuating variables aie thereby projected onto pair-density fluctuations, whose time-dependence follows from that of the transient density correlators q(,)(z), defined in (12). Tliese describe the relaxation (caused by shear, interactions and Brownian motion) of density fluctuations with equilibrium amplitudes. Higher order density averages are factorized into products of these correlators, and the reduced dynamics containing the projector Q is replaced by the full dynamics. The entire procedure is written in terms of equilibrium averages, which can then be used to compute nonequilibrium steady states via the ITT procedure. The normalization in (10a) is given by the equilibrium structure factors such that the pair density correlator with reduced dynamics, which does not couple linearly to density fluctuations, becomes approximated to ... 

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