Geometrical representation

Figure 4.15 Geometrical representation of the temperature variation of the actual volume (solid line) and the occupied volume (broken line). The shaded difference indicates the free volume which decreases to a critical value at T .

Figure 4-426 shows a geometric representation of the terms in the above equation. 

In the geometrical representation of the partial pressures of I., we have on the right the straight line PiQ corresponding with (1), and on the left one or other of the three initial curves 0a, Ob, 0c, corresponding with the equation 7ri = Pi and the inequalities 7Ti < Pi, 7Ti > Pi, respectively. 

Every n vector can be represented as a point in an -dimensional coordinate space. The n elements of the vector are the coordinates along n basis vectors, such as defined in the previous section. The null vector 0 defines the origin of the coordinate space. Note that the origin together with an endpoint define a directed line segment or axis, which also represents a vector. Hence, there is an equivalence between points and axes, which can both be thought as geometrical representations of vectors in coordinate space. (The concepts discussed here are extensions of those covered previously in Sections 9.2.4 to 9.2.5.)... 

Summarizing, we find that, depending on the choice of a and p, we are able to reconstruct different features of the data in factor-space by means of the latent vectors. On the one hand, if a = 1 then we can reproduce the cross-products C between the rows of the table. On the other hand, if p equals 1 then we are able to reproduce the cross-products between columns of the table. Clearly, we can have both a = 1 and P = 1 and reproduce cross-products between rows as well as between columns. In the following section we will explain that cross-products can be related to distances between the geometrical representations of the corresponding rows or columns. 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

Fig. 34.12. Geometrical representation of TTFA target in is accepted target in2 is rejected.

Figure 1. Orthogonal decomposition of a three-dimensional Hilbert space geometrical representation of the two orthogonal subspaces. 

Fig. 5.2. Geometrical representation of a complete two level factorial matrix (three influence factors) with experiments in the centre point (0,0,0)... 

Analytic Geometry Part 2 - Geometric Representation of Vectors and Algebraic Operations... 

For volumes V) and Vn we consider the integrals as sums over small volumes, A Vj, defined by the motion of their corresponding surface elements, AS). Figure 3.3 provides a (two-dimensional) geometric representation of this process in which AS, moves over AS... 

The independent displacements (AN, AQ) (perturbations) in this canonical geometric representation give rise to the first differential of its thermodynamic potential W(N, Q) ... 

It should also be realized that the generalized softness matrix of Equations 30.12 and 30.16 represents the compliant description of the electronic coordinate N coupled to the system geometric relaxations (see Section 30.3). Indeed, the relaxed geometry global softness of the geometrical representation,... 

The MEC can also be introduced in the combined electron-nuclear treatment of the geometric representations of the molecular structure (Nalewajski, 1993, 1995, 2006b Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008). Consider, for example, the generalized interaction constants defined by the electronic-nuclear softness matrix S. The ratios of the matrix elements in SMif = S/l s- to define the following interaction constants between the nuclear coordinates and the system average number of electrons ... 

The two-reactant coupled approach (Nalewajski and Korchowiec, 1997 Nalewajski et al., 1996, 2008 Nalewajski, 1993, 1995, 1997, 2002a, 2003, 2006a,b) can also be envisaged, but the relevant compliant and MEC data would require extra calculations on the reactive system A—B as a whole, with the internal coordinates Q now including those specifying the internal geometries of two subsystems and their mutual orientation in the reactive system. The two-reactant Hessian would then combine the respective blocks of the molecular tensors introduced in Section 30.2. The supersystem relations between perturbations and responses in the canonical geometric representation then read ... 

The Figure 5-25 attempts to provide a geometrical representation of the situation. 

FIGURE 2.2 Data matrix and scatter plot Geometrical representation of objects and variables defined by a two-dimensional A -matrix with 10 objects. 

Fig. 12. Potential energy diagram for the unimolecular decomposition of SiH4 to SiHj and H, showing the activation energy barrier . The sketch above is a geometrical representation of the events (1 to r) along the reaction coordinate, which is the Si-H distance.

The geometrical representation of the Muggianu, Kohler and Toop equations is best summarised using the figures below. 

Figure 2.2. A geometric representation of functions for H2 in terms of vectors for R = Req. The small vectors labeled (a) and (b) are, respectively, the covalent and ionic components of the eigenvector. The vectors with dashed lines are the symmetrically orthogonahzed basis functions for this case.

It is important to realize that the above geometric representation of the H2 Hilbert space functions is more than formal. The overlap integral of two normalized functions is a real measure of their closeness, as may be seen from... 

We can think of these representations as shadows of hypercubes on 2-D pieces of paper. Luckily, we don t have to build the object to compute what its shadow would look like. (The computer code I used to create these forms is listed in Appendix I.) Projections of higher-dimensional worlds have stimulated many traditional artists to produce geometrical representations with startling symmetries and complexities (Figs. 4.18 to 4.20). 

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