Level Two Matrix

Once the hierarchy has been completed, matrices are constructed with the criteria labels on each axis. There will be one Level Two matrix and a number of associated matrices for the... 

Considering the example of the Level Two matrix in Table 9.5, and assuming that Element 1 is weakly more important than Element 2 and strongly more important than Element 3. Then, the matrix in Table 9.5 may be represented as seen in the matrix below ... 

The probability of occurrence and severity make up the two elements in Level Two as seen in Figure 9.4. These two elements are compared against each other to determine the weighting vector of each element The comparison scale in Table 9.5 is used to determine the importance of the two elements. Considering the goal of the analysis, it is decided that both these elements are equally important to a safety assessment, hence, the Level Two matrix is determined as ... 

The weighting vector and normalised vector are determined by considering the weighting vector obtained in the Level Two matrix and are shown below ... 

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

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

Fig. 15.1 (a) Energy levels and dipole matrix elements for the resonant collision of two atoms in the s and s states resulting in the production of two atoms in the p and p states, (b) Energy levels and matrix elements for the radiative collision in which an s and an s atom collide to produce atoms in the p and d states. The production of the d state is via the virtual p state which is detuned from the real p state by an energy A. 

Before examining the effect of an applied magnetic field it is instructive and hopefully helpful to look at the matrix of the above five terms. For each of the three fine-structure components with a given J value there are two parity states, labelled e and /. According to the now accepted convention [68] for integral J values, levels with parity - -(—l) 7 are called e levels and levels with parity —(— If 7 are called / levels. The matrix is as follows. 

At the quantum-mechanical level two subsystems are distinct, when each is described by "its own" wavefunction or density matrix. In the pure-state description the M. wavef unction is the "symmetry-adapted" (inter-reactant antisymmetrized) product of the subsystem wavef unctions, each individually normalized and antisymmetric (15,17),... 

Cherkaoui A, Hibbs J, Emonet S, Tangomo M, Girard M, Francois P, Schrenzel J. Comparison of two matrix-assisted laser desorption ionization-time of flight mass spectrometry methods with conventional phenotypic identification for routine identification of bacteria to the speeies level. J Clin Microbiol. 2010 48(4) 1169-75. doi 10.1128/JCM.01881-09. 

The hierarchy records the flow of detail from the problem statement (Goal) to broad issues (Level Two) and more specific levels (Level Three). While the concerns on a particular level are not equally important, they should be on the same order of magnitude. This feature in AHP allows decisions to be made involving different orders of magnitude criteria, by placing each criterion in its proper matrix in the objective hierarchy. Figure 9.2 shows an example of the hierarchy represented diagrammatically. 

A more intuitive, and more general, approach to the study of two-level systems is provided by the Feynman-Vemon-Flellwarth geometrical picture. To understand this approach we need to first introduce the density matrix. 

A3.13.1). From [38]. The two-level structure (left) has two models I I = const and random signs (upper part), random V.j but V < V.j < (lower part). The right-hand side shows an evolution with initial diagonal density matrix (upper part) and a single trajectory (lower part). 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

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