Frequency matrix

A representative consequence matrix is shown in Table 1.5. The matrix has four levels of consequence covering worker safety, public safety, the environment, and economic loss. There are no rules as to how many levels should be selected, nor does any major regulatory body insist on a particular size of matrix. However, many companies choose four levels. Three levels do not provide sufficient flexibility and differentiation, but five levels imply a level of accuracy that is probably not justified. The steps in Table 1.5, from Low to Very severe, are roughly in order of magnitude, i.e., each increased level is about ten times more serious than the one before it. Some companies choose not to put an economic value in Table 1.5 because it creates a perceived monetary value for safety. 

Once the consequences associated with an incident have been identified, the next step is to estimate the frequency with which the incident may occur. A representative frequency matrix is shown in Table 1.6. As with the consequence matrix, four value levels are provided. The use of just three levels is probably too coarse, but five levels or more implies a degree of accuracy that probably could not be justified. Precision is not the same as accuracy. 

As with the consequence matrix, each step in Table 1.6 is roughly an order of magnitude greater than the one before it. 

1 Reportable or equivalent None Limited impact that is readily corrected 10,000 to 100,000 

Moderate, 2 Hospitalization or lost-time injury Minor medical attention Report to agencies and take premeditative action 100,000 to 1 million 

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The selection rules for isotropic Raman spectra ji = jf = j greatly simplify the formalism. The frequency matrix has only diagonal elements... 

Ramsey obtained ay, by first-order and op by second-order perturbation theory (76) variational treatments give similar results (14, 71, 73). The term wp is sometimes called the second-order paramagnetic term and sometimes the high-frequency term (14), because of the dependence of the (temperature-independent) paramagnetism in molecules on the high-frequency matrix elements of the orbital moments (91). 

The values of C (r) 2 and IC2WI2 obtained from (7.19) and (7.20) are compared in Figs. 9 and 10. The amplitudes and periods of the temporal evolution predicted by the two approaches to the system dynamics are seen to agree quite well. The differences seen in the amplitudes shown in Fig. 9 are a consequence of the replacement of the exact eigenfrequencies of the Rabi frequency matrix with a typical eigenfrequency from the P subspace. 

The frequency matrix Qy and the memory function matrix Ty, in the relaxation equation are equivalent to the Liouville operator matrix Ly and the Uy matrix, respectively. The later two matrices were introduced by Kadanoff and Swift [37] (see Section V). Thus the frequency matrix can be identified with the static variables (the wavenumber-dependent thermodynamic quantities) associated with the nondissipative part, and the memory kernel matrix can be identified with the transport coefficients associated with the dissipative part. 

The velocity autocorrelation function can be obtained from the relaxation equation [Eq. (76)], where Cv(z) = Cjt(q = 0z). Here the suffix s stands for single-particle property. For zero wavenumber, there is no contribution from the frequency matrix [that is, D v(q = 0) = 0] and the memory function matrix becomes diagonal. If we write (z) = Tfj (q = 0z), then the VACF in the frequency plane can be written as... 

To write down the expression for the dynamic structure factor, we need explicit expressions for the components of the frequency matrix, memory function matrix, and the normalization matrix C(q). 

The definition of the classical scalar product, discussed earlier in the derivation of the viscosity, is used in the derivation of the frequency and the normalization matrix. The normalization matrix is diagonal, and its matrix elements are the following Cpp — NS(q)/kBT and C = N/m. The diagonal components of the frequency matrix are zero due to time inversion symmetry. The off-diagonal elements are the following Tlpi = q and Slip = qkBT/ mS(q) = (a>q2)/q. 

Using the properties of correlation functions under time inversion and spatial symmetry operations, one can show [17] that in general the structure of the longitudinal (l + 2) x (l + 2) hydrodynamic frequency matrix is as follows,... 

Since the elements of the hydrodynamic frequency matrix (20) are expressed via the static correlation functions (see (8)), constructed on the densities of conserved variables and its first time derivatives, this allows us to express them via the so-called generalized /c-dcpcndent thermodynamic quantities, namely, one gets [17],... 

The streaming operator is substantially unchanged compared to Eq. (2.58) (except for an additional contribution to the first rank interaction potential). TTie collisional operator is defined in terms of an orientational dependent friction matrix (or collisional frequency matrix in the present dimensionless formulation) as... 

The two methods will provide very similar parameters if an accurate frequency matrix, is available. 

Fig. 1. Position-specific scoring matrix model representation. The information contained in a set of aligned target sites (A) is initially encoded into a count matrix (B) in which each column represents the number of sequences that contain each of the four bases in that position. For practical purposes, the count matrix is further transformed into some form of log-frequency matrix (C). Any of the three forms of information can be graphically represented as a LOGO of symbols (D).

We have just shown that for the particular set of variables given by Eq. (12.2.5), the matrix 0(q) is a null matrix and correspondingly the frequency matrix O(q) is a null matrix that is,... 

The vertex-path incidence matrix is also termed the path-layer matrix (Skorobogatov and Dobrynin, 1988 Diudea, 1994,2003), when the paths are organized with respect to length. The VP matrix is analogous to the so-called cardinality layer matrix (Todeschini and Consonni, 2000, 2009), which has also been called the path-layer matrix (Skorobogatov and Dobrynin, 1988), the distance-frequency matrix (Diudea and F rv, 1988), and the path-sequence matrix (Todeschini and Consonni, 2000, 2009). Diudea and coworkers (Diudea et al., 1991, 1994 Diudea and Ursu, 2003), Dobrynin (1993), and others (e.g., Hu and Xu, 1996 Yang et al., 2002) derived a number of layer matrices. 

We now construct the G.L.E. for these orientational variables. Since all the coupled variables are functions only of angle, the frequency matrix is identically zero. The memory function matrix for this problem will now be shown to have elements... 

Note that f. is a small quantity, but Nf is of the order of unity, often amounting to 0.1 to 0.5 in fluids of small but non-spheripal molecules. The frequency matrix has, as usual, diagonal elements equal to zero and non-zero off-diagonal elements given by ... 

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