Normalized basis

Normal Individual and Process Unit Basis for PR Sizing Considerations -The following single contingencies should be considered as the normal basis for evaluating overpressure that can result from electric power failures ... 

We have called the vibrational quantum numbers here Vj, v2, v3 in order to distinguish them from the local quantum numbers, va, vl , vc. Note that, in view of the presence of the missing label, %, the normal basis is not very convenient for calculations. The spectrum corresponding to Eq. (4.59) is shown in Figure 4.8. There are fewer examples of molecules for which the dynamical symmetry of the normal chain II, provides a realistic zeroth-order approximation. The normal behavior arises when the masses of the three atoms are comparable, as, for example in XY2 molecules with mx = mY. More examples are discussed in the following sections. 

In real calculations, it is advisable to use normalized basis functions in order to avoid problems with numerical instabilities. Therefore, the overlap matrix elements, Su, are defined using normalized basis functions. After simplification, we obtain... 

Using the same method for the integrals gk gk) and gi gi), we find the overlap integral for the normalized basis functions to be... 

Absolute Magnitude of Specific Activities. In the Tewa and Zuni atoll samples absolute magnitude of specific activities for particles of the same kind (e.g., irregulars) are not too different when a refractory chain such as Chain 144 is compared. The absolute magnitudes on a normalized basis are around 3 X 1015 to 5 X 1015 equivalent fissions per gram. Bravo (15 megatons on coral) and Lacrosse (40 kilotons on coral) also give similar values. 

If a liquid form of the enzyme is added, such as lipases in broth samples from microbial cultures, calculate specific activity on a normalized basis by substituting a ml enzyme sample added (instead of mg protein) using the equation above. 

When the true intrinsic rank of a data matrix (the number of factors) is properly determined, the corresponding eigenvectors form an orthonormal set of basis vectors that span the space of the original data set. The coordinates of a vector a in an m-dimensional space (for example, a 1 x m mixture spectrum measured at m wavelengths) can be expressed in a new coordinate system defined by a set of orthonormal basis vectors (eigenvectors) in the lower-dimensional space. Figure 4.14 illustrates this concept. The projection of a onto the plane defined by the basis vectors x and y is given by a. To find the coordinates of any vector on a normalized basis vector, we simply form the inner product. The new vector a, therefore, has the coordinates a, = aTx and a2 = aTy in the two-dimensional plane defined by x and y. 

Thus, using normalized basis functions, the potential energy matrix elements, Vi, are ... 

The known compositional ranges of vent fluids are summarized in Figure 9 and Table 2. Because no two vents yet discovered have exactly the same composition, these ranges often change with each new site. As discussed in Section 6.07.2.2, vent fluids are modified seawater characterized by the loss of magnesium, SO4, and alkalinity and the gain of many metals, especially on a chloride normalized basis. 

I enriched with respect to seawater on a Cl-normalized basis depleted with respect to seawater on a Cl-normalized basis enriched and depleted... 

There have been recent efforts to predict, or at least rationalize, the x parameters of these and other polyolefin-polyolefin blends. Bates et al. (1992) and Fredrickson et al. (1994) suggest that the x parameter is correlated to a difference in statistical segment length of the polymer molecules, on a volume-normalized basis. The volume normalization is required because the definition of the statistical segment length depends on how the monomer unit... 

The definitions (22) and (23) don t require ortho-normalized basis functions. But calculated results fairly depends on basis functions. The absolute value of partitioned energy with orthonormalized basis functions is usually larger and seems more reasonable than the counterpart with non-orthonormalized basis functions, like the case of Okada s bond order. The basis functions orthonormalized by Eq. (19) and Eq. (20) are used for the energy partition calculations in the following sections. But systematic and theoretical studies are necessary to clarify the dependence of bond parameters on the basis functions and the optimization of them. 

Figure 11.2 shows that the use of aeidie filler helps to preserve elongation when the filled material is exposed to y-radiation. Both aeidie and basie ealeium earbonate were evaluated. The aeidie version was obtained through surfaee treatment of normally basie ealeium earbonate. Due to lower aeid/base interaetion, Young modulus and yield stress of the eompounds eontaining the aeidie filler were lower but elongation was less affeeted. 

Figure 10.3. Normalized basis weight data for a roll of paper sheet.

Figure 10.4. Histogram of normalized basis weight data with comparison to normal distribution.

For benchmark accuracy calculations to be done on polyatomic molecules, without time-consuming explorations and expertise, it is necessary to have a means of systematic improvement not provided by the normal basis sets. It was shown earlier that for the... 

The text by Hehre et al. explains the procedures used to obtain molecular wavefunctions. Let the molecular orbitals Tj be defined in terms of N normalized basis functions by... 

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