Expectation values, overview

Table 9.28 gives an overview of the initiating events and their contributions to the expected frequency of the undesired event filling level too high . The expected frequency of the undesired event, Hj, is obtained by summing the expected values of all minimal cut sets in which the corresponding initiating event appears. 

Figure 1.21. Monte Carlo simulation of six groups of eight normally distributed measurements each raw data are depicted as x,- vs. i (top) the mean (gaps) and its upper and lower confidence limits (full lines, middle) the confidence limits CL(s ) of the standard deviation converge toward a = 1 (bottom, Eq. 1.42). The vertical divisions are in units of 1 a. The CL are clipped to +5a resp. 0. .. 5ct for better overview. Case A shows the expected behavior, that is for every increase in n the CL(x,nean) bracket /r = 0 and the CL(i t) bracket a - 1. Cases B, C, and D illustrate the rather frequent occurrence of the CL not bracketing either ii and/or ff, cf. Case B n = 5. In Case C the low initial value (arrow ) makes Xmean low and Sx high from the beginning. In Case D the 7 measurement makes both Cl n = 7 widen relative to the n 6 situation. Case F depicts what happens when the same measurements as in Case E are clipped by the DVM.

Figure 1.26. Confidence limits of the standard deviation for p = 0.05 and/- 1. .. 100. The /-axis is logarithmically transformed for a better overview. For example, at n = 4, the true value Ox is expected between 0.62 and 2.92 times the experimental Sx. The ordinate is scaled...

The project portfolio enables an overview on the ongoing research activities. Numerous economic and technical parameters have been proposed to provide a meaningful picture. Examples are attractiveness, strategic fit, innovation, gross/net present value, expected profits, R D expenditures, development stage, probability of success, technology fit, and realization time. Most of these parameters cannot be determined quantitatively, at least during the early phases of a project. 

A pictorial overview of the substrate scope is presented in Figure 2.3. More detailed results for synthetically useful reactions are presented in the next sections. Besides a-keto acid derivatives, a-keto acetals, a-keto ethers, and some trifluoromethyl ketones have been shown to give high ee values with Pt-based catalysts. Pd-based catalysts give moderate enantioselectivities for most a,P-unsaturated acids and up to 94% ee for selected pyrones. Nevertheless, for the synthetic chemist, the substrate scope is still relatively narrow, and it is not expected that new important substrate classes will be found soon. On the other hand, the chemoselectivity of this system has not yet been exploited to its full value, and this might hold a potential for synthetically useful applications in the future. 

In dealing with heteronuclear couplings the general relationship Jcx. = 7c 7X "Kqk can be used. Kcx depends on the electron density distribution along the n bonds. This provides the basis for an estimation of J values in situations where bonding is expected to change little with variation in X, e.g., when D is substituted for H. Tables 6-8 show an overview of representative carbon couplings. 

One of the most important parameters controlling iodine volatility is sump water pH not only will the I2 hydrolysis equilibrium and the iodine partition coefficient be affected by this parameter, but the product yields of radiolytic reactions and the extent of formation of organoiodine compounds as well. Because of the lack of practical experience, the sump water pH to be expected under severe accident conditions has to be calculated on the basis of assumed concentrations of potential sump water ingredients. In Table 7.17. (according to Beahm et al., 1992) an overview of substances to be expected in the sump water, which would effect a shift in solution pH either to lower or to higher values, is given. Besides these chemical substances, radiation may also affect sump water pH irradiation of trisodium phosphate solution (5.3 kGy/h) was reported to decrease the pH from an initial value of 9.0 to about 4.0 after 60 hours of irradiation (Beahm et al., 1992). It is obvious that in such a complicated system definition of the sump water pH to be expected in a real severe reactor accident is a difficult task. Nonetheless, a model for calculation has been developed by Weber et al. (1992). 

It would be quite impossible to give a short and complete overview of the various semiempirical methods that have evolved over time. One must really get into the mathematical details of the method to understand what approximations have been made in each case and what kinds of empirical data have been included. In many of fhese mefhods, it is common to omit integrals that are expected (either from experience or for theoretical reasons) to have negligible values. Certain integrals are stored in a table and are not calculated each time the program is applied. For instance, the frozen core approximation is often used. This approximation assumes that the completed shells of the atom do not differ from one atom to another in the same period. All the core calculations are stored in a table, and they are simply looked up when needed. This makes the computation much easier to perform. 

The relative standard deviation, RSD, was calculated for each level of the standards (0, 5, 10 and 15 pg Cu 1 ), which can be considered as a rough overview of the uncertainty associated with the predictions at each level they were 299.9%, 5.5%, 2.1% and 1.9%, respectively. The high value for the zero concentration must be considered with care because there was no analyte in these samples and, therefore, the denominator was very low [as detailed above, the standard deviations for the four levels of concentration (0.175,0.212,0.206 and 0.284, respeetively) were not statistically different]. As expected, the higher the concentration, the lower the variability in the predictions. 

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