Lower-bound estimate

The solution of the equation (4.2.26) cannot be found in an analytical form and thus some approximations have to be used, e.g., variational principle. Its formalism is described in detail [33, 57, 58] for both lower bound estimates and upper bound estimates. Note here only that there are two extreme cases when a(r)/D term is small compared to the drift term, reaction is controlled by defect interaction, in the opposite case it is controlled by tunnelling recombination. The first case takes place, e.g., at high temperatures (or small solution viscosities if solvated electron is considered). 

On the other hand, employing the Kohn variational principle [67] two lower bound estimates may be obtained [59, 60]... 

In Fig. 4.7, another case of A0, H pairs is shown when half the Bohr radius of the electron activator centre is typically about 1 A (e.g., for Tl° in KC1). Here a sum of two upper estimates (curves 5 and 6) practically coincide at all temperatures with the exact radius (curve 1) and with lower-bound estimates (curve 7 and 8) above 70 K. 

Fig. 4.7. Temperature dependence of the effective radius of H, A0 recombination in KBr controlled by an elastic interaction, diffusion and tunnelling. Curve 1 - exact result, 2 - effect of tunnelling and annihilation, 3 - isotropic attraction and annihilation, 4 - pure annihilation. Variational estimates upper bound when (i) tunnelling dominates (equation (4.2.32) - curve 5) or an elastic interaction dominates (equation (4.2.34) - curve 6). Curve 7 - lower bound estimate, equation (4.2.36), when an elastic interaction is a predominant factor.

An increase of the standard deviation at r 3 due to small number of survived particles, demonstrates a limited possibility of the direct statistical simulations for a system with a variable number of particles. However, certain conclusions could be drawn even for such limited statistical information. Say, if for equal concentrations the analytical theory based on the superposition approximation seems to be quite adequate, for unequal concentrations its deviation from the computer simulations greatly increases in time. The superposition approximation gives the lower bound estimate of the actual kinetic curves tia( ) but if for d = 2 shown in Fig. 5.8 the deviation is considerable, for d, = 1 (Fig. 5.7) it is not observed, at least for the reaction depths considered. 

More concrete conclusions could be drawn for the linear approximation applicability it is adequate for small reaction depths r < 1, whereas at r > 1 it is in serious error. In its turn, errors of the superposition approximation are essentially less, the relevant lower bound estimate is quite acceptable to fit theoretical parameters to the experimental curves. 

As it took place for the tunnelling recombination, divergence in results is not large. It will be shown in Chapter 6 that the reaction depths studied here are enough to establish appearance of the new asymptotic kinetic laws. The superposition approximation giving a lower bound estimate of the kinetics, reproduces correctly the kinetics at long times. Results of the linear approximation are not plotted since they diverge considerably from the statistical simulations. 

The analysis of the diffusion-controlled computer simulations confirms once more conclusions drawn above for the static reactions of immobile particles. In particular, the superposition approximation gives the best lower bound estimate of the kinetics reaction, n = n(i). Divergence of computer simulations and analytical theory being negligible for equal concentrations become essential for large depths and when one of reactants is in excess. The obtained results allow us to use the superposition approximation for testing the applicability of simple equations of the linear theory in those cases when computer simulations because of some reasons cannot be performed. Examples will be presented in Chapter 6. 

Cancer potency is taken as the upper 95% confidence limit (UCL) on the linear term qi, which is called the cancer slope factor (CSF) or q, in units of (mg/kg-day), derived using maximum likelihood estimate techniques. Cancer risks at low doses equals dose X q " and the dose associated with a specific risk equals risk/ j . The calculated risks are upper bound estimates and calculated doses are lower bound estimates. These are further summarized below. 

Figure 6.15. Damage cost estimates per ton of NOx emission. Best estimates or geometric average of upper and lower bound estimates, inflated to 2001 based on Consumer Price Index.

Note that, in practice, the mean and variance of the distribution (/u. and a) are not precisely known, and are estimated from n measurements. As such the lower-bound estimate of the mean, at a prescribed confidence level, used to represent the mean of the distribution, and some multiple of the standard deviation (or standard error) is used to define the design allowable, namely,... 

The 1.7-eV CFs F excitation threshold follows from extrapolation of low-pressure data for hot H-for-H and F-for-F substitution reactions. The other excitation enemes represent lower-bound estimates obtained from standard heats of formation. Cf. References 25 and 45 ... 

Results from monitoring data for AFBi for hazelnuts, almonds and pistachios were reported as 29.4, 9,4 and 7.4 pg/kg, respeotively, with less than 3% of values below the reporting limits (LOD and LOQ), assumed to be zero for a lower-bound estimate, For the retail food survey, average AFBi and AFT levels for pistachios were reported to be 5.8 and 6.1 pg/kg, respeotively, and for almonds, 0.02 and 0.03 pg/kg, respectively the reporting limit values (LOD and LOQ) were assumed to be zero for a lower-bound estimate, where 96% of the results for pistaohios and 92% of the results for almonds were less than the LOD or LOQ. 

It is worthwhile to reemphasize that our estimate of the reduction in workers compensation costs from engaging in these HRM practices is probably a lower bound estimate of the potential benefits. To the extent that the workplace is safer, either because physical risks have been reduced or because workers are taking more appropriate safety precautions, then some other accident costs are likely to be reduced as well. Uncompensated wage loss and pain and suffering associated with... 

Figure 8. Convergence behavior of the beam model (blue line) with lower bound estimation (magenta line) and relative error (green line).

Figure 6.34. Mechanical modulus of various polymer blend phase arrangements. The cube shows a unit cell model of an idealized IPN structure. The shaded area is determined by upper- and lower-bound estimates for this model. " ...

Stegeman et al [37] have proposed a figure of merit T=2pX/n2 for a nonlinear directional coupler, with T values < 1 implying useful materials. In the case of poly(di-n-hexylsilane), our lower bound estimate on n2 yields a T value ranging from 0.1 - 1.0 for the wavelength range 580-650 nm. 

Preliminary measurements using a prism coupler yield a repeatable shift in a waveguide mode corresponding to an n2 value of =60 x 10 (cm /MW), 4 times larger than that predicted by our lower bound estimate [8]. However, spurious effects due to heating, photoinduced birefiringencc, etc. at this time cannot be conclusively ruled out, and this result is presented only as a comparison to the theoretical value. Further measurements are necessary to confirm these preliminaiy results and determine the utility of the polysilanes for practical nonlinear optical devices. 

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