Uncertainty exponent

For simple self-similar fractals /(e) scales like a powerlaw /(e) e", where v is called the uncertainty exponent. In this case the quantities needed in (2.3.8) for calculating the fractal dimension are easily available k = f e) N, m = 1/e. Therefore,... 

There is now agreement between experiment and theory on the Ising exponents. Indeed it is now reasonable to assume that the theoretical values are better, smce their range of uncertainty is less. 

Many of the earlier uncertainties arose from apparent disagreements between the theoretical values and experimental detemiinations of the critical exponents. These were resolved in part by better calculations, but mainly by measurements closer and closer to the critical point. The analysis of earlier measurements assumed incorrectly that the measurements were close enough. (Van der Waals and van Laar were right that one needed to get closer to the critical point, but were wrong in expectmg that the classical exponents would then appear.) As was shown in section A2.5.6.7. there are additional contributions from extended scaling. 

Moreover, some uncertainty was expressed about the applicability to fluids of exponents obtained for tlie Ising lattice. Here there seemed to be a serious discrepancy between tlieory and experiment, only cleared up by later and better experiments. By hindsight one should have realized that long-range fluctuations should be independent of the presence or absence of a lattice. 

Table A2.5.1 shows the Ising exponents for two and tluee dimensions, as well as the classical exponents. The uncertainties are those reported by Guida and Ziim-Justin [27]. These exponent values satisfy the equalities (as they must, considering the scalmg assumption) which are here reprised as functions of p and y ...

The small uncertainties in the calculated exponents seem to preclude the possibility that the d = 3 exponents are rational numbers (i.e. the ratio of integers). (At an earlier stage this possibility had been suggested, since not only die classical exponents, bnt also tlie rational numbers pre-RG calculations had suggested p = 5/16 and y = 5/4.)... 

There has for long been uncertainty concerning the appropriate value to be used for the exponent of the Schmidt number in equation 10.225. SHERWOOD, PlGFORD and WlLKE 27 ... 

That is, there would be a 10% error, or uncertainty, in the answer. Note that even though terms in the denominator have a negative exponent, the maximum error due to these terms is still cumulative, because a given error may be either positive or negative i.e., errors may either accumulate (giving rise to the maximum possible error) or cancel out (we should be so lucky ). 

The late William Horwitz, who is the leading exponent of top down measurement uncertainty, offers the following four approaches to estimate measurement uncertainty (Horwitz 2003) ... 

As pointed out by Livingstone and Imboden (1993), the change in the exponent aSc (Eq. 20-24a), when moving from the SSR to the RSR, has the blemish that the abrupt transition between different Sc-dependences must result in a discontinuity in viw for all Schmidt Numbers different from the reference number of 600. In view of the other uncertainties involved, this should, however, be of little practical significance. 

Table 3-1 summarizes rules for propagation of uncertainty. You need not memorize the rules for exponents, logs, and antilogs, but you should be able to use them. 

One can show that the block length distribution function /b( ) is characterized for asymptotically large l by the power-law decay of its density /bCO a ta with the exponent a = 3/2 [88]. The exponent a estimated from the simulation data (a % 1.6) is, up to the experimental uncertainty, quite close to that predicted by the exact analytical model. As has been noted above, such a power-law decay is a characteristic of the Levy probabilistic processes [36]. 

We now contrast the behavior of the wave packet in equation (1.31) with that of the wave packet in (1.20). At any time t, the maximum amplitudes of both occur at x = vgt and travel in the positive x-direction with the same group velocity vg. However, at that time t, the value of fiTdx, /) is 1 /e of its maximum value when the exponent in equation (1.31) is unity, so that the half width or uncertainty Ax for MT,(x, /) is given by... 

There is even more uncertainty in estimating the heat-transfer coefficient at the wall of the tube than in estimating the effective thermal conductivity in the bed of catalyst. The measurement is essentially a difficult one, depending either on an extrapolation of a temperature profile to the wall or on determining the resistance at the wall as the difference between a measured over-all resistance and a calculated resistance within the packed bed. The proper exponent to use on the flow rate to get the variation of the coefficient has been reported as 0.33 (C4), 0.47 (C2), 0.5 and 0.77 (HI), 0.75 (A2), and 1.00 (Ql). 

The size distribution of sixfold-ordered regions, n, also resembles that of the 2D WCA liquid, being well described by the Fisher droplet model. We fit rtj to Eq. (3.26) and obtained a power law exponent of = 1.35 0.02 and a correlation size of = (3 6) x 10 (the large uncertainty in is due to poor statistics in the tail of n ). This value of is similar to the values obtained for the dense time-averaged WCA liquid. The small s part of is shown in Fig. 71, together with the fit to Eq. (3.26). A extended tail (out to 5-600) is observed in the large 5... 

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

It may be concluded that the value of a will generally be 2 or 3, although intermediate values, or values slightly larger than 3, may also occur. The dependence of the modulus on cp is strong for D = 2.35, and a values of 2 and 3, the exponent of cp equals 3.1 and 4.6, respectively. Such relations are shown in Figure 17.13c (page 716). Because of the uncertainties in the relation between modulus and volume fraction, it is better to derive D values from permeability results. 

Fig. 4.7. Variation of the mean exponent with the interaction parameter, (a) for the square lattice (d — 2) (b) for the simple cubic lattice (d = 3). The dashed line gives the uncertainty range. (From Fisher and Hiley.8)...

Because or limited variation in experimental values of the Schmidt number, some uncertainty is associated with its exponent. Also, the Reyanlds number is based on gas velocity relative to the pipa wall rather than the liquid surface. The abeve correlation and data typically lie some 20% higher than the Chilton-Colbum prediction. Wetted-wall column studies by Jacksou and Caegieske4 and Johnstone and Pigford, 9 employing Re bsted on gas velocity relative to the liquid, generally are in reasonable agreement with the Chilton-Colbum analogy. 

An explanation of the erroneous estimate of stoichiometric number may be that Equations 6 and 7 contain a number of flexible parameters such as v, Kbutane, Kbutenes and the exponents of partial pressures of both butane and butenes. In addition the equation is nonlinear with respect to these parameters, and the parameters are highly correlated. In those situations it is extremely difficult to obtain precise estimates of parameters, even with well-developed nonlinear regression computer programs. Uncertainty associated with the estimate of stoichiometric number may arise from the fact that the assumptions made when Equations 6 and 7... 

---