Error, maximum apparent Errors

Suppose therefore that we have to multiply 3 34i6 by a 55, and suppose each of these numbers to have the maximum apparent error then the relative error in the first number is S in 230,000, and the error in the second number is 5 in 2600. Evidently, therefore, the result of the multiplication will also have an error of about 5 in 2600, or 0 2 per cent Consequently it would be quite incorrect to perform the multiplication in the ordinary manner, and write the result as S 971080 for this result has a derived error of 0 2 per cent, or of about i unit in the second place of decimals. All the figures after this are therefore meaningless, and should be discarded, the result being written 5 97. 

This example will illustrate the method. We have still to consider, however, the number of figures in the result. If we consider the two numbers to have the maximum apparent error, we see that the greatest error in divisor or dividend is about 5 in 30,000, or i in 6000. The number of significant figures in the result should therefore be such that the error is... 

The performance of the above schemes can be validated by the results of Figures 2.4(a) and 2.4(b), where the variation of the maximum L2 error norm, i.e., the global error, as a function of the At/Ah number and computational time are, respectively, illustrated. It becomes apparent that the implicit higher order schemes outperform Yee s technique, achieving fast convergence in terms of a reasonable CPU overhead. 

We have chosen ethylene and water as substances to correlate with our equation of state. The reasons for picking these substances are their considerable practical and theoretical importance, and because of the apparently poor state of correlation in the critical region for ethylene in the IUPAC tables (15) and for water in the steam tables (16). Tables I and II present the results of the correlation. The agreement between observed and calculated pressures is excellent. The maximum percentage errors are 0.005% for ethylene and 0.09% for water. Tables III and IV present the values of the various parameters used in the correlation. For... 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

When using the selectivity constant or coefficient (k) mentioned by ISE suppliers, one must be sure that if the ion under test and the interfering ion have different valence the exponent in the activity term according to Nikolski has been taken into account it has become common practice to mention the interferent concentration that results in a 10% error in the apparent ion concentration these data facilitate the proper choice of an ISE for a specific analytical problem. Often maximum levels for no interference are indicated. 

Distortion of miscibility gaps. In the case of some miscibility gaps such intersections are accompanied by an even more marked distortion (Fig. 8.9). This is now often called the Nishizawa Horn, due to the extensive work of Nishizawa and coworkers (1979, 1992) on this effect, but it is interesting to note that the effect had previously been noted by Meijering (1963). Here, too, the apparent presence of more than one maximum in a miscibility gap was believed to represent experimental error before it was shown to have a sound theoretical foun tion. 

An interesting experiment is to allow oxidative phosphorylation to proceed until the mitochondria reach state 4 and to measure the phosphorylation state ratio Rp, which equals the value of [ATP] / [ADP][PJ that is attained. This mass action ratio, which has also been called the "phosphorylation ratio" or "phosphorylation potential" (see Chapter 6 and Eq. 6-29), often reaches values greater than 104-105 M 1 in the cytosol.164 An extrapolated value for a zero rate of ATP hydrolysis of log Rf) = 6.9 was estimated. This corresponds (Eq. 6-29) to an increase in group transfer potential (AG of hydrolysis of ATP) of 39 kj/mol. It follows that the overall value of AG for oxidation of NADH in the coupled electron transport chain is less negative than is AG. If synthesis of three molecules of ATP is coupled to electron transport, the system should reach an equilibrium when Rp = 106 4 at 25°C, the difference in AG and AG being 3RT In Rp = 3 x 5.708 x 6.4 = 110 kj mol-1. This value of Rp is, within experimental error, the same as the maximum value observed.165 There apparently is an almost true equilibrium among NADH, 02 and the adenylate system if the P/O ratio is 3. 

Figure 12. Apparent growth temperatures for various Altiplano carbonates based on clumped isotope thermometry, plotted as a function of estimated maximum burial depth. Symbols discriminate among soil carbonates from sections near Callapa, Corque and Salla and lacustrine carbonates from near Tambo Tambillo, as indicated by the legend. The heavy solid line indicates an estimated burial geotherm, assuming a surface temperature of 20 °C and a gradient of 30 °C per km. The dashed lines define a 10° offset from this trend, which we consider a reasonable estimate of its uncertainty. Carbonates deposited within the last 28.5 Ma and buried to 5000 meters or less exhibit no systematic relationship between apparent temperature and burial depth, and show no evidence for pervasive resetting of deeply buried samples. Error bars are la (when not visible, these are approximately the size of the plotted symbol).

The data apparently indicate that the Q dinitrile was a primary product, and the overall selectivity at low benzene conversion (less than 7%) was around 20% the selectivity to mucononitriles (the three isomers) rapidly declined when the benzene conversion was increased above 7-8%, and finally became nil. The C4 unsaturated dinitriles, maleonitrile and fumaronitrile, reached a maximum selectivity of around 20% and 10% respectively, at conversions lower than 10%. It is not clear whether they were primary or secondary products, because of the errors made in yield and selectivity calculation for low benzene conversions (and also errors made in data extrapolation from figures). It is worth noting that the C balance was much lower than 100%. Therefore, it is likely that some products were not detected. 

---