Maximum value

The maximum value of P, for any R, occurs as Py tends to -oo. Study of the Ft functions above reveals that for Ft to be determinate,... 

Thus the maximum value of P for any value of R, Pmax is given by... 

The procedure most commonly employed (NF M 07-047 or ASTM D 2274) Is to age the diesel fuel for 16 hours while bubbling oxygen into it at 95°C. The gums and sediment obtained are recovered by filtration and weighed. There is no official French specification regarding oxidation stability however, in their own specifications, manufacturers have set a maximum value of 1.5 mg/100 ml. 

It is worthwhile to mention that the distribution of naphthenic acids is not uniform in a crude oil since a maximum value is observed in the fractions distilled between 400 and 450°C and whose average specific gravity is 0.950 (Figure 8.2). 

If there is insufficient data to describe a continuous probability distribution for a variable (as with the area of a field in an earlier example), we may be able to make a subjective estimate of high, medium and low values. If those are chosen using the p85, p50, pi 5 cumulative probabilities described in Section 6.2.2, then the implication is that the three values are equally likely, and therefore each has a probability of occurrence of 1/3. Note that the low and high values are not the minimum and maximum values. 

The cumulative cashflow was used to derive ultimate cash surplus - the final value of the cumulative cashflow maximum exposure - the maximum value of the cash deficit payout time - the time until cumulative cashflow becomes positive. 

To find now the optimum excitation frequency, we calculate the first derivative of Equ. (3.1) to find the maximum value of the response field of cracks in different depths (Fig. 3,1). For example a crack (20 x 0.6 x 0,2 mm) in a depth of 9 mm in an aluminium sample (a = 20 MS/m) could be found with highest SNR when using a frequency of 260 Hz Here a double-D... 

The changing of welding method to the vibro-contact has caused a lowering of own tensile stress to 180 MPa, whereas in the joint the stress is close to zero After annealing the maximum value of stress has reached 20 MPa... 

The combination of contrast and granularity produces a signal to noise ratio which allows for direct comparison of various films. The classes have minimum values for eontrast and maximum values for graininess. The ASTM classification system employs the same parameters as the European Standard EN584-1 and ISO CD (see Table 1). 

It has been pointed out [138] that algebraically equivalent expressions can be derived without invoking a surface solution model. Instead, surface excess as defined by the procedure of Gibbs is used, the dividing surface always being located so that the sum of the surface excess quantities equals a given constant value. This last is conveniently taken to be the maximum value of F. A somewhat related treatment was made by Handa and Mukeijee for the surface tension of mixtures of fluorocarbons and hydrocarbons [139]. 

Earlier in tiiis seetion it was shown that, when a eonstraint, e.g. fixed /, was released in a system for whieh U, V and n were held eonstant, the entropy would seek a maximum value eonsistent with the remaining... 

Figure C3.6.1 (a) WR single-handed chaotic attractor for k 2 = 0.072. This attractor is projected onto tire (c, C2) plane. The maximum value reached by c (t) is 54.1 and tire minimum reached by <7 " 2.5. The vertical line, at Cj = 8.5 for < 1, shows the position of tire Poincare section of tire attractor used later, (b) A projection, onto tire cft- ),cft2)) plane, of tire chaotic attractor reconstmcted from tire set of delayed coordinates cft),cft ),c (t2), where t = t + and t2 = t -I- T2, for 0 < t < 00, and fixed delays = 137 and T2 = 200. Note tliat botli cft ) and cftf) reach a maximum of P and a minimum of <"T""so tliat tire tliree-dimensional reconstmcted attractor is...

The origin of a torsional barrier can be studied best in simple cases like ethane. Here, rotation about the central carbon-carbon bond results in three staggered and three eclipsed stationary points on the potential energy surface, at least when symmetry considerations are not taken into account. Quantum mechanically, the barrier of rotation is explained by anti-bonding interactions between the hydrogens attached to different carbon atoms. These interactions are small when the conformation of ethane is staggered, and reach a maximum value when the molecule approaches an eclipsed geometry. 

On the other hand, if we want to characteri2e objects which are described by the rather fuzzy statement "numbers dose to three", we then need a membership function which describes the doseness to three. An adequate membership function could be the one plotted in Figure 9-25 m x) has its maximum value of m x) = 1 for value x = 3. The greater the distance from x to 3 gets, the smaller is the value of m x). until it reaches its minimum m x) = 0 if the distance from x to 3 is greater than say 2, thus for x > 5 or x < 1. 

The zero prevents negative similarity. The pair of segments with the maximum similarity i< found by first locating the matrix element with the maximum value of Hj j and then tracing... 

The second-order kappa index is determined by the count of two-bond paths, written P. The maximum value is expressed by a star shape = (A — 1) (A — 2) fl) and the mini-... 

Just as e takes its maximum value when x is at a minimum, the right side of proportion (3-3) is a maximum when its exponent is a minimum. To minimize a fraction with a constant denominator, one minimizes the numerator... 

For example, in the e ease, one ean apply the projeetor to the determinant with the maximum value to obtain... 

The rate constant for the collision of two species is given by the following approximate formula, and corresponds to a maximum value of the rate constant of reaction ( 2.5, 3.3) ... 

Another method for finding the end point is to plot the first or second derivative of the titration curve. The slope of a titration curve reaches its maximum value at the inflection point. The first derivative of a titration curve, therefore, shows a separate peak for each end point. The first derivative is approximated as ApH/AV, where ApH is the change in pH between successive additions of titrant. For example, the initial point in the first derivative titration curve for the data in Table 9.5 is... 

In general, the sensitivity of FIA is less than that for conventional methods of analysis for two principal reasons. First, as with chemical kinetic methods, measurements in FIA are made under nonequilibrium conditions when the signal has yet to reach its maximum value. Second, dispersion of the sample as it progresses through the system results in its dilution. As discussed earlier, however, the variables that influence sensitivity are known. As a result the FIA manifold can be designed to optimize the sensitivity of the analysis. 

Onc-Factor-at-a-Timc Optimization One approach to optimizing the quantitative method for vanadium described earlier is to select initial concentrations for ITiOz and 1T2S04 and measure the absorbance. We then increase or decrease the concentration of one reagent in steps, while the second reagent s concentration remains constant, until the absorbance decreases in value. The concentration of the second reagent is then adjusted until a decrease in absorbance is again observed. This process can be stopped after one cycle or repeated until the absorbance reaches a maximum value or exceeds an acceptable threshold value. 

Figure 3.1 la is a plot of this solution, showing how the strain develops exponentially with time. The maximum value of the compliance is attained as t hence the upper limits of compliance is designated J(°°). Since the deformation... 

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