Straight-line approximation

This instability can be eliminated by calculating the slope of the straight-line approximation not at x, where the value of y is already known, but at the future time x + delx. Therefore, instead of... 

In order to simplify calculation of 1/L, Golder s (1972) stability classes can be approximated by a single straight line of ML against surface roughness. The parameters for the straight-line approximations to the stabUity classes are shown in Table V. 

Coefficients for Straight-Line Approximation to Golder s Plot as a Function of Stability Classes... 

Figure 9. Dotted lines represent the theoretical Equation 48 relating MW/MD to b for various values of z. Solid lines are determined from straight-line approximations of the lines in...

Pressure. The effect of pressure is (by a straight-line approximation of the Kelvin equation)... 

The terms represent successive approximations to a function at small values of its argument by the value of the function at the origin, the straight-line approximation of the function at the origin, the parabolic approximation of the function at the origin, and so forth. 

Figure 77 Current-field characteristics [in log(j) F3 4) against F1/2 representation] of the DL LEDs consisting of variable concentration of TPD HTL (%TPD PC)(70nm) and a 100% evaporated Alq3 ETL (60 nm) sandwiched between an ITO anode and a Mg cathode. The slopes of straight lines approximating the experimental plots (points), a, in (cm/V)1 2 are given in the bottom-right corner. After Ref. 303. Copyright 2001 Institute of Physics (GB).

The method outlined in the preceding obviously can become very tedious mathematically, and a computer solution is normally necessary. The method also has limitations based on choice of scale and incremental steps for the variables, extrapolation past the region where the straight line approximates the surface, and inability to handle surfaces that are not unimodal. 

T) the phase angle is zero, and above 10/T the phase angle is —90°. For a series of lags the individual straight-line approximations may be added graphically. 

The theoretical description of the total cross section is easier in the semi classical limit since the angular range is indeed restricted to 9 = 0 so that the transition approximation gives results which are in quantitative agreement with those calculated by quantum mechanics. Using the parabola approximation for the phase shifts in the maximum or the straight line approximation for the deflection function at the zero point we have... 

That is, the corrector uses the predictor value at h to estimate the slope at ti which is averaged with the slope at t0 to provide the slope of the straight line approximation. A variation of this method is discussed at some length by Hamming (66). 

A plot of the slopes of the lines representing the data for (Re) r values vs. (Re) (Figure 5) falls very nearly on the straight-line approximation shown. 

Next, we note that if P is large, the distance between space-time points is very small and it is reasonable to make a straight-line approximation to the path connecting any two intermediate points (see Fig. 2). This approximation is convergent if we weight each straight line so that it contributes the proper amount to the kernel thus we obtain... 

Phase diagrams, which describe the equilibrium relations that exist between phases in mixtures, are often very complex as is the case with the nickel-aluminum diagram for example. However, if one restricts attention only to a relatively narrow range of concentration and, in particular, to dilute solutions, then one can simplify this description of these relationships markedly by using straight line approximations as shown in Figures l(a, b). ... 

Figure 4. Instanton path for tunneling between pairs of minima in effective potential at Z oo limit for with R = 5. Panel at left show contour maps of effective potential in spheroidal coordinates, at right maps in cylindrical coordinates. Heavy dots show exact instanton paths long-dashes indicate parabolic approximation short-dashes straight-line approximation.

Figure 4 shows contour plots of the effective potential and compares the exact instanton path obtained from our algorithm with the straight-line approximation [16] and with a paraboUc approximation. 

D17. On a Robeson plot (a log-log plot of selectivity versus oxygen permeability in Barrers) the upper bound for separation of oxygen from nitrogen plots as a straight line. Approximate values of the end points are for Pq2 = 0.0001 Barrers, 002. 2 = 42 and for Pq2 = 10,000 Barrers, ao2-N2 ... 

Jacobs pointed out that Ivey s relation is really a straight line approximation to a curve. He derived a relationship for the change in J -center energy with lattice parameter... 

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