Errors proportional

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

The integral and derivative modes are normally used in conjunction with the proportional mode. Integral action (or automatic reset) gives an output which is proportional to the time integral of the error. Proportional plus integral (PI) action may be represented thus ... 

We have already observed phase modulation in the acquisition of data in a ID spectrum, where finite rf power results in off-resonance magnetizations giving rise to a phase error proportional to the off-resonance frequency (see Eq. 2.55), and similar effects occur in the t2 dimension of 2D spectra. Here our interest is in modulation that arises in the f, dimension. A simple example (but with no utility as a 2D pulse sequence) is 90°, t2, which resembles a ID experiment in which... 

The scanty results in the literature for percentages of the slag that has reacted (Table 9.3) show wide discrepancies. The extents to which these result from differences between slags, experimental errors, proportions in the... 

Beam shear Over large OPDs, or in critical applications, imperfections in the wavefronts convolved with misalignment of test and reference beams can introduce errors proportional to the wavefront shear. 

This is a low-order (two-point) approximation with error proportional to AX, but it is quite accurate as long as AX is sufficiently small. As an... 

The necessity of using weighted fits can be seen visually from Fig. 1, which shows envelopes of probable error for data obeying the linear Lineweaver-Burk equation, for the cases where the initial rates have either equal standard errors or the standard errors proportional to initial rates of reaction. When the standard error is proportional to Uoi the variance is proportional to u and the estimation of kinetic parameters by linear regression becomes more reliable (Fig. 1). 

Extrapolating (k —fc)-times, yields derivative approximations with an approximation error proportional to - the minimal required order and... 

Proportion of type 1 errors Proportion of program beneficiaries who are not poor... 

Proportion of type 2 errors Proportion of the poor who do not benefit from the program... 

Up to an error proportional to the square of quantities assumed to be small, these expressions are identical to those for perfectly elastic spheres as given, for example, in Section 16.34 of Chapman and Cowling (1970). Consequently, so also are the expressions for the pressure tensor and the energy flux vector calculated by employing the velocity distribution function (4.7) and Enskog s extension of the assumption of molecular chaos. 

By this analysis, we see that at every time step, there is a new local error, proportional to (Af)2, introduced into our numerical state trajectory. ... Over the course of... 

Both of these one-sided difference approximations have errors proportional to Ax, the spacing between successive grid points. Subtracting the two,... 

This equation has an error proportional to (Ax) as opposed to (Ax) for the second-order central difference of Eq. (5.4). More accurate formulations of a derivative expression such as this are necessarily going to involve function evaluations at more points to gather more information on the function. Thus one approach is to pursue such higher order derivative expressions. 

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