Errors constant

We name Kerr the position error constant.1 For the error to approach zero, Kcrr must approach infinity. In Example 5.1, the error constant and steady state error are... 

In many control texts, we also find the derivation of the velocity error constant (using R = s-2) and acceleration error constant (using R = s-3), and a subscript p is used on what we call Kerr here. 

Based on the analysis presented in ref. 108 we conclude that the error constant s smallest value can be obtained for a = 0. [... 

Actually, in most cases, a correction is not necessary because the error resulting from the buoyancy will cancel out in percent composition calculations. The same error will occur in the numerator (as the concentration of a standard solution or weight of a gravimetric precipitate) and in the denominator (as the weight of the sample). Of course, all weighings must be made with the materials in the same type of container (same density) to keep the error constant. 

There are several ways to reduce both type I and type II errors available to researchers. First, one can select a more powerful statistical method that reduces the error term by blocking, for example. This is usually a major goal for researchers and a primary reason they plan the experimental phase of a study in great detail. Second, as mentioned earlier, a researcher can increase the sample size. An increase in the sample size tends to reduce type II error, when holding type I error constant that is, if the alpha error is set at 0.05, increasing the sample size generally will reduce the rate of beta error. 

Quantitation by the standard addition technique Matrix interferences result from the bulk physical properties of the sample, e.g viscosity, surface tension, and density. As these factors commonly affect nebulisation efficiency, they will lead to a different response of standards and the sample, particularly with flame atomisation. The most common way to overcome such matrix interferences is to employ the method of standard additions. This method in fact creates a calibration curve in the matrix by adding incremental sample amounts of a concentrated standard solution to the sample. As only small volumes of standard solutions are to be added, the additions do not alter the bulk properties of the sample significantly, and the matrix remains essentially the same. Since the technique is based on linear extrapolation, particular care has to be taken to ensure that one operates in the linear range of the calibration curve, otherwise significant errors may result. Also, proper background correction is essential. It should be emphasised that the standard addition method is only able to compensate for proportional systematic errors. Constant systematic errors can neither be uncovered nor corrected with this technique. 

The reality can well be approximated by keeping an unknown systematic error constant in each simulation, but vary it between simulations. Examples are probe diameters, temperatures and temperature gradients, calibration-uncertainties of material measures, etc. 

Xt is interssfdng to caaapare the static velocity error constants for the tso systess designed above. 

Two methods of the same order differ by their error constant, which is defined to be the constant Cp i of the first non vanishing term in the Taylor expansion of l x t),h)/a l). (The normalization hy = cr(l) is performed to get a constant which is independent of the scaling of the method.) For the implicit two-step Adams... 

Figure 4.1 Absolute value of error constants for different multistep methods. (For fc = 1 the Adams-Moulton method is the implicit Euler method.)...

Dielectric constants of metals, semiconductors and insulators can be detennined from ellipsometry measurements [38, 39]. Since the dielectric constant can vary depending on the way in which a fihn is grown, the measurement of accurate film thicknesses relies on having accurate values of the dielectric constant. One connnon procedure for detennining dielectric constants is by using a Kramers-Kronig analysis of spectroscopic reflectance data [39]. This method suffers from the series-tennination error as well as the difficulty of making corrections for the presence of overlayer contaminants. The ellipsometry method is for the most part free of both these sources of error and thus yields the most accurate values to date [39]. 

Figure B2.5.19. The collisional deactivation rate constant /c, (O3) (equation B2.5.42 ) as a fimction of the vibrational level v". Adapted from [ ]. Experimental data are represented by full circles with error bars. The broken curve is to serve as a guide to the eye.

Figure B3.2.11. Total energy versus lattice constant of gallium arsenide from a VMC calculation including 256 valence electrons [118] the curve is a quadratic fit. The error bars reflect the uncertainties of individual values. The experimental lattice constant is 10.68 au, the QMC result is 10.69 (+ 0.1) an (Figure by Professor W Schattke).

The results of the optimization for 9 small test proteins, both for the potential with constant weights 1 and with the optimized weights, are given in Table 1. The optimized weights lead to smaller errors the resulting potentials have minima within 1.3-4.7A of the PDB geometry, with one exception that has an error of 8.5A. 

We first note errors in total energy means that are not greater than 0.5% for all LN versions tested. Individual energy components show errors that are generally less than 1%, with the exception of the van der Waals energy that can reach 4% for large k2. Of course, this discussion of relative errors reflects practical rather than mathematical considerations, since constants can be added to individual terms without affecting the dynamics. The relative errors... 

Jacobian is a constant. Since our knowledge on the properties of the errors is limited, our guess better be simple. 

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

The essential assumption of this manuscript is the existence of a constant variance of Gaussian errors along the trajectory. While we attempted to correlate the variance with the high frequency motions, many uncertainties and questions remain. These are topics for future research. 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

From the derivation of the method (4) it is obvious that the scheme is exact for constant-coefficient linear problems (3). Like the Verlet scheme, it is also time-reversible. For the special case A = 0 it reduces to the Verlet scheme. It is shown in [13] that the method has an 0 At ) error bound over finite time intervals for systems with bounded energy. In contrast to the Verlet scheme, this error bound is independent of the size of the eigenvalues Afc of A. 

By using an effective, distance-dependent dielectric constant, the ability of bulk water to reduce electrostatic interactions can be mimicked without the presence of explicit solvent molecules. One disadvantage of aU vacuum simulations, corrected for shielding effects or not, is the fact that they cannot account for the ability of water molecules to form hydrogen bonds with charged and polar surface residues of a protein. As a result, adjacent polar side chains interact with each other and not with the solvent, thus introducing additional errors. 

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