Minimum deviation

It is only natural that computers present a powerful method for determining yield stress with the help of Eq. (3). A corresponding program of computations is based on such a selection of constants b, n, and m that provides a minimum deviation of the theoretical curve from experimental points. 

In the REC model, the ligand is modelled through an effective point charge situated in the axis described by the lanthanide-coordinated atom axis, at a distance R, which is smaller than the real metal-ligand distance (Figure 2.6). To account for the effect of covalent electron sharing, a radial displacement vector (Dr) is defined, in which the polar coordinate R is varied. At the same time, the charge value (q) is scanned in order to achieve the minimum deviation between calculated and experimental data, whereas 9 and cp remain constant. 

Na+ CH30COCH3 214) RoNa+ = 3.00 60.0 minimum deviates from the ester resp. amide plane). 

Figures 7 and 8 plot deviations of total energies from FCI results for the various methods. It is clear that the CASSCF/L-CTD theory performs best out of all the methods smdied. (We recall that although the canonical transformation operator exp A does not explicitly include single excitations, the main effects are already included via the orbital relaxation in the CASSCF reference.) The absolute error of the CASSCF/L-CTD theory at equilibrium—1.57 mS (6-31G), 2.26 m j (cc-pVDZ)—is slightly better than that of CCSD theory—1.66m j (6-31G), 3.84 m j (cc-pVDZ) but unlike for the CCSD and CCSDT theories, the CASSCF/L-CTD error stays quite constant as the molecule is pulled apart while the CC theories exhibit a nonphysical turnover and a qualitatively incorrect dissociation curve. The largest error for the CASSCF/L-CTD method occurs at the intermediate bond distance of 1.8/ with an error of —2.34m (6-3IG), —2.42 mE j (cc-pVDZ). Although the MRMP curve is qualitatively correct, it is not quantitatively correct especially in the equilibrium region, with an error of 6.79 mEfi (6-3IG), 14.78 mEk (cc-pVDZ). One measure of the quality of a dissociation curve is the nonparallelity error (NPE), the absolute difference between the maximum and minimum deviations from the FCI energy. For MRMP the NPE is 4mE (6-3IG), 9mE, (cc-pVDZ), whereas for CASSCF/ L-CTD the NPE is 5 mE , (6-3IG), 6 mE , (cc-pVDZ), showing that the CASSCF/L-CTD provides a quantitative description of the bond breaking with a nonparallelity error competitive with that of MRMP.

Measurement of refraction angles, such as the angle of minimum deviation of a prism n is obtained from Snell s law. This requires samples of high transparency (k = 0). 

A liquid prism was created on a PDMS chip for detection based on absorption and refractive index shift. The liquid prism was formed by filling a hollow triangular-shaped chamber with a liquid sample. Excitation and emission were arranged at the minimum deviation configuration. At a low concentration of fluorescein (< 100 pM), excitation light from an optical fiber was absorbed by the molecule, but there was no shift in the excitation maximum. In this absorption-only mode, the LOD of fluorescein was 6 pM. At higher concentration (i.e., 100-1000 pM), there is an additional shift in the excitation maximum This leads to a much sharper decrease in the measured intensity, which is more than can be accounted for simply by the absorption effect [714]. 

We have investigated the transient spectra of DMEP solutions in the delay interval —2 to 10 ps for further excitation frequencies in the range 3430-3555 cm-1. The data (not shown) are consistently fitted with a set of Lorentzian-shaped spectral lines (transient holes and inverse holes) by determining the minimum deviation numerically (65). Some results are summarized in Table 3. The transition frequency of holes (v = 0 v =... 

In preparing the tabulation, a literature search was conducted to identify data source publications (1-13). Both experimental values for the property under consideration and parameter values for estimation of the property are included in the source publications. The publications were screened and copies of appropriate data were made. These data were then keyed into the computer to provide a database of values for compounds for which experimental data are available. The database also served as a basis to check the accuracy of the estimation methods. Upon completion of data collection, estimation of the values for the remaining compounds was performed. The numerous point values were processed using a computer program for minimum deviation. 

As usual, the least squares procedure is employed to determine the values of the regression coefficients a, b, c, d, etc., giving the minimum deviation between the observed data and the model. Also, we impose the criterion that subsequent... 

Microscopic medium effects are usually regarded as small and the question of which data points to include in the linear correlation is best dealt with by inspection. The minimum deviation between observed and predicted rate constants is generally accepted to be between one and two orders of magnitude if a different mechanism is to be assigned. 

These equations are obtained by writing an expression for the sum of the squares of the deviations of the data points from the line of Eq. (H), differentiating this equation with respect to both a and b, setting the differentiated equations equal to zero to apply the minimum-deviation restriction, and then solving the two equations for the values of a and b. 

Simultaneously satisfying each of these objectives is never possible therefore, tuning is a compromise. For example, tuning for minimum deviation from setpoint for normal disturbances is contrary to tuning the controller to remain stable for major disturbances. That is, if the controller is tuned for normal disturbances, the closed-loop system may go unstable when a major disturbance enters the process. On the other hand, if the controller is tuned for the largest possible disturbance, control performance is likely to be excessively sluggish for normal disturbance levels. 

For this system we have already published earlier results with the SS-MRCEPA methods. For the sake of completeness, we summarize the comparative performance of the various SS-MRPT methods vis-a-vis the SS-MRCEPA methods. Similar to the H4 model, for this system the performance of the SS-MRCEPA(O) with respect to the PCI values is slightly better than the CEPA(I) counterpart [59]. Por example, for the SS-MRCEPA (0) method, the maximum and minimum deviations are of the order of 0.2 and 1.8 mH, respectively (except at reaction point P, the deviation is of the order of 4.6 mH) with the small average deviation of the order 1.5 mH while that for the SS-MRCCEPA (I) method. 

The measurement accuracy was experimentally determined by positioning the tracer in known locations inside the bed (Moslemian, 1987). The apparent tracer positions were calculated from the linear regression formulation Eq. 9.13, based on the measured detector count rates and the predetermined calibrations. Sufficient data were taken at each location to allow for statistical determination of mean and standard deviations of the tracer position and velocity. In general, the axial errors were often greater than the radial errors because of the longer axial distance of the bed that the detectors had to monitor. For an empty bed, the mean axial error in determining the tracer position was 4.7 mm and the mean radial error was 3.9 mm The corresponding standard deviations were 1.6 mm in the axial direction and 1.2 mm in the radial direction. These deviations were due to the statistical nature of the radiation detection and are the minimum deviations obtainable for the tracer position. The measured mean axial and radial velocities were approximately zero (<1 cm/s) at all locations inside the bed. However, the standard deviations of the velocities were 7.6 cm/s in the... 

Water Solute in Hydrocarbon-Rich Vapor and Liquid. The pure component parameters of water solute AP(TC) and a were determined by using Equations 5 and 16 to fit the gas-phase volumetric properties of steam (5) and the second virial cross coefficients of steam and light gases such as methane, ethane, and nitrogen (6). The least-squares minimization technique was used to find the parameters that gave the minimum deviations between calculated and experimental pressures and second virial cross coefficients. (Table I lists the parameters for pure steam and of other compounds used in this study.)... 

Determining the volumetric probing deviation P and comparing it to MPEp A calibrated test sphere, with a nominal diameter between 10 and 50 mm, is measured using 25 measurement points distributed on its surface, as prescribed by the standard. A Gaussian least squares sphere is calculated based on these 25 coordinate measurements. The value of P is the distance between the maximum and minimum deviation from the calculated sphere. The value describes the deviation of form measurements, e.g., flatness or cylindricity deviation in single-point modus. 

The value THP is the distance between the maximum and minimum deviation regarding the form error of the CMM in scanning mode. 

The function of a die is to accept the available melt from an extruder and deliver it to takeoff equipment as a shaped profile (film, sheet, pipe, filament, etc.) with minimum deviation in cross-sectional dimensions and a uniform output by weight, at the fastest possible rate. A well-designed... 

Figure 3-7 shows a trace of the path of a monochromatic beam of light through a prism at the angle of minimum deviation. At minimum deviation the path through the prism is parallel to the base of the prism and angles I l = 1 2 and rj = r2-Snell s law gives... 

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