Deviations propagating

Figure 4. Plots of sediment-accumulation rate versus age for the cores in Figure 3. Error bars represent one standard deviation propagated from counting uncertainty. The apparent increase in recent sediment accumulation in Little Rock Lake is atypical of other cores from this site.

Appendix 6. Equations for Count Rates, Standard Deviation, Propagation of Error, and Weighted Averages... 

When we have N measures for the exit variables in a process, the technical problem of identification of the unknown parameter resides in solving the equation (p) = 0. From the theoretical viewpoint, all the methods recommended for the solution of the transcendent equation can be used to determine parameter p. The majority of these methods are of iterative type and require an expression or an evaluation of the (p) derivate. When we evaluate the derivate numerically, as in the case of a complex process model, then important deviations can be introduced into the iteration chain. Indeed, the deviation propagation usually results in an increasing and non-realistic value of the parameter. This problem can be avoided by solving the equation (p) =0 by integral methods such as the method of minimal function value (MFV). When (p) values are only obtained in the area of influence of parameter p, the MFV method is reduced to a dialogue with the mathematical model of the process and then the smallest (p) value gives the best value for the parameter. 

There are several cases where NMR spectroscopy has been used to investigate copolymers which deviate from the terminal model for copolymerisation (see also chapter 3). For example, Hill and co-workers [23, 24] have examined sequence distributions in a number of low conversion styrene/acrylonitrile (S/A) copolymers using carbon-13 NMR spectroscopy. Previous studies on this copolymer system, based on examination of the variation of copolymer composition with monomer feed ratio, indicated significant deviation from the terminal model. In order to explain this deviation, propagation conforming to the penultimate (second-order Markov) and antepenultimate (third-order Markov) models had been proposed [25-27]. Others had invoked the complex participation model as the cause of deviation [28]. From their own copolymer/comonomer composition data. Hill et al [23] obtained best-fit reactivity ratios for the terminal, penultimate, and the complex participation models using non-linear methods. After application of the statistical F-test, they rejected the terminal model as an inadequate description of the data in comparison to the other two models. However, they were unable to discriminate between the penultimate and complex participation models. Attention was therefore turned to the sequence distribution of the polymer. 

It performs control equipment, Deviation propagate safety measures... 

The probability of consequence event in the scenario tree is calculated according to equation (5). The probability of consequence events is obtained by summing the probability each scenario including the safety measures. Two deviations propagate in the scenario tree in Figure 7. The deviations are... 

Breindl et. al. published a model based on semi-empirical quantum mechanical descriptors and back-propagation neural networks [14]. The training data set consisted of 1085 compounds, and 36 descriptors were derived from AMI and PM3 calculations describing electronic and spatial effects. The best results with a standard deviation of 0.41 were obtained with the AMl-based descriptors and a net architecture 16-25-1, corresponding to 451 adjustable parameters and a ratio of 2.17 to the number of input data. For a test data set a standard deviation of 0.53 was reported, which is quite close to the training model. 

Propagation of uncertainty allows us to estimate the uncertainty in a calculated result from the uncertainties of the measurements used to calculate the result. In the equations presented in this section the result is represented by the symbol R and the measurements by the symbols A, B, and C. The corresponding uncertainties are sr, sa, sb) and sq. The uncertainties for A, B, and C can be reported in several ways, including calculated standard deviations or estimated ranges, as long as the same form is used for all measurements. 

Determine the density at least five times, (a) Report the mean, the standard deviation, and the 95% confidence interval for your results, (b) Eind the accepted value for the density of your metal, and determine the absolute and relative error for your experimentally determined density, (c) Use the propagation of uncertainty to determine the uncertainty for your chosen method. Are the results of this calculation consistent with your experimental results ff not, suggest some possible reasons for this disagreement. 

The propagation of polymer chains is easy to consider under stationary-state conditions. As the preceding example illustrates, the stationary state is reached very rapidly, so we lose only a brief period at the start of the reaction by restricting ourselves to the stationary state. Of course, the stationary-state approximation breaks down at the end of the reaction also, when the radical concentration drops toward zero. We shall restrict our attention to relatively low conversion to polymer, however, to avoid the complications of the Tromms-dorff effect. Therefore deviations from the stationary state at long times need not concern us. 

We now identify the increments Ax and Ay as deviations in x and y. Then Eq. (2-61) reveals, to a good approximation, how these deviations are propagated into the deviation AF in the function F. Squaring Eq. (2-61) gives... 

As for the dependence of the polymerization rate V on the monomer concentration some authors have also found first-order kinetics (84, 90, 96, 99), but sometimes deviations from the first order were observed (38, 51, 88) that may be connected with a change in the number of propagation centers with monomer concentration. 

Of all the requirements that have to be fulfilled by a manufacturer, starting with responsibilities and reporting relationships, warehousing practices, service contract policies, airhandUng equipment, etc., only a few of those will be touched upon here that directly relate to the analytical laboratory. Key phrases are underlined or are in italics Acceptance Criteria, Accuracy, Baseline, Calibration, Concentration range. Control samples. Data Clean-Up, Deviation, Error propagation. Error recovery. Interference, Linearity, Noise, Numerical artifact. Precision, Recovery, Reliability, Repeatability, Reproducibility, Ruggedness, Selectivity, Specifications, System Suitability, Validation. 

Restricted diffusion, correlated motion of spins, or any deviation from a free behavior of the molecules will result in a propagator shape different from a Gaussian one. A wide range of studies have dealt with such problems during the last two decades and NMR has turned out to be the method of choice for quantifying restricted diffusion phenomena such as for liquids in porous materials or dynamics of entangled polymer molecules. 

It is not always possible to tell strictly the difference between random and systematic deviations, especially as the latter are defined by random errors. The total deviation of an analytical measurement, frequently called the total analytical error , is, according to the law of error propagation, composed of deviations resulting from the measurement as well as from other steps of the analytical process (see Chap. 2). These uncertainties include both random and systematic deviations, as a rule. 

A realistic uncertainty interval has to be estimated, namely by considering the statistical deviations as well as the non-statistical uncertainties appearing in all steps of the analytical process. All the significant deviations have to be summarized by means of the law of error propagation see Sect. 4.2. 

Where no KIE is present, the measurement of the 5 value is a result of two measurements, each with their own associated precision. During correction for the added derivative carbon, where cr is the standard deviation associated with a given 5 determination, the errors propagate according to Equation (14.4). 

A seemingly innocent change in the procedures, e.g. in parts, in staffing, was, almost without exception, the alteration that initiated the propagation and escalation of the deviation. Had the corresponding control loop not been ineffective then the escalation most likely would have been most likely prevented. 

In a real experiment with MatLab, the mean CSr was 79.8 ppm and the standard deviation 20.4ppm. The computed mean l/CSr was 0.0136ppm-1 and the standard deviation 0.0045 ppm -. These estimates are significantly biased relative to the linear propagation theory which would predict 0.0125 and 0.0031, respectively. This example shows that linear propagation should be applied with utmost care when a variable depends on another through a strongly non-linear relationship. <=... 

The orientation of primary pores is in general in the <100> direction for all the PS formed on all types of (100) substrates.7,14,84 For the PS with dendritic structure, as shown in Figure 12, pores propagate along (100) direction even on the (110) and (111) substrates.20,84,94 The branched pores of non-dendritic types formed on (100) substrate may not be strictly perpendicular to the primary pores but deviate to various extents from the <100> direction toward to the source of holes.12,14,95 Ronnebeck et al.95 found that the macro pores formed on... 

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