Distribution long-tailed

This function is shown in Figure 15.9. It has a sharp first appearance time at tflrst = tj2. and a slowly decreasing tail. When t > 4.3f, the washout function for parabohc flow decreases more slowly than that for an exponential distribution. Long residence times are associated with material near the tube wall rjR = 0.94 for t = 4.3t. This material is relatively stagnant and causes a very broad distribution of residence times. In fact, the second moment and thus the variance of the residence time distribution would be infinite in the complete absence of diffusion. 

Every minimization departs from an initial estimation for the vector field. The minimizations were carried out with a starting configuration obtained by randomizing the coefficients aj and a the resulting vector field has no preferential orientation and the distribution of curvature in the simulation box exhibits a long tail mainly due to abrupt changes in the direction of the stream lines (see Fig. 3.2A). 

Fig. 3.2.B). For small goal curvatures it is difficult to eliminate the long-tail part of the curvature distribution.

FIG. 10. Theoretical calculations reveal that in the case of adsorption of Xe on Ni the resonance associated with Xe(6s) state is broadened significantly with a long tail that extends to the Ni Fermi level. STM images are determined by the LDOS at the Fermi level. Although the contribution of Xe to the LDOS is small, it significantly extends the spatial distribution of the electronic wave function further away from the surface thereby acting as the central channel for quantum transmission to the probe tip. (From Ref. 71.)... 

The disadvantage of R-factor is that the same R factor value may not indicate the same level of fit depending on the noise in the experimental data. The other difference is that the R-factor is based on an exponential distribution of differences. This makes the R factor a more robust GOT against possible large differences between theory and experiment. The exponential distribution has a long tail compared to the normal distribution. R-factor is used extensively in crystallographic methods. 

The figure only shows the far left-hand side of the distribution, which is very long tailed (i.e, Zipfian) in nature. In the part shown, it can be seen that, for instance, there are 2118 of the three-point pharmacophores that are represented by only one molecule and 1396 of those that have only two molecules to represent them. Because of the Zipfian nature of the distribution, the figure is not the best way to convey the contents of the entire data set. This is better represented in Table 3.1. In this representation, the curve of Figure 3.2 is in effect summed in logarithmic portions from the right-hand... 

The distribution of the particle size must be as tight as possible, especially for hard materials such as alumina. A typical particle size distribution curve for alumina used in tungsten slurries is shown in Fig. 2a (data taken by the laser scattering method). However, from time to time the particle size distribution in a slurry may be out of control, so that a long tail will appear at the large particle end (Fig. 2b). These large particles may become a source of CMP scratches. 

Fig. 2. Typical particle size distribution of silica in oxide CMP slurry (a) normal and (b) abnormal with a long tail.

These large groups of particles are not desirable in CMP slurry. They will cause scratches and show an additional peak on the particle size distribution curve (see Fig. 6). To fix these problems, milling and/or slurry filters can be used. Milling is used at the point of slurry manufacture, and filtration is used at the point of use (filtration can also be used to fix the long tail problems mentioned in Fig. 2b), as discussed in the following. 

This is usually indicated by a residence time distribution which rises to a maximum very much before o = 1 and is followed by a long tail due to the exchange of tracer with this stagnant region. This particular fault is especially serious in wash tanks and skim tanks where the ratio of height to diameter is small. 

Regions of stagnancy are often caused by baffles and by interference due to pipes and fittings in corners and other places where abrupt changes in flow paths can occur. It is evidenced principally by long tails in the residence time distribution curve and in extreme cases, by a mean residence time which is very much shorter than that calculated by the volume divided by the flow rate. 

Finally, if the 95th percentile is highly unstable or uncertain because of high variability (for example, if the distribution has a long tail), a more stable percentile would be preferable. 

Retention data that after a possible delay in concentration show a sharp decline followed by a long tail would be modeled by is 2 and h(t > h > h+. The condition h- > h+ ensures that the drift of the random walk (or diffusion) is away from the reflecting barrier. Figure 9.9 illustrates the probability profiles in the distribution and elimination compartments when m = 20, is = 15, h+ = 0.1,... 

Erlang- and phase-type distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where particles move between a series of compartments, so that phase-type compartmental retention-time distributions can be incorporated simply by increasing the size of the system. This class of distributions is sufficiently rich to allow for a wide range of behaviors, and at the same time offers computational convenience for data analysis. Such distributions have been used extensively in theoretical studies (e.g., [366]), because of their range of behavior, as well as in experimental work (e.g., [367]). Especially for compartmental models, the phase-type distributions were used by Faddy [364] and Matis [301,306] as examples of long-tailed distributions with high coefficients of variation. 

The six aligned sequences in Fig. 1 were used to build amino add parsimony trees by PROTPARS in PAUP.13 An exhaustive search of the 105 alternative unrooted trees was performed the resulting distribution of possible trees was skewed14 positively, with a long tail (not shown) containing the shortest tree. This shortest unrooted tree (Fig. 2) requires 601 amino acid replacements the next shortest tree requires 625 events. In this case, the most parsimonious tree appears rather trustworthy. 

The distribution of collocated differences for the daily and weekly measurements were used to detect outliers. The distributions for the three precipitation types at each site were symmetrical with long tails. In this study, the tails were truncated at 3a from the mean for each observable, and pairs of samples with the extreme (i.e. minimum and maximum) differences were rejected from parametric statistical analyses according to the following criteria ... 

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