Root count

Fundamental Theorem of Algehra Eveiy polynomial of degree n has exactly n real or complex roots, counting multiplicities. 

Kinetic polynomial as the generalized overall reaction rate equation 2.2.4.7 The root count. The reader interested first in deriving explicit reaction rate equation may omit this section and start to read Section 2.2.4.T)... 

Estimate (43) could be generalized to the multi-route mechanism — we can drop at least P reactions from Bezout root count. 

In older, less systematic nomenclature, the root counted all the carbons. As an example, butane meant C4H10, with the suffix (word ending) ane indicating an alkane. However, there is more than one structure with... 

By about 24 days almost all of the stocks showed some leafing. For each of the sample groupings the ten best stocks, i.e., ten stocks showing the most leafing, were removed from the perlite and the roots counted. The top achievers were the polymers and the lowest the monomers (Table 5). Again, visual observation showed that the roots associated with the polymer-dipped stocks were markedly superior with respect to size and length. Callus formation was consistently good with the polymer-dipped samples. In fact, it appears that some roots formed v ithout the need for prior callus formation. 

Flow cytometer cell counts are much more precise and more accurate than hemocytometer counts. Hemocytometer cell counts are subject both to distributional (13) and sampling (14—16) errors. The distribution of cells across the surface of a hemocytometer is sensitive to the technique used to charge the hemocytometer, and nonuniform cell distribution causes counting errors. In contrast, flow cytometer counts are free of distributional errors. Statistically, count precision improves as the square root of the number of cells counted increases. Flow cytometer counts usually involve 100 times as many cells per sample as hemocytometer counts. Therefore, flow cytometry sampling imprecision is one-tenth that of hemocytometry. 

The richer a plant part is in nutrients and the slower the drying process, the higher the bacterial count of the resulting drug root drugs, which of course are to begin with more heavily contaminated, always have a higher bacterial count than do flowers, for example, which are a less suitable as a nutrient medium. 

FMEA is particularly suited for root cause analysis and is quite useful for environmental qualification and aging analysis. It is extensively used in the aerospace and nuclear ]iowei indiistrii-s but seldom used in PSAs, Possibly one reason for this is that FMEA, like parts count. ,s not chrectlv suita lundant systems such as those that occur in nuclear power plants Table i 4... 

Traditionally, column efficiency or plate counts in column chromatography were used to quantify how well a column was performing. This does not tell the entire story for GPC, however, because the ability of a column set to separate peaks is dependent on the molecular weight of the molecules one is trying to separate. We, therefore, chose both column efficiency and a parameter that we simply refer to as D a, where Di is the slope of the relationship between the log of the molecular weight of the narrow molecular weight polystyrene standards and the elution volume, and tris simply the band-broadening parameter (4), i.e., the square root of the peak variance. 

Consider, for definiteness, a set of otherwise identical lowest-level components of a system, so that the hierarchy is a tree of constant depth. Since we assume that the components are all identical, the only distinction among the various nodes of the hierarchy consists of the structure of the subtrees. Now suppose we have a tree T that consists of /3 subtrees branching out from the root at the top level. We need to determine the number of different interactions that can occur on each level, independent of the structure of each subtree i.e. isomorphic copies of trees do not contribute to our count. We therefore need to find the number of nonisomorphic subtrees. We can do this recursively. 

Under the simplest conditions, the standard counting error is approximately equal to the square root of the total number of counts or... 

The applicability of the Poisson distribution to counting statistics can be proved directly that is, without reference to binomial theorem or Gaussian distribution. See J. L. Doob, Stochastic Processes, page 398. The standard deviation of a Poisson distribution is always the square root of its mean. 

These data are plotted in Figure 10-3 about the Gaussian curve for which the standard deviation is the square root of the mean. The data of Rutherford and Geiger, which were obtained by counting alpha-particles, are plotted about the same curve. In the figure, both sets of data fit the Gaussian about equally.well. 

For the usual accurate analytical method, the mean f is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating at random. The standard deviation, s, depends upon these small causes and may assume any value mean and standard deviation are wholly independent, so that an infinite number of distribution curves is conceivable. As we have seen, x-ray emission spectrography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve, not an error curve in the strictest sense there is no true value of N such as that presumably corresponding to a of Section 10.1—there is only a most probable value N. 

Seed germination bioassay of root exudates. Bioassay results are presented as a 23 week mean for each germination count time (Table III, IV, V, VI). Means were separated by LSD after data normalization by the inverse sine transformation. 

Tossing a mental coin, the decision was to analyze the case of noise proportional to the square root of the signal. This, as you will recall, is Poisson-distributed noise, characteristic of the noise encountered when the limiting noise source is the shot noise that occurs when individual photons are detected and represent the ultimate sensitivity of the measurement. This is a situation that is fairly commonly encountered, since it occurs, as mentioned previously, in UV-Vis instrumentation as well as in X-ray and gamma-ray measurements. This noise source may also enter into readings made in mass spectrometers, if the detection method includes counting individual ions. We have, in... 

Fibre class Role Activation threshold Conduction velocity Cell body diameter Cell count in dorsal root ganglia... 

Each abundance was divided by the abundance of that element (except for Rh) in Type / carbonaceous chondrites. Rh abundances were divided by Rh abundances in other types of chondrites as Cl values were not available. Errors in the LBL measurements reflect 1 a values of the counting errors, except for the Au error. The latter is the root-mean-square deviation of six measurements, because the six values were not consistent within counting errors. The Os measurement was on a HNO,-insoluble residue that had been fired to 800°C. Key , this work and O, previous work of Ganapathy. 

The Tb abundance in meteorites is assumed to be 0.5 ppm. Errors for the Danish Cretaceous (3 samples) and Tertiary (3 samples) HNO,-insoluble residues are root-mean-square deviations. Errors for the HNO,-insoluble residues from the Gubbio and Danish boundary layers are 1 a values of the counting errors. Key , Gubbio boundary layer residue O, Danish Cretaceous residues < >, Danish Tertiary residues and , Danish... 

Figure 8. Root Mean Square Error (RMSE) and Revisit Count for one vs. two step ahead beam and waveform scheduling.

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