Root mean square measure

The RMS (root mean square measurement of the molecular radius) versus the molar mass gives an indication of polymer molecular shape and branching, where a value of 1 indicates an unbranched, rod shaped molecule 0.5 a random coil 0.3 a spherical structure and 0.1 a highly branched molecule. Guayule stem rubber seems to form a random coil with few branch points, whereas Hevea rubber appears to be a more compact and branched polymer [Table 5 (12)]. Guayule latex rubber from the roots and stem bases seems more like the Hevea rubber in shape than the stem latex rubber, but it is not known whether this is a reflection of its location or of its greater age. 

The ternary diagrams shown in Figure 22 and the selectivi-ties and distribution coefficients shown in Figure 23 indicate very good correlation of the ternary data with the UNIQUAC equation. More important, however, Table 5 shows calculated and experimental quarternary tie-line compositions for five of Henty s twenty measurements. The root-mean-squared deviations for all twenty measurements show excellent agreement between calculated and predicted quarternary equilibria. 

This subroutine also prints all the experimentally measured points, the estimated true values corresponding to each measured point, and the deviations between experimental and calculated points. Finally, root-mean-squared deviations are printed for the P-T-x-y measurements. 

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

For multi-dimensional potential energy surfaces a convenient measure of the gradient vector is the root-mean-square (RMS) gradient described by... 

Time series plots give a useful overview of the processes studied. However, in order to compare different simulations to one another or to compare the simulation to experimental results it is necessary to calculate average values and measure fluctuations. The most common average is the root-mean-square (rms) average, which is given by the second moment of the distribution. 

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

Figure 14 Measures of disorder m the acyl chains from an MD simulation of a fluid phase DPPC bilayer, (a) Order parameter profile of the C—H bonds (b) root-mean-square fluctuation of the H atoms averaged over 100 ps.

Which measure of scatter is likely to be larger, the mean absolute error or the root-mean-square error ... 

The root-mean-square error (RMS error) is a statistic closely related to MAD for gaussian distributions. It provides a measure of the abso differences between calculated values and experiment as well as distribution of the values with respect to the mean. 

Audible sound has a frequency range of approximately 20 Hertz (Hz) to 20 kilohertz (kHz) and the pressure ranges from 20 x 10 N/M to 200 N/M. A pure tone produces the simplest type of wave form, that of a sine wave (Figure 42.1). The average pressure fluctuation is zero, and measurements are thus made in terms of the root mean square (rms) of the pressure variation. For the sine wave the rms is 0.707 times the peak value. 

Amplitude can be measured as the sum of all the forces causing vibrations within a piece of machinery (broadband), as discrete measurements for the individual forces (component), or for individual user-selected forces (narrowband). Broadband, component, and narrowband are discussed in Section 43.8 Measurement classifications. Also discussed in this section are the common curve elements peak-to-peak, zero-to-peak, and root-mean-square. 

Another characteristic of a polymer surface is the surface structure and topography. With amorphous polymers it is possible to prepare very smooth and flat surfaces (see Sect. 2.4). One example is the PMIM-picture shown in Fig. 7a where the root-mean-square roughness is better than 0.8 ran. Similar values are obtained from XR-measurements of polymer surfaces [44, 61, 62], Those values compare quite well with observed roughnesses of low molecular weight materials. Thus for instance, the roughness of a water surface is determined by XR to 0.32 nm... 

This important result is used to find the root mean square speeds of the gas-phase molecules at any temperature (Fig. 4.25). We can rewrite this equation to emphasize that, for a gas, the temperature is a measure of mean molecular speed. From... 

Experience has shown that correlations of good precision are those for which SD/RMS. 1, where SD is the root mean square of the deviations and RMS is the root mean square of the data Pfs. SD is a measure equal to, or approaching in the limit, the standard deviation in parameter predetermined statistics, where a large number of data points determine a small number of parameters. In a few series, RMS is so small that even though SD appears acceptable, / values do exceed. 1. Such sets are of little significance pro or con. Evidence has been presented (2p) that this simple / measure of statistical precision is more trustworthy in measuring the precision of structure-reactivity correlations than is the more conventional correlation coefficient. 

Some alternative method had to be devised to quantify the TCDD measurements. The problem was solved with the observation, illustrated in Figure 9, that the response to TCDD is linear over a wide concentration range as long as the size and nature of the sample matrix remain the same. Thus, it is possible to divide a sample into two equal portions, run one, then add an appropriate known amount of TCDD to the other, run it, and by simply noting the increase in area caused by the added TCDD to calculate the amount of TCDD present in the first portion. Figure 9 illustrates the reproducibility of the system. Each point was obtained from four or five independent analyses with an error (root mean square) of 5-10%, as indicated by the error flags, which is acceptable for the present purposes. 

Figure 9. Linearity of response and reproducibility. The error flags indicate the root mean square error for five measurements at each value. The average relative error is about 10%.

Figure 3. Partition coefficient of freely jointed chains between the bulk solution and a cylindrical pore. The chains have different numbers of mass-points (n) and different bond lengths, and are characterized by the root-mean-square radius of gyration measured in units of the pore radius. See text for details.

The root-mean-square distance Vr separating the ends of the polymer chain is a convenient measure of its linear dimensions. The dissymmetry coefficient will be unity for (VrV 0< l and will increase as this ratio increases. 

Apart from their utility in determining the correction factor 1/P( ), light-scattering dissymmetry measurements afford a measure of the dimensions of the randomly coiled polymer molecule in dilute solution. Thus the above analysis of measurements made at different angles yields the important ratio from which the root-mean-square... 

Thus we may retain the root-mean-square end-to-end distance as a measure of the size of the random-coiling polymer chain, and the parameter jS required to characterize the spatial distribution of polymer segments (not to be confused with the end-to-end distribution) can be calculated from It should be noted that the r used here... 

Statishcal criteria of Eq. (24) are too good the standard deviation, which was created on the basis of different measurements by various authors, is much less than even the experimental error of determinahon. This could be due to mutual intercorrelation of descriptors leading to over-ophmistic statistics [18]. Another reason may be the lack of diversity in the training set. The applicahon of the solvation equation to data extracted from the MEDchem97 database gave much more modest results n = 8844, = 0.83, root mean square error = 0.674, F = 8416... 

Often, one needs to compare different 3D structures or conformations of a molecule. That is done internally by the 3D stmcture generation program to weed out too similar conformations of fragments. Another aspect is the need of the computational chemist to compare different generated or experimental structures. A well-established measure is the so-called root mean square (RMS) value of all atom-atom distances between two 3D structures. The RMS value needed here is a minimum value achieved by superimposing the two 3D structures optimally. Before calculating the RMS, the sum of interatomic distances is minimized by optimizing the superimposition in 3D. 

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