Root mean square from

The average value and root mean square fluctuations in volume Vof the T-P ensemble system can be computed from the partition fiinction Y(T, P, N) ... 

Example Crippen and Snow reported their success in developing a simplified potential for protein folding. In their model, single points represent amino acids. For the avian pancreatic polypeptide, the native structure is not at a potential minimum. However, a global search found that the most stable potential minimum had only a 1.8 Angstrom root-mean-square deviation from the native structure. 

Note that has the significance of being the mean value of the square of the deviations of individual Mi values from the mean M. Accordingly, a is sometimes called the root mean square (rms) deviation. 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

Figiire 8-38 ihust rates the typical spread of values of the controhed variable that might be expected to occur under steady-state operating conditions. The mean and root mean square (BMS) deviation are identified in Fig. 8-38 and can be computed from a series of n observations Cl, C9,. . . c as fohows ... 

The mean value of each of the distributions is obtained from these high, modal, and low values by the use of Eq. (9-101). If the distribution is skewed, the mean and the mode will not coincide. However, the mean values may be summed to give the mean value of the (NPV) as 161,266. The standard deviation of each of the distributions is calculated by the use of Eq. (9-75). The fact that the (NPV) of the mean or the mode is the sum of the individual mean or modal values implies that Eq. (9-81) is appropriate with all the A s equal to unity. Hence, by Eq. (9-81) the standard deviation of the (NPV) is the root mean square of the individual standard deviations. In the present case s° = 166,840 for the (NPV). 

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.

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. 

Nitrile rubbers, including fiber-reinforced varieties, are used both as radial shaft-seal materials and as molded packing for reciprocating shafts. They have excellent resistance to a considerable range of chemicals, with the exception of strong acids and alkalis, and are at the same time compatible with petroleum-based lubricants. Their working temperature range is from —1°C to 107°C (30°F to 225°F) continuously and up to 150°C (302°F) intermittently. When used on hard shafts with a surface finish of, at most, 0.00038 mm root mean square (RMS), they have an excellent resistance to abrasion. 

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... 

The importance of the root mean square speed stems from the fact that is proportional to the average kinetic energy of the molecules, [Pg.284]

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... 

How does the root mean square speed of gas molecules vary with temperature Illustrate this relationship by plotting the root mean square speed of N, as a function of temperature from T = 100 I< to T = 300 K. ... 

Root-mean-square end-to-end distance, which effectively takes account of the average distance between the first and the last segment in the macromolecule, and is always less that the so-called contour length of the polymer. This latter is the actual distance from the beginning to the end of the macromolecule travelling along the covalent bonds of the molecule s backbone. Radius of gyration, which is the root-mean-square distance of the ele-... 

Table XIV lists comparative SD and /values for fittings of all the sets of Table Xlll with each of the scales of Table V, the FandR values of Swain, and with the single substituent parameter treatment, po y These statistics, coupled with structural considerations, we believe support the usefulness and uniqueness of a scale of limited generality. In general, the / values of Table XIV for the Or scale are smaller than those of the other scales by factors of from 2 to 10. The root-mean-square F values for the other scales are from 2.25 (< j (BA)) to 3 to 4 (S L,, cr (yv)) times that for. Because this analysis has demonstrated that Swain s F and R are generally inferior for the discriminating data for all four types, there appears little to encourage proliferation of these parameters.

The value should be that of single polymer chain elasticity caused by entropic contribution. At first glance, the force data fluctuated a great deal. However, this fluctuation was due to the thermal noise imposed on the cantilever. A simple estimation told us that the root-mean-square (RMS) noise in the force signal (AF-lS-b pN) for an extension length from 300 to 350 nm was almost comparable with the thermal noise, AF= -21.6 pN. 

Turbulent mass burning rate versus the turbulent root-mean-square velocity by Karpov and Severin [18]. Here, nis the air excess coefficient that is the inverse of the equivalence ratio. (Reprinted from Abdel-Gayed, R., Bradley, D., and Lung, F.K.-K., Combustion regimes and the straining of turbulent premixed flames. Combust. Flame, 76, 213, 1989. With permission. Figure 2, p. 215, copyright Elsevier editions.)... 

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. 

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