Root-mean-square value

We have assumed that there are M values of x, aird i/ iir the data sets. This correiatior function can be normalised to a value between —1 and +1 by dividing by the root-mean-square values of z and y ... 

If the magnitude of the fluctuating velocity component is the same in each of the three principal directions, the flow is termed isotropic. If they are different the flow is said to be anisotropic. Thus, if the root mean square values of the random velocity components... 

Table IV lists comparative SD and / values for fittings of all of the reactions of Table II and III with each of the gr scales derived in this paper. The comparison includes figures for fittings with F and R values of Swain and Lupton (S L) and fitting with the Hammett equation. We believe the results given in Table IV provide a clear confirmation of the uniqueness and limited generality of the o/2(ba) scale. Very consistently, the fit achieved by the or (BA) scale is shown in Table IV to be superior by significant factors to that achieved by any of the other scales or by the simple op treatment. This fact is clearly reflected in the overall / values and the similarly weighted root-mean-square / values, F = y/lfnif lN, sum taken over all reactions. The value of F is. 067 for the basis sets of Table II (compare with overall / of. 058). The comparable F values are. 140 for Or, . 088 for S L, and. 155 for 0(p) with the data differences as explained in Table IV. For all sets of Table IV, the corresponding figures are. 073 for o (ba). -143 for a%, . 097 for S L, and. 209 for 0(py...

The statistical distribution of r values for long polymer chains and the influence of chain structure and hindrance to rotation about chain bonds on its root-mean-square value will be the topics of primary concern in the present chapter. We thus enter upon the second major application of statistical methods to polymer problems, the first of these having been discussed in the two chapters preceding. Quite apart from whatever intrinsic interest may be attached to the polymer chain configuration problem, its analysis is essential for the interpretation of rubberlike elasticity and of dilute solution properties, both hydrodynamic and thermodynamic, of polymers. These problems will be dealt with in following chapters. The content of the present... 

Thus the root-mean-square value of the projection is given by... 

On the other hand, the correction factor by which W r) is altered through this refined treatment, namely, exp[ —(9n/20)(r/r, ) ] from Eq. (16), depends both on n and on r/Vm If the distance of separation of the ends of the chain lies in the vicinity of its root-mean-square value, i.e., if r / then... 

Secondly come methods depending on W asastjema s Criterion (36) that the distance D should be split into structural radii in the ratio of the Electron Cloud radii y at approximately their root-mean-square values yrms-... 

The root-mean-square value of the radial velocity fluctuation [ C y] /C varies only slightly with fraction voids and appears to be independent of particle density. 

A typical freely jointed chain will therefore be quite compact since the root-mean-square value of its end-to-end length, J(ft2) = f N b, will be small compared with its length if it were stretched out (i.e., Nb, when N is large). Figure 10.9 shows a simulated molecule of polyethylene, (-CH2-CH2-)/v, which approximates a freely jointed configuration. 

VAR (var). A unit of reactive power. The. reactive power at the por t of entry of a single-phase two-wire circuit when the product of (a) the rms (root mean square) value in amperes of the sinusoidal current, (b) the rms value in volts of the voltage, and (c) the sine of the angular phase difference by which the voltage leads the current is equal to J. (The SI llnil of reacLive power.)... 

Fig. 5.16. AFM images (lxl pm2) of the surface morphology and root mean square value of the surface roughness Rrms for 1pm ZnO films deposited at Ts = 200° C at different oxygen partial pressure...

For copper, the electrode burn-off versus arcing time has been measured for different short-circuit currents up to 15 kA (root mean square value), see Fig. 6.156. 

The observed ft . are minimized root mean square values based upon the spectroscopic data reported by Ciampolini et al. [66]. The individual crystal field contributions ft .(CF) are listed for each calculated ft together with the particular ligand polarization contribution ft (LP). The calculated total ft .(tot) sums the crystal field (CF) and ligand field (LF) contributions with the CF-LP cross-term. 

If a flow in the tank is turbulent, either because of high power levels or low viscosity, then a typical velocity pattern at a point would be illustrated by Fig. 3. The velocity fluctuation i can be changed into a root mean square value (RMS), which has great utility in estimating the intensity of turbulence at a point. So in addition to the definitions above, based on average velocity point, we also have the same quantities based on the root mean square fluctuations at a point. We re interested in this value at various rates of power dissipation, since energy dissipation is one of the major contributors to a particular value of RMS v. ... 

Figures 33.13 shows the topography of the two plasma polymer layers deposited under different conditions on a polished iron surface. Both films show a similar topography as observed by atomic force microscopy, but the film deposited on O2 plasma-pretreated polished iron showed a little more grainy surface than (Ar + H2) plasma-pretreated sample. In Figure 33.14 the root mean square value is plotted against the film thickness. The grainy surface (O2 plasma pretreated), which showed a higher deposition rate, increased the roughness as the thickness increased as expected.

The results indicate that some surfactants can definitely lead to better surface finish upon polish. Table 5 gives the average RMS (root mean square) values, Ra, of surface roughness obtained with the Dektak profilometer. The measurements are carried out on both the copper disks and 4 inch copper wafers. Dramatic improvements are noticed in polished surface uniformity when surfactants Brij 35 and SDS are employed in the slurry. The destabilizing... 

The root-mean-square value can be calculated from the probability distribution, as shown in Appendix C.6. For a symmetrical distribution, this would be... 

Here, N is the total number of time steps R, the root mean square value of the data about its mean, is aimed at reflecting the size of the manifold, and A,/ is... 

Although Eq. (f) is derived assuming energy conservation in the electron-transfer step, it does not require that the inner-sphere and solvent modes conserve energy separately. Instead one mode can borrow energy from the other. This is allowed for by Ae. (The average value of A is zero and its root mean square value is 16.7 kJ mol for [Fe(H20) p -[Fe(H20)jP exchange at room T. 

It is the fluctuating element of the velocity in a turbulent flow that drives the dispersion process. The foundation for determining the rate of dispersion was set out in papers by G. 1. Taylor, who first noted the ability of eddy motion in the atmosphere to diffuse matter in a manner analogous to molecular diffusion (though over much larger length scales) (Taylor 1915), and later identified the existence of a direct relation between the standard deviation in the displacement of a parcel of fluid (and thus any affinely transported particles) and the standard deviation of the velocity (which represents the root-mean-square value of the velocity fluctuations) (Taylor 1923). Roberts (1924) used the molecular diffusion analogy to derive concentration profiles... 

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