Mean square formulae

The mean squared formula for diffusion tells us that a micron-sized particle will diffuse a distance of around 1 nm in 1 /r in water at room temperature. But this formula ignores the inertia of the particle and of the fluid displaced by it. It has been shown both theoretically and experimentally (see, e.g., [7] and references therein) that at such short timescales, both the inertia of the particle and the fluid displaced by it matter and that the central assumption in the mean squared formula - that of uncorrelated kicks - does not hold. In a short enough time,... 

The dlffuslvltles parallel to the pore walls at equilibrium were determined form the mean square particle displacements and the Green-Kubo formula as described In the previous subsection. The Green-Kubo Formula cannot be applied, at least In principle, for the calculation of the dlffuslvlty under flow. The dlffuslvlty can be still calculated from the mean square particle displacements provided that the part of the displacement that Is due to the macroscopic flow Is excluded. The presence of flow In the y direction destroys the symmetry on the yz plane. Hence the dlffuslvltles In the y direction (parallel to the flow) and the z direction (normal to the flow) can In principle be different. In order to calculate the dlffuslvltles the part of the displacement that Is due to the flow must of course be excluded. Therefore,... 

The molecular susceptibility Xm gives therefore directly the root-mean-square radiusyrms of the electron-cloud in 1918 and earlier years this formula provided one of the best ways of estimating atomic radii" (14) which are obviously electron-cloud radii y and not structural radii r. 

This handy energy formula requires only the NMR spectra of the molecules under semtiny. A comparison made for a group of 19 molecules indicated a root-mean-square deviation of 0.25 kcal/mol relative to experimental data, whereas the rms deviation amounts to 0.21 kcal/mol for calculations made with the theoretical Ai and A2 parameters, with nonbonded energies deduced directly from Eq. (10.3). 

K, the static disorder is certainly maintained. The results are presented as plots of formula in Fig. 7. The deviations from linearity of the plots is small enough to support such method of analysis. The slopes of the curves give the 5a values tabulated in Table 4. It follows that in the (1 x l)Co/Cu(lll) case the anisotropy of surface vibrations clearly appears in the measured values of 8a and 5aT There are two reasons for such anisotropy the first is a surface effect due to the reduced coordination in the perpendicular direction. cF is a mean-square relative displacement projected along the direction of the bond Enhanced perpendicular vibrational amplitude causes enhanced mean-square relative displacement along the S—B direction. The second effect is due to the chemical difference of the substrate (Fig. 8). S—B bonds are Co—Cu bonds and the bulk Co mean-square relative displacement, cr (Co), is smaller than the bulk value for Cu, aJ(Cu). Thus for individual cobalt-copper bonds, the following ordering is expected ... 

From the mean square (MS) values and the formulas for the expected mean square (see Table 3.30) one can calculate the variance components. For the example of Figure 3.5 this gives ... 

For acyclic graphs only, if in the above formula only distances between endpoints are taken into account, the related endpoint mean square topological index Df1 results. 

Formulas for the mean-square radii of various branched and ringed polymer molecules are developed under the usual assumptions regarding the statistics of chain configuration. For branched molecules, it Is shown that the mean square radii vary less rapidly with molecular weight than for strictly linear molecules, while for systems containing only rings and unbranched chains the variation is more rapid than for the linear case. 

Recently a new analytic approach has been initiated by Edwards36 using a self-consistent field treatment similar to that of Hartree for atomic wave functions. (Edwards also gives a more rigorous justification in terms of functional integration.) The resulting formula for mean square end-to-end distance is... 

If one now chooses the appropriate induced dipole model, Eqs. 4.1 through 4.3, or a suitable combination of these, with N parameters po, >7, R0,. .., and one has at least N theoretical moment expressions available, an empirical dipole moment may be obtained which satisfies the conditions exactly, or in a least-mean-squares fashion [317, 38]. We note that a formula was given elsewhere that permits the determination of the range parameter, 1/a, directly from a ratio of first and zeroth moments it was used to determine a number of range parameters from a wide selection of measured moments [189]. In early work, an empirical relationship between the range parameter and the root, a, of the potential is assumed, like 1/a 0.11 a. That relationship is, however, generally not consistent with recent data believed to-be reliable. 

Root-mean-square deviation o, defined according to the formula... 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

Non-uniform scalar quantization. While usually non-uniform scalar quantization is applied to reduce the mean squared quantization errors like in the well known MAX quantizer, another possibility is to implement some default noise shaping via the quantizer step size. This is explained using the example of the quantization formula for MPEG Layer 3 or MPEG-2 Advanced Audio Coding ... 

Assuming that the deposition process is dominated by equilibration rather than kinetic trapping effects, Rivetti et al. have performed statistical analysis of the chain conformation in terms of the mean-square end-to-end distance, , and the mean-square bend angle, <02> [493]. Both values are well described theoretically for both two-dimensional and three-dimensional states [494-498]. For a two dimensional system, the following formula can be used ... 

If we desire to study the effects of two independent variables (factors) on one dependent factor, we will have to use a two-way analysis of variance. For this case the columns represent various values or levels of one independent factor and the rows represent levels or values of the other independent factor. Each entry in the matrix of data points then represents one of the possible combinations of the two independent factors and how it affects the dependent factor. Here, we will consider the case of only one observation per data point. We now have two hypotheses to test. First, we wish to determine whether variation in the column variable affects the column means. Secondly, we want to know whether variation in the row variable has an effect on the row means. To test the first hypothesis, we calculate a between columns sum of squares and to test the second hypothesis, we calculate a between rows sum of squares. The between-rows mean square is an estimate of the population variance, providing that the row means are equal. If they are not equal, then the expected value of the between-rows mean square is higher than the population variance. Therefore, if we compare the between-rows mean square with another unbiased estimate of the population variance, we can construct an F test to determine whether the row variable has an effect. Definitional and calculational formulas for these quantities are given in Table 1.19. 

ISOLDE facility at CERN [EBE85a]. Systematical data on magnetic dipole moments, electric quadrupole moments and isotope shifts were obtained in the mass range A=105 to A=127. The mean square charge-radii, deduced from measured isotope shifts, show clearly onset and disappearance of deformation, if compared to a simple two-parameter formula (see Fig.2). The lat-... 

At the same time, we remark that if we estimate the difference between the two undertaken fittings in terms of statistical moments, the deviations that occur are rather moderate. Indeed, evaluating mean volume and mean-square volume from formulas using the numbers from Table I (entries 1 and 2), one gets for the... 

Some multilayered HTSC, for example Bi2Sr2CaCu20x, show an anisotropy of the elastic moduli inherent for layered crystals, and negative thermal expansion in a direction within the layer [16], which can be described by formula (2). At the same time for multilayered structures such as HTSC 1-2-3, where the interlayer interaction between all layers is of the same order, the intralayer interaction essentially varies from one layer to another layer. Local anisotropy of chain type is characteristic for layers with weak intralayer interaction (a layer of the rare earth and a layer of chains Cu-O). In these layers the root-mean-square displacement of atoms in a direction within the plane is beyond the classical limit at lower temperatures, and is appreciably higher than the root-mean-square... 

The formulae (1.20) allow one to estimate the mean square radius of gyration of the macromolecule... 

Equations (3.17), (3.25) and (3.29) define the dependence of the parameters on the length of a macromolecule due to empirical evidence. The above-written relations are applicable to all linear polymers, whatever their chemical structure is. One can also define these quantities as functions of concentration. Indeed, one can see that the parameters B and E can be written as functions of a single argument. Actually, since the above kinetic restrictions on the motion of a macromolecule are related to the geometry of the system, the only parameters in this case are the number of macromolecules per unit volume n and the mean square end-to-end distance (f 2), while (see formulae (1.4) and (1-33))... 

There are no major difficulties in calculating the mean square normal coordinate when more general formulae (4.28) and (4.29) for the functions Mu(s) and /ij/(s) are used. In this case three sets (branches) of relaxation times... 

We use the formula to write down the general expression for the mean square displacement of a particle in an arbitrary viscoelastic liquid... 

The mean square displacement A of the centre of mass of a chain (thick solid line) and the mean square displacement Ajg/2 of the central particle are measured in units of the intermediate length . The curves are calculated according to formulae (5.5) and (5.13) for the values of the parameters B = 100 x = 10-2. The displacement of the centre of mass does not depend on parameter ip, but the mean square displacement of the internal particles does. The values of parameter ip are shown at the curves for Ajg/2- The picture demonstrates the existence of the universal intermediate scale . Adapted from the papers of Pokrovskii and Kokorin (1985) and Kokorin and Pokrovskii (1990). 

Formula (2.8) is an obviously convenient expression for the distribution function. From it one can find the mean square deviation of xo. Obviously the mean value of x is x0, from the symmetry of Eq. (2.8). Then, using Eq. (2.1), we have... 

Thermal noise — originates from the thermal agitation of charge carriers (- electrons, -> ions, etc.) in a - resistor. It exists even in the absence of current flow and can be described by the formula (/thermal = (4kB TRAf)1/2. [/thermal is the average amplitude of this noise (also denoted [/rms (or Vrms), see also - root mean square), k is the -> Boltzmann constant, R is the resistance, and A/ is the bandwidth of measurement frequencies. 

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