Root mean squar vector

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

Derivation of the Gaussian Distribution for a Random Chain in One Dimension.—We derive here the probability that the vector connecting the ends of a chain comprising n freely jointed bonds has a component x along an arbitrary direction chosen as the x-axis. As has been pointed out in the text of this chapter, the problem can be reduced to the calculation of the probability of a displacement of x in a random walk of n steps in one dimension, each step consisting of a displacement equal in magnitude to the root-mean-square projection l/y/Z of a bond on the a -axis. Then... 

Root Mean Square Error of Cross Validation for PCA Plot (Model Diagnostic) As described above, the residuals from a standard PCA calculation indicate how the PCA model fits the samples that were used to construction the PCA model. Specifically, they are the portion of the sample vectors that is not described by the model. Cross-validation residuals are computed in a different manner, A subset of samples is removed from the data set and a PCA model is constructed. Then the residuals for the left out samples are calculated (cross-validation residuals). The subset of samples is returned to the data set and the process is repeated for different subsets of samples until each sample has been excluded from the data set one time. These cross-validation residuals are the portion of the left out sample vectors that is not described by the PCA model constructed from an independent sample set. In this sense they are like prediction residuals (vs. fit). 

The b vector chosen by the validation procedure can be employed prospectively to predict concentrations of the analyte of interest in independent data. Similar to the calculation of RMSECV, the root mean square error of prediction (RMSEP) for an independent data set is defined as the square root of the sum of the squares of the differences between predicted and reference concentrations. 

Calculate die estimated concentration vector for component A, (i) using one and (ii) using two PLS components. What is the root mean square error for prediction in each case ... 

Note that in Eq. 6.34 the mean-square displacement is used, rather than the root-mean-square displacement. For a one-dimensional random walk, the mean-square displacement is given by 2Dt, and for a two-dimensional random walk, 4Dt. Since the jump distance (a vector) is A, if the jump frequency is now defined as F = n/t (the average number of jumps per unit time), then on combining Eq. 6.33 and 6.34 gives ... 

The photoelectron wave vector is k, the magnitude of the amplitude for backscattering from the th atom is f k) and A is the electron mean free path. The phase shift 5j is the sum of contributions from the p-wave shift in the absorbing-atom s potential and the shift in the backscattering amplitude from the th atom. The final exponential term is a Debye-Waller-type factor representing the probability distribution caused by lattice vibrations and disorder, Oj being the root mean square deviation from Rj (also known as the thermal correlation factor ). 

Quantitative structure-activity relationship Root mean square error Receiver-operating characteristic Recursive partitioning Support vector machine TOPological Substructural Molecular Design Topological polar surface area... 

The quality of ACARS data had been examined by many researchers, for example, Mamrosh (1998), Schwartz and Benjamin (1995), and Lord et al. (1984). Estimated wind vector accuracy was about 1.8 m/s and estimated temperature accuracy was about 0.50°C. When ACARS was compared to radiosondes, root-mean-square (RMS) deviations were 7.4° in direction and 3.3 m/s in speed. In comparing ACARS ascent/descent winds and temperatures with radiosondes, it was found that temperature differences were less than 20°C on 94% of all occasions, and less than 10°C greater than 68% of the time. Wind speed rms deviations were 4.1 m/s, while direction rms differences were 35° (mostly due to light and variable wind situations). 

Whereas the complete RDE vector describes a probability distribution, the individual vector components are related to the relative frequencies of atom distances in the molecule. Thus, the individual g(r) values are plotted in a frequency dimension whereas r lies in the distance dimension (Eigure 5.1). The smoothing parameter B can be interpreted as a temperatnre factor that is, is the root mean square... 

We start by developing the quadratic terms in Eq. (A2.80), for a given atom. The vector components along the Cartesian coordinate scheme used to generate the root mean square displacement vectors are developed (a superscripted T denotes matrix transpose). 

The mode specific vibrational root mean squared displacement vector is thus replaced by the mean square displacement tensor of that atom in that specific mode, it has units of A. The tensor is represented in Eq. (A2.85) by a matrix, in more compact notation it is written, for a specific vibrational mode and atom Bi. The colon-operator ( ) or tensor contraction operation is found by taking the trace (Tr) of the product of the two tensors, see Eq. (2.50). Further we write the total mean square displacement tensor of an atom as the sum over all the individual vibrational contributions. 

The superposition problem can be described as follows Given two sets X, Y with n vectors each, find a transformation T = (Q, t) minimizing the root-mean-square deviation between X and the transformed vector Y ... 

As mentioned previously, CTRs arise as a result of the abrupt termination of a crystal lattice, and the diffuse diffracted intensity connects Bragg points in reciprocal space. In this case, the scattering vector is normal to the surface, and as a result, this technique is very sensitive to surface and interface roughness but not to in-plane atomic correlations. Thus, it yields information that is complementary to that obtained by grazing incidence diffraction. The most important feature of CTR is the characteristic decay of the scattered intensity described by Eq. (38). For surfaces that are not perfectly terminated (i.e., rough) the intensity will decay faster than predicted by this equation, and this can be used as a measure of root-mean-square surface roughness. 

Fundamental frequency of the resonator Correlation function for surface roughness Root mean square height of a roughness Wave vector of shear waves in quartz, (Uy pq//rq Correlation length of surface roughness Thickness of the liquid film Thickness of interfacial layer Molecular dynamics Pressure in a liquid Quartz crystal microbalance Hydrodynamic roughness factor Electrochemical roughness factor Coordinates (normal and lateral)... 

The radius of gyration is essentially the root mean-square radius of a macromolecule (Tanford, 1961). Let R be a vector locating the center of mass of a molecule and let r be a vector locating segment i with mass mi. Then by definition... 

One of the characteristic dimensions of polymer coils is the root-mean-square end-to-end distance ( r ) ), which for a linear chain of n bonds is calculated by considering the backbone bonds as vectors (6 ) [49]. 

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