Scaling factor, definition

Diffraction patterns may always be multiplied by a constant factor without changing the physics of the system. In MSLS this is a scaling factor which is needed to scale the observed and calculated intensities to the same order of magnitude. This scaling factor is one of the relinable parameters. However, one has to take care with the definition of this factor. It should be defined in such a way that it is not dependent on the change of any other parameter in the refinement procedure, in particularly the crystal thickness and the absorption parameter. Therefore the actual scaling factor, C, was expressed as function of c, the parameter which was used in the refinement process ... 

If sufficient data are available, substance-specific PBPK models should always be given preference over the use of general scaling factors. However, PBPK models were considered not to replace all of the sub-factors in the interspecies comparison and should, by definition, only include toxicokinetic differences. A further extrapolation factor for toxicodynamic differences between the species needs to be discussed. 

We have assumed x >= 0 without any loss of generality since the problem definition is independent of translation or scaling factors. Thus,... 

Suppose A is positive definite, symmetric, and a reasonable approximation to the inverse hessian H. Select an arbitrary positive scale factor px and set... 

Both of the xp and Xm scales are empirical approximations based on incomplete experimental data. The theoretical definition of absolute electronegativity, x = a/Eg = /Xm — > Xv has been demonstrated to account for both empirical scales. The scale factor x varies with periodic shell and //, represents the number of valence energy-level vacancies. 

A precise definition of reinforced plastics can be difficult (or impossible) to formulate because of the scale factor. At the atomic level all elements... 

For a two-parameter treatment of solvent effects (with two independent solvent vectors), only two critical subsidiary conditions must be defined in order to force the two solvent parameters to represent physically significant solvent properties. Four other trivial arbitrary conditions have to be defined in order to fix zero reference points and scale-unit sizes. However, for a three-parameter treatment (with three independent solvent vectors), already six critical subsidiary conditions must be defined, in addition to the six trivial reference or scale-factor conditions. On the contrary, singleparameter treatments require no definition of critical subsidiary conditions, but only one reference (zero) condition and one standard (unit) condition, whose arbitrary assignment changes only the reference solvent and the scale-unit size (265, 276). 

A simple inequality follows from the definition of the scaling-nesting semisimilarity measure For three objects X, Y, and Z, all of unit volume, Sf xY) scales so a version A, of it fits within Y, and s yz) scales y so a version of it fits within Z. Consequently, a scaling factor xYfyrz) certainly reduces X so that a version X of it fits within Z. However, by definition, Sf z) is the largest scaling factor of X that allows X to fit within Z. Consequently,... 

A definition of the unconstrained scaling-nesting similarity measures is given below the constrained measures can be derived easily from these by applying the appropriate constraint while determining the maximum scaling factor. 

In a linear system Ax = b where the matrix A is symmetric and positive definite, the solution is obtained by minimizing the quadratic form (12.331). This implies that the gradient, / (x) = Ax — b, is zero. In the iteration procedure an approximate solution, x +i, can be expressed as a linear combination of the previous solution and a search direction, p, which is scaled by a scaling factor am-... 

To gain insight into the structure of the dimensionless reaction-hased design equations, recall tiie definition of the characteristic reaction time, Eq. 3.5.1, ter = Co/ro. It follows that the scaling factor is ta/Cq) = I/tq, where tq is the reference rate of a selected chemical reaction. Substituting fliis relation into Eqs. 4.4.4,... 

The scaling factor fPm ., /Pi (r . )1 is included in this definition in order to provide proportionality among the actual density contributions from various molecules M,. 

Instead of trying to use the covariance itself as a standard for comparing the degree of statistical association of different pairs of variables, we apply a scale factor to it, dividing each individual deviation from the average by the standard deviation of the corresponding variable. This results in a sort of normalized covariance, which is called the correlation coefficient of the two variables (Eq. (2.9)). This definition forces the correlation coefficient of any pair of random variables to always be restricted to the [—1,+1] interval. The correlations of different pairs of variables are then measured on the same scale (which is dimensionless, as can be deduced from Eq. (2.9)) and can be compared directly. 

While a detector distribution with varying sign is physically acceptable, we may easily see that the fundamental solution of the critical adjoint equation must be nonoscillating (either positive definite or negative definite) if the fundamental neutron distribution is positive definite. The fundamental adjoint solution describes the relative contribution of neutrons at different places in the system when the ultimate progeny of these neutrons have distributed themselves into the fundamental mode and are all positive of course. Whatever the detector distribution at that time, therefore, the effect of the earlier neutron upon it varies with the location of that neutron according to some scale factor, but may not vary in sign. 

In this respect, while noting the above scaling condition (4.335) as being quite restrictive for the scaling factor q, an additional constrain that takes into consideration the number of valence electrons is to be regarded. This aim is to be accomplished observing that the valence formulation atomic electronegativity fits with the definition of chemical action of Eq. (4.277) for the Coulombic potential (March, 1993)... 

The scale and the mode of drawing are preset in the program three scale factors, 15.5 X 9.5, 31 X 19, and 62 x 38 A, and four modes, to paint circles and print out, to paint circles but not to print out, open circles and print out, and open circles but not to print out. The most appropriate selection of the scale factor and the mode of display is ascertained by successive trials. Successive displays of the sections parallel to the original one with input of the number of planes and the interval in A, or display of non-parallel sections with input of the plane definition are available. In these processes, DWA-SAN cycles run automatically and the work file for each section is memorized into the disk. 

A common problem with both Gerber 274D and drill formats (usually Excellon I and II) is the lack of definition of both units and scale factor of nnmbers (ExceUon is a precision drill machines manufacturer). Each coordinate is given as a collection of digits, and the translator, or user, must define the units (in in. or mm) and the location of the decimal points in the format. Another problem stems from the use of arbitrary numbers ( Dcodes for aperture numbers on an aperture wheel) to describe the width of lines and size of pads. Accurate translation of these files depends on the correct definition of an apertnre wheel, which is described separately. Figure 18.8 shows a photo-plotter schematic that explains the sonrce of the terminology. 

It follows that Hartree-Fock atomic wavefunctions must satisfy the virial relation. Such solutions are, by definition, the best (lowest energy) attainable in a single deter-minantal form. Best includes all conceivable variation, linear or nonlinear, so all improvements achievable by scale factor variation are already present at the Hartree-Fock level, and r) = 1. 

The approximation we make in the transformation process is to neglect both the off-diagonal terms in the transformed Hamiltonian and the residual coupling in the wave function. At the same time as neglecting 8, we also renormalize rlf. This will introduce a scaling factor, which we will absorb into the definition of and O. With these approximations, the relations between the wave functions become... 

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