Uniform scaling

The use of a uniform scale in partial pivoting can also significantly reduce round off eiTors (Gerald and Wheatley, 1984). 

The analysis of stresses in the laminae of a laminate is a straight-fonvard, but sometimes tedious, task. The reader is presumed to be familiar with the basic lamination principles that were discussed earlier in this chapter. There, the stresses were seen to be a linear function of the applied loads if the laminae exhibit linear elastic behavior. Thus, a single stress analysis suffices to determine the stress field that causes failure of an individual lamina. That is, if all laminae stresses are known, then the stresses in each lamina can be compared with the lamina failure criterion and uniformly scaled upward to determine the load at which failure occurs. 

This leads to a remarkable improvement over the uniformly scaled frequencies, and many programs are available for this task. 

The vibrational spectra of 1,2-dithiole-3-thione 46 and 1,2-dithiol-3-one 47 were computed at the DFT and MP2 levels (Scheme 31) [98VS77]. Most remarkably, the uniformly scaled MP2 fundamentals are in better agreement with experimental data than the corresponding DFT frequencies. 

Low cycle fatigue loading of 9.5%Cr steel at 650°C in air has been shown to enhance uniform scale formation as well as promote nodular scale forma-... 

Rao and Singh32 calculated relative solvation free energies for normal alkanes, tetra-alkylmethanes, amines and aromatic compounds using AMBER 3.1. Each system was solvated with 216 TIP3P water molecules. The atomic charges were uniformly scaled down by a factor of 0.87 to correct the overestimation of dipole moment by 6-31G basis set. During the perturbation runs, the periodic boundary conditions were applied only for solute-solvent and solvent-solvent interactions with a non-bonded interaction cutoff of 8.5 A. All solute-solute non-bonded interactions were included. Electrostatic decoupling was applied where electrostatic run was completed in 21 windows. Each window included 1 ps of equilibration and 1 ps of data... 

Here the reaction of a neutral base with water results in the formation of a cation. To compare acids and bases on a uniform scale, it is convenient to use the acidity constant of the conjugate acid (i = BH+) as a measure of the base strength ... 

Recall from Section 2.10 that the characteristic feature of extensive properties XL is their uniform scaling with respect to the size of the system, expressible in terms of a multiplicative positive scale factor A. Re-sizing the macroscopic system merely means that all extensive properties are multiplied by the common scale factor A,... 

The majority of the location factors shown in Figure 29 are self-explanatory. While no uniform scale is required when using the pairwise comparison approach in a site selection situation, it has proven helpful to create a joint definition of each attribute that also contains an indication of minimum requirements and ideal state together with all stakeholders involved in the site selection process. To this end, verbal scale descriptions were defined for all sub-objectives. The global scale to use the model for site controlling (see below) was then created based on these definitions. 

On the other hand, upon uniform scaling of the electron density, Eq.(2), Ec[n] does not scale homogeneously. Instead, Ec[n] satisfies the following high-density relationship [10-13] ... 

The dominant role of the h -hardest mode, a = 15, immediately follows from its wx value and the On vector coefficients (see Fig. 2). The same conclusion, therefore, must follow from the relevant resolution of the pure CT-vector, given by the 0 transformation, uniformly scaled by the factor 1 /m. This feature of the a = 15 mode is also seen in the (AIM + PNM) resolution of the FF vector, shown in Fig. 3b, where all but last CT-active modes mainly describe the CT-induced polarization (explicitly extracted in Fig. 3c). 

The spectrum below was obtained with a more drastic alteration in conditions. What can cause the non-uniform scaling of the /-splittings ... 

In order to use Eq. (6) in electrochemical studies of ion solvation, the problems related to the liquid junction potential have been presented in Sec. 2.2.2. Equation (11) may also be used in such studies, but the measured potentials should be expressed versus the same solvent-independent reference electrode. Such electrodes, which give a basis for the formation of a uniform scale of electrode potentials in different solvents, are available. A scale of this kind is also needed for a correlation of equilibrium potentials (E° 1/2) of electrode systems in various solvents. 

Consider two bounded and closed 3D objects, A and B°. If V(A) and V(B°) denote the volumes of these objects, then first a uniform scaling... 

The uniform scaling s(B°) can be characterized by a single scaling factor s ... 

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