Variables variation

The prescription proposed in the original Meister-Kroll-Groot [138,139] theory for hard spheres requires the determination of the local density and the averaged density as two independent variational variables by minimizing the grand potential with respect to these variables. The modification introduced by Rickayzen et al. [143,144] arises from another definition of the average density... 

For the GS, the two Hohenberg-Kohn (HK) theorems legitimize the density p(r) (a function of only 3 coordinates) as the basie variational variable henee, all terms in the GS eleetronie energy of a quantum system are frmetionals of the density ... 

The orthonormalization conditions reduce the number of independent variation variables as compared to this estimate, but do not reduce so to say the number of numbers to be calculated throughout the diagonalization procedure. 

Individual or biological variation Variability due to inherited influences, environmental and host influences... 

Each measure of an analysed variable, or variate, may be considered independent. By summing elements of each column vector the mean and standard deviation for each variate can be calculated (Table 7). Although these operations reduce the size of the data set to a smaller set of descriptive statistics, much relevant information can be lost. When performing any multivariate data analysis it is important that the variates are not considered in isolation but are combined to provide as complete a description of the total system as possible. Interaction between variables can be as important as the individual mean values and the distributions of the individual variates. Variables which exhibit no interaction are said to be statistically independent, as a change in the value in one variable cannot be predicted by a change in another measured variable. In many cases in analytical science the variates are not statistically independent, and some measure of their interaction is required in order to interpret the data and characterize the samples. The degree or extent of this interaction between variables can be estimated by calculating their covariances, the subject of the next section. 

The next step in this intra-orbit optimization procedure is the creation of an auxiliary functional [p(f, s) W] made up from the energy functional p r, s) W] plus the auxiliary conditions which must be imposed on the variational variables. Notice that there are many ways of carrying out this variation. In Sections 3.1.1,3.1.2, and 3.1.3, we treat the intra-orbit variation with respect to p(f, s), [p(r, s)]1/2 and the set of N orthonormal orbitals i(r) =1, respectively. Clearly, by setting 6[p(r, a) M] = 0, one obtains the Euler-Lagrange equations corresponding to each one of the above cases. 

As with the pi-electron model we want to treat the orbital coefficients as variation variables and use something like the Clementi-Raimondi-Slater atomic orbitals for the basis functions, or at least something like them which are easy to integrate. We want to minimize the energy by varying the values of the but we also want to maintain the orthonormality of the linear combination of basis functions as orthonormal one-electron orbitals. They are formed from linear combinations of basis... 

Equations 3.60 and3.61 are written in terms of variation variables. Variation variables represent a change from or about a steady-state level of the variable. The gain was... 

In terms of variation variables, the change in steam flow can be modelled as follows ... 

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