Dummy variables variable

Before attempting to answer this question, let us first summarize the procedure of section 11.3 in a slightly modified form. Equations (11.20) and (11.21) provide a set of simultaneous ordinary differential equations to determine the pressure and the composition, represented by mole fractions Xi,..,Xn in terms of the dummy variable. If at least one of the x s varies monotonically with X, so that its derivative never vanishes, we may use this x in place of X as an Independent variable. Without loss of generality this x may be labelled x, so we may divide equation (11.20) and each equation (11.21) for r = 2,...,n-l, by equation (11.21)... 

The notation < i j k 1> introduced above gives the two-electron integrals for the g(r,r ) operator in the so-called Dirac notation, in which the i and k indices label the spin-orbitals that refer to the coordinates r and the j and 1 indices label the spin-orbitals referring to coordinates r. The r and r denote r,0,( ),a and r, 0, ( ), a (with a and a being the a or P spin functions). The fact that r and r are integrated and hence represent dummy variables introduces index permutational symmetry into this list of integrals. For example,... 

Thus, the value of a definite integral depends on the limits a, b, and any selected variable coefficients in the func tion but not on the dummy variable of integrations. Symbolically... 

The problem [Eq. (15)] is a minimax optimization problem. For the case (as it is here) where the approximating function depends linearly on the coefficients, the optimization problem [Eq. (15)] has the form of the Chebyshev approximation problem and has a known solution (Murty, 1983). Indeed, it can be easily shown that with the introduction of the dummy variables z, z, z the minimax problem can be transformed to the following linear program (LP) ... 

Once the particular branching process that specifies the probability measure on the set of macromolecules of a polymer specimen has been identified, the statistical method provides the possibility to determine any statistical characteristic of the chemical structure of this specimen. In particular, the dependence of the weight fraction of a sol on conversion can be calculated by formulas [extending those (55)] which are obtainable from (61) provided the value of dummy variable s is put unity ... 

The variable s is a dummy variable in the sense that it does not enter die final result. Thus, if the exponential function in Eq. (94) is expanded in a power series in s, the coefficients of successive powers of s are just the Hermite polynomials divided by u . It is not too difficult to show that Eqs. (93) and (94) are equivalent definitions of the Hermite polynomials. 

Plackett and Burman [1946] have developed a special fractional design which is widely applied in analytical optimization. By means of N runs up to m = N — 1 variables (where some of them may be dummy variables which can help to estimate the experimental error) can be studied under the following prerequisites and rules ... 

An example of a Plackett Burman plan for Z = 2 levels, m = 7 influence factors (including dummy variables) and, therefore, N = 8 runs is given in Table 5.10. 

Table 5.10. Plackett-Burman design matrix lor N 8 experiments and consequently m = 7 factors (including dummy variables) at two levels... 

From equation 14.3-19, with MA defined by equation 4.3-4. From equation 13.52, with 3.4-10 or 14.3-20 and 13.4-2. dE is an exponential integral defined by E x) = y le y dy, where y is a dummy variable the integral must be evaluated numerically (e.g., using E-Z Solve) tabulated values also exist. 

Note that s is a dummy variable the value of the integral depends only on the value of the upper limit. Tables of the error function are available and values can be calculated from power series [Dwight (1961), Kreyszig (1988)]. The error function has the properties erf(0) = 0 and erf(°°) = 1. Equation 10.31 can be written in terms of the error function as... 

The summation over the dummy variables n, m.. . can be performed explicitly and, setting... 

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