Multiplier terms

If there is more than one constraint, one additional multiplier term is added for each constraint. The optimization is then performed on the Lagrange function by requiring that the gradient with respect to the x- and A-variable(s) is equal to zero. In many cases the multipliers A can be given a physical interpretation at the end. In the variational treatment of an HF wave function (Section 3.3), the MO orthogonality constraints turn out to be MO energies, and the multiplier associated with normalization of the total Cl wave function (Section 4.2) becomes the total energy. 

In this least squares method example the object is to calculate the terms /30, A and /J2 which produce a prediction model yielding the smallest or least squared differences or residuals between the actual analyte value Cj, and the predicted or expected concentration y To calculate the multiplier terms or regression coefficients /3j for the model we can begin with the matrix notation ... 

The zth row of the pre-factor is multiplied term-by-term with the jth column of the post-factor and added to yield the ijth element of the product. 

For a uni-univalent electrolyte, the multiplier term in (6.24) cancels out and the liquid junction potential can be calculated from the equivalent ionic conductivities Xi of the two solutions, from the Sargent equation. 

Units energy per unit mass of fluid flowing [J kg-1, m2 s 2], To get rates [J s-1, W] multiply terms by mass flow rate, G]... 

Lagrange multipliers Terms in a method used to find the maximum or minimum of a function that is subject to constraints. The function is written so that... 

If there is more than one constraint, one additional multiplier term is added for each... 

For fixed normalization the Lagrange multiplier terms in 8Ts vanish. If these constants are undetermined, it might appear that they could be replaced by a single global constant pt. If so, this would result in the formula [22] 8Ts = J d3r p, — v(r) 8p(r). Then the density functional derivative would be a local function vr(v) such that STj/Sp = Vj-(r) = ix — v(r). This is the Thomas-Fermi equation, so that the locality hypothesis for vT implies an exact Thomas-Fermi theory for noninteracting electrons. 

Based on equation (3.328) increasing R is connected with a steep decrease in U due to the EJE multiplying term in the... 

Individual pathlength data is used as a scalar multiplier term for correction of individual or groups of spectra. 

Alternatively, the terms can be rewritten by multiplying terms together, although the number of independent dimensionless parameters stays the same. Thus, Eq. (6) can be rewritten, for example, as... 

If we compare equation (6-17) and (6-21), we see that the differences occur in the multiplier term (1.62 versus 0.023), Reynolds number power ( versus 0.8), and D/L power ( versus 0). The Prandtl number power (5) is the same for both. A phenomenological approach to these effects indicate that the change in flow from laminar to turbulent is reflected both in the larger power on the Reynolds number and the elimination of the D/L effect (transformation from layers to vortices and eddies). Also, the change in flow does not alter the Prandtl number power. 

These expressions are obtained directly from the functional form of the I agrangian as given in (14.1.43), by collecting terms of order n. In this process, we have made repeated use of the fact that the projection operator P projects away the zero-order parameters and multipliers We now observe that, for the optimal choices of C(a) and C(a), the l agrangian (14.1.43) and the original energy expression (14.1.39) coincide for all a since the multiplier terms in the... 

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