First variation

The first variation to be approved was the American Organic Standard, a private sector standard for North America developed in 2003 by the industry through the Organic Trade Association (lOAS 2003). 

Variation ofX Ligand. Most of the rate, spectroscopic, and structural information available on Co(III) compounds has been obtained for neutral LCo(DH)2X compounds. In reviewing this chemistry, we will consider first variations in the X ligand and then variations in the L ligand. 

Using familiar methods of the calculus of variations, one can set the first variation of the energy with respect to the orbitals and linear coefficients to zero. This leads to a set of Fock-like operators, one for each orbital. Gerratt, et al use a second-order stabilized Newton-Raphson algorithm for the optimization. This gives a set of occupied and virtual orbitals from each Fock operator as well as optimum <7,-s. 

There are two major sources of nutritional variation in full-fat soy. The first variation depends on the variety of seed used. Varieties grown in the United States show major differences in their fat and protein content when compared with varieties grown in Brazil, Argentina, and other countries. The second major source of variation is the equipment characteristics and the processing techniques. Also, the energy level in full-fat soy is influenced by the processing tech-... 

To prove this, let us consider the first variation (see Appendix D) of the minimum support functional ... 

The problem of minimization of the misfit functional (3.7) can be solved using variational calculus. Let us calculate the first variation of / (m) ... 

The problem of minimization of the weighted misfit functional can be solved by calculating the first variation of this functional and setting it equal to zero ... 

Now we should calculate the first variation of the parametric functional with respect to the perturbation of the parameters... 

The absolute value of the first variation of the misfit functional can be estimated using equation (5.8) and the Schwarz inequality (A.38) ... 

We now consider somi ai)])roaches to finding t,he mininiuni of the last functional with respect t.o A . Let us calculate the first variation of 4i(A ) and equate it to zero at a minimum point ... 

Note that, according to Theorem 81 of Appendix D, the first variation of the rni.sfit functional at the minimum must he equal to zero ... 

IaiI us find t he minimum of the last functional with respect to k . T calculate now the first variation of (A- ) ... 

To solve the problem of minimization (5.74) we have, as usual, to calculate the first variation ... 

The steepest descent method for nonlinear regularized least-squares inversion To solve the problem of minimization of the parametric functional using the steepest descent method, let us calculate the first variation of P (m,d), assuming that the operator A(m) is differentiable, so that... 

Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models... 

The first variation of the magnetic field 6Hx can be calculated from SEy using Maxwell s equations ... 

Here 8Ey, SH are the first variations of the residual electric and magnetic fields ... 

According to the integral representations (9.25) and (9.26), the first variations of the anomalous electric and magnetic fields can be calculated as... 

The first variation of the energy flow functional at the point (ct(, + Act) can be evaluated using linearization of the forward modeling operator for the predicted electric field Ey r c b + Act) ... 

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