Method alternating direction

The matte can be treated in different ways, depending on the copper content and on the desired product. In some cases, the copper content of the Bessemer matte is low enough to allow the material to be cast directly into sulfide anodes for electrolytic refining. Usually it is necessary first to separate the nickel and copper sulfides. The copper—nickel matte is cooled slowly for ca 4 d to faciUtate grain growth of mineral crystals of copper sulfide, nickel—sulfide, and a nickel—copper alloy. This matte is pulverized, the nickel and copper sulfides isolated by flotation, and the alloy extracted magnetically and refined electrolyticaHy. The nickel sulfide is cast into anodes for electrolysis or, more commonly, is roasted to nickel oxide and further reduced to metal for refining by electrolysis or by the carbonyl method. Alternatively, the nickel sulfide may be roasted to provide a nickel oxide sinter that is suitable for direct use by the steel industry. 

The alternating direction method can be used for elhptic problems by using sequences of iteration parameters (Refs. 106 and 221). The method is well suited to transient problems as well. 

To avoid such small time steps, which become smaller as Ax decreases, an implicit method could be used. This leads to large, sparse matrices rather than convenient tridiagonal matrices. These can be solved, but the alternating direction method is also useful (Ref. 221). This reduces a problem on an /i X n grid to a series of 2n one-dimensional problems on an n grid. 

In the alternate approach, the amide formation is performed on para-nitrobenzenesulfonyl chloride (91). Reduction by either chemical or catalytic methods affords directly the desired product (90). ... 

Since the migration reaction is always toward the end of a chain, terminal alkenes can be produced from internal ones, so the migration is often opposite to that with the other methods. Alternatively, the rearranged borane can be converted directly to the alkene by heating with an alkene of molecular weight higher than that of the product (17-14). Photochemical isomerization can also lead to the thermodynamically less stable isomer. ... 

THE ALTERNATING DIRECTION METHOD (THE LONGITUDINAL-TRANSVERSE SCHEME)... 

Then problems (29)"(30) and (31)-(32) can be solved by the alternating direction method in just the same way as was done in Section 1 for problem (46)-(47) using Gaussian elimination along the rows as well as along the columns of the grid W/j. 

Thus, three different examples of interest bring out the indisputable merit of the alternating direction method and unveil its potential. From what has been said above it follows that the factorized schemes find some range of applications solely in rectangles and parallelepipeds and no more. The only exception is the case B = BiB, Ba = E + tRc, where Ri and R2 are triangle operators, by means of which it is possible to generate a lower-order approximation only under the condition r = 0 h" ). 

The problem statement. First of all, it should be noted that it is impossible to generalize directly the alternating direction method for three and more measurements as well as for parabolic equations of general form. Second, economical factorized schemes which have been under consideration in Section 2 of the present chapter are quite applicable under the assumption that the argument x = (xq, x, ., Xp) varies within a parallelepiped. 

Alternating direction methods as additive schemes. The starting point for special investigations is the alternating direction method developed in Section 1 ... 

In the two-dimensional case the iterative alternating direction method or the direct decomposition method turns out to be more economical, but for the multidimensional Dirichlet problem ATM is the most economical one among other available methods. This advantage is stipulated by the special structure of the operator A (see Chapter 4, Section 5) ... 

The alternating direction method for solving the difference Dirichlet problem in a rectangle. So far we have considered the Dirichlet problem for Poisson s equation in a rectangle G = (0 < < / , a = 1, 2) ... 

By analogy with the nonstationary heat conduction equation the iterative process (2) is referred to as the alternating direction method (ADM), the convergence of which can be established on account of the... 

An alternative approach to the finite element approach is one, introduced as a concept by Courant as early as 1943 [197], in which the total energy functional, implicit in the finite element method, is directly minimized with respect to all nodal positions. The approach is conjugate to the finite element method and merely differs in its procedural approach. It parallels, however, methods often used in atomistic modeling schemes where the potential energy functional of a system (e. g., given by the force field ) is minimized with respect to the position of all (or at least many) atoms of the system. A simple example of this emerging technique is given below. 

For all these reasons, some European regulations foresee the use of methods alternative to animal testing, such as the REACH legislation [2], and actually the cosmetics directive [3] foresees the complete ban of animal tests for cosmetics by 2013. 

However, this request of more animal testing faces several issues. There is an ethical concern on the millions of animals used every year for experiments. These tests are also those more expensive, and thus this poses questions about the costs for these experiments and the resources to cover them. Many of these tests, especially the chronic ones, require long times, years in some cases. The number of available laboratories in Europe to cover this potential request is insufficient. For all these reasons, some European regulations foresee the use of methods alternative to animal tests, such as the REACH legislation, and actually the cosmetics directive foresees the complete ban of animal tests for cosmetics by 2013. 

Further developments of this method were directed at broadening the range of organosilicon compounds capable of participation in cross-coupling, as well as at developing alternative (fluoride-free) activation methods. 

The Department of the Environment UK [155] has described a number of alternative methods for the determination of total oxidised nitrogen (nitrate and nitrite) in aqueous solution, while specific methods for nitrate and nitrite are also included. Among the methods for total oxidised nitrogen, one is based on the use of Devarda s alloy for reduction of nitrate to ammonia, and another uses copperised cadmium wire for reducing nitrate to nitrite, which is determined spectrophotometrically. Nitrate may also be determined spectrophotometrically after complex formation with sulfosalicylic acid or following reduction to ammonia, the ammonia is eliminated by distillation and determined titrimetrically. Other methods include direct nitrate determination by ultraviolet spectrophotometry, measurements being made at 210 nm, and the use of a nitrate-selective electrode. Details of the scope, limits of detection, and preferred applications of the methods are given in each case. 

---