Factoring scheme

Bartoli recently discovered that by switching from azide to p-anisidine as nucleophile, the ARO of racemic trans- 3-substituted styrene oxides could be catalyzed by the (salen)Cr-Cl complex 2 with complete regioselectivity and moderate selectivity factors (Scheme 7.36) [14]. The ability to access anti-P-amino alcohols nicely complements the existing methods for the preparation of syn-aryl isoserines and related compounds [67] by asymmetric oxidation of trans-cinnamate derivatives [68]. 

In sharp contrast, Bartoli showed that the (salen) Co catalyst system could be applied to the kinetic resolution of terminal epoxides with unprotected tert-butyl carbamate as nucleophile with extraordinarily high selectivity factors (Scheme 7.40) [72]. Excellent yields and selectivities are also obtained with use of ethyl, Cbz,... 

In this regard, it should be taken into account that scheme (33)-(34) is equivalent to the factorized scheme (20)-(21) with the member = (a (a ) in special cases either in scheme (33) the operator A2(t)... 

It is plain to show that the scheme in view is stable and generates an approximation of + With this aim, it seems worthwhile to design a factorized scheme for the same reason as before. The starting point is to eliminate y pi/2 from equations (46) in the process of transformations... 

Especial attention is being paid to the factorized schemes with the members... 

It is worth recalling here that the first economical schemes were intended for the elimination of intermediate values with no problems. The main idea behind this approach is to involve factorized schemes in integer steps , a key role of which is to relate the values y and in some or other convenient ways. 

The boundary conditions. As one might expect, stability and approximation take place for the factorized scheme (1). In this view, it seems reasonable to adopt equations (6) or (lO)-(ll) as a perfect computational algorithm in designing the factorized scheme (1). But this equivalence can be established only with consistent boundary conditions and needs certain clarification. 

We begin by placing the factorized scheme (1) with A y — Vx x rectangular grid = (ij h, h )] with steps and in the specified... 

Constructions of economical factorized schemes. Using the regularization method behind, we try to develop the general method for constructing stable economical difference schemes on the basis of the primary stable scheme... 

Thus, B > B > 0.5 tA, meaning the stability of the factorized scheme (22). The operators Ra are so chosen as to satisfy the condition of approximation, too. The forthcoming example helps clarify what is done. 

For this, we have occasion to use the two-layer factorized scheme... 

Under such an approach the factorized scheme is of no less than first-order accuracy in r. In a similar way an economical factorized scheme can be designed in the p-dimensional case when... 

Example 3 As a matter of experience, the intention is to use a higher-accuracy scheme similar to (12) being used for the heat conduction equation. As we have stated in Section 1, the factorized scheme... 

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" ). 

Three-layer factorized schemes. Being concerned with economical three-layer schemes, we confine ourselves here to... 

A similar procedure works for the boundary conditions imposed for as was done for a two-layer factorized scheme and for this reason it is omitted here. 

Stability of the factorized scheme (35) can be established on account of the general theorems from Chapter 6, Section 3, due to which it follow s from the foregoing that the conditions... 

A particular case where R = a-A, cr = 0.5 (cr +cr ), is showing the gateway to the future research, whose aims and scope are connected with the general method for constructing three-layer economical factorized schemes by means of the regularization principle of difference schemes. A simple example... 

In that case the factorized scheme (39) is stable under condition (40) ... 

In this direction, the intention is to construct in a similar manner the factorized scheme related to another primary scheme... 

Both factorized schemes (44) and (45) generate second-order approximations in T for any a and they are stable under the condition 4cr > 1 -1- e,... 

The algorithm of solving equation (45) was demonstrated before and so it remains only to construct economical factorized schemes associated with problem (42) by means of the operator acting in accordance with the rule... 

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