Elimination Methods

Exposure of p-acetoxysulphones, which are readily accessible from an a-sulphonyl carbanion and an aldehyde, to an excess amount of potassium alkoxide results in elimination of both functional groups in a one-pot reaction, introducing unsaturation in two carbon-carbon bonds. As a consequence, acetylenes or dienes are obtained, depending on the aldehyde (Equations 4 and 5) [15,16]. 

KOMe [93% (all- )]. Notably, /-BuOK does not promote the elimination in this case. The mechanism of this process is different from that with the alkoxysulphones. The initial step is formation of a vinyl sulphone 50. Isomerization of this intermediate to the allylic sulphone 48 via 51 is effected by Michael addition and elimination of methanol. The ineffectiveness of t-BuOK is explained by this mechanism. This bulky reagent cannot abstract the proton from the a-carbon of the sulphonyl group because this is embedded in a sterically highly crowded environment. Moreover, r-BuOK does not participate in the Michael addition because of its weaker nucleophilic character. 

As can be seen from the above examples, the double elimination method is highly efficient and practical for polyene synthesis because labile polyenic moieties are protected as stable p-alkoxy or 5-halo sulphones until the final stage of the synthesis, when the desired skeleton and double bond system are generated in a single step. It should also be emphasized that the desulphonylation process is particularly suitable for synthesis of acid-sensitive retinoids and carotenoids because the reaction is performed under basic conditions. 

In this part of the Chapter, selected synthetic routes are described for the most important isoprenoid building blocks used in the construction of the polyene chain, Le. C(8) to C(8 ) of the carotenoid molecule. 

In view of the vast amount of work that has been published on this topic, a relatively stringent selection has been made. The following criteria have been applied in the selection of synthons and synthetic routes  

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The most important direct solution algorithms used in finite element computations are based on the Gaussian elimination method. 

To describe the basic concept of the Gaussian elimination method we consider the following system of simultaneous algebraic equations... 

The Gaussian elimination method provides a systematic approach for implementation of the described forward reduction and back substitution processes for large systems of algebraic equations. 

SOLUTION ALGORITHMS BASED ON THE GAUSSIAN ELIMINATION METHOD... 

The most frequently used modifications of the basic Gaussian elimination method in finite element analysis are the LU decomposition and frontal solution techniques. 

SOLUTION ALGORITHMS GAUSSIAN ELIMINATION METHOD 205 6.4.2 Frontal solution technique... 

Procedure. Write a program for solving simultaneous equations by the Gaussian elimination method and enter the absorptivity matiix above to solve Eqs. (2-51). Set up and solve the problem resulting from a new set of experimental observations on a new unknown solution leading to the nonhomogeneous veetor b = 0.327,0.810,0.673. ... 

This result is remarkably simple as compared to the usual methods. For a spin-polarised potential V, Kraft, Oppeneer, Antonov and Eschrig (1995) used the elimination method and found the corrections as a sum of 9 terms, which is equivalent to our Eq.(ll). They notice that three terms of their sum have a known physical meaning (spin-orbit, Darwin and mass-velocity corrections), but the other terms have no special name . 

With the aid of effective Gauss method for solving linear equations with such matrices a direct method known as the elimination method has been designed and unveils its potential in solving difference equations,... 

The algorithm presented below as the sequence of applied formulae is called the right elimination method and is showing the gateway for the... 

Stability of the elimination method. Let us stress that the conditions (7i — a yf 0 and 1 — Q cannot be excluded or relaxed during the... 

Returning to the right elimination method, we show that the conditions ai < 1 guarantee that the error 6yi y = — j/j+i arising when... 

The left elimination method. The counter elimination method. Still using the framework of the right elimination method (formulae (10)-(15)) in reverse order, we obtain through such an analysis the computational formulae of the left elimination method ... 

Joint use of the left and right elimination methods refers to the counter elimination method. The essence of this method is to consider a fixed inner... 

Of course, the counter elimination method could be especially effective in an attempt to determine yi merely at only one node i = i. ... 

Solution estimation for difference bormdary-value problems by the elimination method. In tackling the first boundary-value problem difference equation (21) has the tridiagonal matrix of order TV — 1... 

The computational formulae of the right elimination method help derive estimate (38) ... 

Formulae (74)-(75) show that the elimination method is stable. The values and should be known before proceeding to the applications of (70), (74) and (75). For this reason we involve here the second boundary condition (67) and relation (69) for i = N ... 

Hyclic elimination method. We now focus the reader s attention on periodic solutions to difference schemes or systems of difference schemes being used in approximating partial and ordinary differential equations in spherical or cylindrical coordinates. A system of equations such as... 

We give below without proving the algorithm of the cyclic elimination method which will be used in the sequel ... 

Thus emerged the system of algebraic equations with a tridiagonal matrix. Because of this form, the elimination method may be useful (see Chapter 1, Section 1). 

The natural replacement of the central difference derivative u x) by the first derivative Uo leads to a scheme of second-order approximation. Such a scheme is monotone only for sufficiently small grid steps. Moreover, the elimination method can be applied only for sufficiently small h under the restriction h r x) < 2k x). If u is approximated by one-sided difference derivatives (the right one for r > 0 and the left one % for r < 0), we obtain a monotone scheme for which the maximum principle is certainly true for any step h, but it is of first-order approximation. This is unacceptable for us. 

These equations admit the form (39) and can be solved by the elimination method. The main difference from the cylindrical case lies in the selection rules for and d. To obtain the formulae for

[Pg.197]

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