Substitutions direct

A direct substitution strategy can be used for equations of the form 

The difference between a convergent and divergent system is the slope of g x) at the solution condition. When the slope 1, then the method diverges. However, when the slope 1, then the method converges. For the example of equation (2.3.5) 

Therefore, it appears that the direct substitution algorithm will be unstable.. Actual implementation is as follows. 

Sometimes it is possible to reformulate a divergent g x) function to obtain a new convergent expression. For our example, an alternative 

Therefore, with this formulation, the direct substitution algorithm should be stable  

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Direct substitution into the equation given provides the required value for r ... 

In this case we cannot directly substitute Mij into the equilibrium equation as it was done for the previous elastic and inelastic models. So w, Mij cannot be found in consecutive order, in general. 

Theorem 2.30 can also be verified by the direct substitution of the obtained solutions in (2.271), (2.272) and (2.273), respectively. We apply this result to the following example. Let... 

Biomaterials for Cardiovascular Devices. Perhaps the most advanced field of biomaterials is that for cardiovascular devices. For several decades bodily parts have been replaced or repaired by direct substitution using natural tissue or selected synthetic materials. The development of implantable-grade synthetic polymers, such as siHcones and polyurethanes, has made possible the development of advanced cardiac assist devices (see... 

The following rules pertain to electrophilic substitution in pyrroles (35) (/) an electron-withdrawing substituent in the a-position directs substitution to the P and a -positions, (2) an electron-releasing substituent in the a-position directs substitution to the neighboring -position or to the a -position, (J) an electron-withdrawing substituent in the -position leads to substitution in the a -position, and (4) an electron-releasing substituent in the P-position tends to direct substitution into the neighboring a-position. 

There are direct substitutions of possible interest that would not be feasible without drastic changes in the feed system or pressure. Thus if the available substitute for natural gas is, eg, a manufactured gas containing much CO, there would almost always be a mismatch of the WIs unless the fuel could be further modified by mixing with some other gaseous fuel of high volumetric heating value (propane, butane, vaporized fuel oil, etc). Moreover, if there are substantial differences in eg, as a result of the presence of considerable H2 as well as CO in the substitute gas, the variation in dame height and dashback tendency can also make the substitution unsatisfactory for some purposes, even if the WI is reproduced. Refinements and additional criteria are occasionally appHed to measure these and other effects in more complex substitution problems (10,85). 

Reactions with strongly basic nucleophiles such as potassium amide in liquid ammonia may prove much more complex than direct substitution. 2-Chloro-4,6,7-triphenylpteridine reacts under these conditions via an S ANRORC mechanism to form 2-amino-4,6,7-triphenylpteridine and the dechlorinated analogue (78TL2021). The attack of the nucleophile exclusively at C-4 is thereby in good accord with the general observation that the presence of a chloro substituent on a carbon position adjacent to a ring nitrogen activates the position meta to the chlorine atom for amide attack. 

A direct substitution at the 5-position of the intact penicillanic acid ring system has been reported as shown in Scheme 27 (79CC485). 

Equations for derived properties may be developed from each of these expressions. Consider first Eq. (4-190), which is explicit in volume. Equations (4-159), (4-161), and (4-176) are therefore applicable. Direct substitution for Z in Eq. (4-161) gives... 

Oxidation of the 3-(hydroxyalkyl)pyrrole derivative gives a pure 3-acylpyrrole derivative which is difficult to obtain by direct substitution in the pyrrole ring. Acylation of pyrrole yields 1- and/or 2-acetyl pyrrole, whereas acylation of 1-methyl pyrrole forms both 2- and 3-acetyl-1-methyl-... 

The extent to which rearrangement occurs depends on the structure of the cation and foe nature of the reaction medium. Capture of carbocations by nucleophiles is a process with a very low activation energy, so that only very fast rearrangements can occur in the presence of nucleophiles. Neopentyl systems, for example, often react to give r-pentyl products. This is very likely to occur under solvolytic conditions but can be avoided by adjusting reaction conditions to favor direct substitution, for example, by use of an aptotic dipolar solvent to enhance the reactivity of the nucleophile. In contrast, in nonnucleophilic media, in which fhe carbocations have a longer lifetime, several successive rearrangement steps may occur. This accounts for the fact that the most stable possible ion is usually the one observed in superacid systems. 

Substituted allylic halides give mixtures of products resulting from bond formation at both C-1 and C-3 of the allylic system, with the product ratio favoring the product formed by reaction at the less substituted site. The portion of the product formed by reaction at C-1 in allylic systems may result from direct substitution, but it has also been suggested that a... 

Condensation of sodium phenoxide witli 2,2,2-trifluoroethyl iodide gives a product of direct substitution in a low yield, several other ethers are formed by eliminatton-addition reactions [7] Use of mesylate as a leaving group and hex amethyl phosphoramide (HMPA) as a solvent increases the yield of the substitution [S] Even chlorine can be replaced when the condensation is performed with potassium fluoride and acetic acid at a high temperature [9] (equations 6-8)... 

Direct substitution of the carbonyls themselves is of course possible. Besides Group 15 donor ligands, unsaturated hydrocarbons give especially interesting products. The iron carbonyl acetylenes provided early examples of the use of carbonyls in organic synthesis. From them a wide variety... 

Until recently, 3-substituted thiophenes have been rare and difficultly available compounds, as they, with very few exceptions, cannot be prepared by direct substitution of thiophene. 

Other derivatives of this type were studied by Chang and Ulbricht and were prepared by direct substitution or other secondary reactions. 

The 4-position of isothiazole is attacked by electrophilic reagents, and many simple derivatives are thus readily available by direct substitution followed, if necessary, by suitable transformation of the group introduced. For example, bromination of 3-methylisothiazole... 

An attractive feature in this reaction is the possibility of direct substitution and formation of unblocked sugar derivatives containing one or more chlorodeoxy function in essentially two steps. Another facet is the formation of methyl 4,6-dichloro-4,6-dideoxy-hexosides from certain methyl glycosides in one step. Such compounds could be valuable intermediates in the synthesis of dideoxy and diamino sugars of biological importance. 

Frequently, values are known for three of these quantities (perhaps V, n, and T) the other one (P) must be calculated. This is readily done by direct substitution into the ideal gas law. 

Direct substitution of Eq. (8-83) leads to the proof of these, when we recall that the one-particle wave functions satisfy the relation... 

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