Networks piecewise simple

A non-simple network can be broken at the offending point or points into piecewise simple portions (see Section 6.5). 

If only a small minority of reaction steps are non-simple, much benefit can be had by breaking the pathway or network down into "piecewise simple" portions and then applying the methods described in the preceding sections to these [8], To this end the pathway or network is cut at the offending intermediates or steps, as will now be shown. 

Table 6.1. Examples of break-up of non-simple reduced networks into piecewise simple portions (co-reactants and co-products of other steps not shown, and other parts of network assumed to be simple).

As a rule, a reduction to a single, explicit rate equation (plus algebraic equations for stoichiometric constraints and yield ratios) is not achieved. Rather, the equations for the end members of the piecewise simple network portions must be solved simultaneously. Nevertheless, The concentrations of all trace-level intermediates that do not react with one another have been eliminated by this procedure and, in many cases of practical interest, the reduction in the number of simultaneous rate equations and their coefficients is substantial. 

The network can be cut into four piecewise simple portions ... 

Many reactions of practical interest have non-simple pathways or networks, i.e., the concentration of an intermediate rises above trace level or a (forward or reverse) step involves two or more molecules of intermediates as reactants. If the majority of the steps meet the simplicity conditions, a significant reduction in mathematical complexity can still be achieved by cutting the network into piecewise simple portions at the offending steps. In some other instances, the quasi-equilibrium or long-chain approximations can be invoked in order to obtain explicit rate equations although the network is non-simple. On the other hand, if a majority of steps in a network are non-simple, the tools described here are of little use. 

In practice, many reaction systems involve non-trace intermediates, but no obvious non-simple reactions of intermediates. A good strategy in such situations is to cut the overall reaction network into portions at the non-trace intermediate or intermediates (see Section 6.5), then reduce the portions as described for simple networks in Section 6.4.1. Network reduction makes it unnecessary to keep track of trace intermediates (except those reacting in a non-simple manner) and so obviates much of the hard work Trace intermediates are the more troublesome ones in network elucidation because they are difficult or impossible to detect, identify, analyze for, or synthesize, tasks that usually do not pose problems with intermediates that rise above trace level. Often, the network portions will turn out to be "piecewise simple" (see Section 6.5). If not, further cutting at additional nonsimple steps is called for when these become apparent. 

Piecewise simple portions feeding into others may, of course, be reversible. This complicates network elucidation significantly. Often, however, such back-reactions can be blocked by an additive, the omission of a catalyst or co-catalyst, or some other experimental stratagem. Alternatively, the intermediate produced by the portion can be trapped in some fashion this allows at least the forward reaction through the portion to be studied without interference. 

If a pathway or network should turn out to contain a step in which two or more molecules of the same or different intermediates react with one another, it can be cut at the offending species into piecewise simple portions as discussed in Section 6.5. However, it will rarely be possible to study these portions independently because, more often than not, the respective trace intermediates cannot be synthesized for use as starting materials. 

If a pathway or network turns out to be non-simple, a good strategy is to try to break it up into piecewise simple portions that can be studied independently. Whether and how this can be done depends on the reaction at hand. The job is easiest if the portions are irreversible, so that none of them feeds back into a preceding one, and if the non-trace intermediates can be synthesized. 

Break-up into piecewise simple portions and their reduction. The network 11.1 is "simple" except that aldehyde, an intermediate, builds up to higher than trace concentrations. Thus it can be cut at the aldehyde into two piecewise simple portions which share aldehyde and the ligand-deficient catalyst, cat . After reduction the portions can be written... 

The whole progress can be approximated by a set of piecewise linear functions by a threshold bipolar (TB) protocol, which is very simple in its implementation and design philosophy. For a host, when the current Vp< a threshold, the urgent task is to download the most wanted chunks, while if Vp > the threshold, the job is switched to help spread the latest or rarest chunks over the network. We have ever observed some other peers startup procedures in our trace, and all of them can be interpreted easily by the TB protocol. 

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