Condition-based divergence

Abstract In the past decade, it has been extensively demonstrated that multicomponent chemistry is an ideal tool to create molecular complexity. Furthermore, combination of these complexity-generating reactions with follow-up cyclization reactions led to scaffold diversity, which is one of the most important features of diversity oriented synthesis. Scaffold diversity has also been created by the development of novel multicomponent strategies. Four different approaches will be discussed [single reactant replacement, modular reaction sequences, condition based divergence, and union of multicomponent reactions (MCRs)], which all led to the development of new MCRs and higher order MCRs, thereby addressing both molecular diversity and complexity. 

Fig. 8 Schematic representation of the Condition Based Divergence (CBD) strategy to scaffold diversity...

In 2008, Chebanov et al. reported an excellent example of condition base divergence by the MCR of 5-amino pyrazoles 135, cyclic 1,3 diketones 136 and aromatic aldehydes 137 (Scheme 12) [90]. 5-Amino pyrazoles 135 have at least... 

The existence of non-equilibrium combustion products is important to at least two considerations. Firstly, the observed propellant performance may depart substantially from the predicted level. This departure may result in performance either less than or greater than the equilibrium predicted level. A striking example of greater than equilibrium performance is that of hydrazine monopropellant decomposition, table m-A-1. Another is that of ethylene oxide monopropellant, as mentioned in section n. B. 4., in which the equilibrium quantities of condensed carbon never are formed. Secondly, the non-equilibrium composition may have significant effects on the expansion process. In particular, nozzle kinetic calculations based on an assumed equilibrium composition initial condition may diverge significantly from expansions occurring from non-equilibrium initial conditions. 

The strain-based method can be performed by measuring the strain distribution of the intact structure in advance, which acts as the reference. Damage can be then detected when the strain measurement of the structure under loading condition significantly diverges from the reference strain. 

If the structure of the models is more complex and we have more than one independent variable or we have more than two rival models, selection of the best experimental conditions may not be as obvious as in the above example. A straightforward design to obtain the best experimental conditions is based on the divergence criterion. 

The idea behind the computation of the Divergence only among adequate models in Step 2, is to base the decision for the next experiment on the models that are still competing. Models that have been found inadequate should not be included in the divergence computations. However, once the new data point becomes available it is good practice to update the parameter estimates for all models (adequate or inadequate ones). Practice has shown that under conditions of high experimental errors, as additional information becomes available some models may become adequate again ... 

The most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

The present chapter is devoted mainly to one of these new theories, in particular to its possible applications to photon physics and optics. This theory is based on the hypothesis of a nonzero divergence of the electric field in vacuo, in combination with the condition of Lorentz invariance. The nonzero electric field divergence, with an associated space-charge current density, introduces an extra degree of freedom that leads to new possible states of the electromagnetic field. This concept originated from some ideas by the author in the late 1960s, the first of which was published in a series of separate papers [10,12], and later in more complete forms and in reviews [13-20]. 

When these conditions are not fulfilled the Newton-Raphson procedure may converge only slowly or even diverge. It is then necessary to introduce manual procedures, which drive the solution to the local minimum. A number of such methods have been devised, which in many cases work well. They are based on... 

The specific proposal is that the hydroxide attacks the 2- or the 4-position of the co-ordinated pyridine to form a hydroxy-substituted 1,2- or 1,4-dihydropyridine (Fig. 8-17). These hydroxy species are known as pseudo-bases. This behaviour is fully in accord with the known behaviour of A-alkylpyridinium cations. Although the pseudo-bases of simple A-alkylpyridinium cations are not dominant solution species under aqueous conditions, those derived from a variety of other nitrogen heterocycles are readily formed and are well-known. The suggestions had the advantage of linking the apparently divergent fields of heterocyclic and co-ordination chemistry by explaining some well-documented anomalies in the reactivity of pyridine complexes. 

Following the Morbidelli and Varma criterion, several other methods have been proposed in recent years in order to characterize the highly sensitive behavior of a batch reactor when it reaches the runaway boundaries. Among the most successful approaches, the evidence of a volume expansion in the phase space of the system has been widely exploited to characterize runaway conditions. For example, Strozzi and Zaldivar [9] defined the sensitivity as a function of the sum of the time-dependent Lyapunov exponents of the system and the runaway boundaries as the conditions that maximize or minimize this Lyapunov sensitivity. This has put the basis for the development of a new class of runaway criteria referred to as divergence-based approaches [5,10,18]. These methods usually identify runaway with the occurrence of a positive divergence of the vector field associated with the mathematical model of the reactor. 

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