Point convergence

As the feed composition approaches a plait point, the rate of convergence of the calculation procedure is markedly reduced. Typically, 10 to 20 iterations are required, as shown in Cases 2 and 6 for ternary type-I systems. Very near a plait point, convergence can be extremely slow, requiring 50 iterations or more. ELIPS checks for these situations, terminates without a solution, and returns an error flag (ERR=7) to avoid unwarranted computational effort. This is not a significant disadvantage since liquid-liquid separations are not intentionally conducted near plait points. 

In the local region, optimization methods may be characterized by their rate of convergence. Let xk be a sequence of points converging to x ... 

Due to the dissipative nature of these dynamics, trajectories initiated from two nearby points converge rapidly if they possess the same noise history. As a consequence, a new trajectory generated from a reactive old trajectory has a high probabihty to be reactive as well. Such moves in noise space therefore have a high acceptance probability. This algorithm is summarized in Scheme 1.6. 

It is important to observe in Figure 4-11 that, contrary to the prediction of eq 3.21, the plotted points converge not to unity but a value considerably smaller... 

How is DNA synthesis terminated at the end of each replication unit One possibility is that there exist sites in each replication unit where DNA synthesis from growing points converging from either direction is stopped. This may be likely if many adjacent replication units have nonoverlapping times of replication. The alternative consideration is that replication is terminated when converging growing points on one DNA molecule meet, but this would be difficult in the situation described above where adjacent replication units are replicated in nonoverlapping time periods. 

In the first case, the decay to the periodic solution can be checked through a Poincare map. When the limit cycle is reached, the mapping points converge to one point in state space. 

The following theorem can be used to establish point convergence of a given Fourier series [1,4]. 

At this point, convergence should be achieved. However, the simulation is not quite finished because a lean solvent makeup input stream needs to be added to ensure that solvent losses through the vapor product and CO2 pipeline is replenished. 

It is natural to conjecture that the phenomenon set forth with Figure 4.1 does not happen and that the critical point converges, in the limit T oo, to the critical point of the quenched model, but this is an open problem. 

---