Loop-free networks

To prove the above formulated uniqueness theorem, we consider an arbitrary but loop-free network N and assume that it has a unique steady state for arbitrary but fixed values of its external variables A, . .. A. As mentioned earlier, an external variable can be visualized as an infinitely large capacitance. Now we convince ourselves that the uniqueness remains valid if one of the external variables, say Ap is replaced by a 0-junction to which an internal variable X and an additional reaction 2-port element R are connected. The new element R may possibly involve further internal or external variables Y and Z, but its connection to N is assumed to generate no closed loop together with the elements of N ... 

What actually can be derived on the basis of a network representation is a negative criterion for limit cycles which says that loop-free networks not only have a unique steady state as argued in Section 7.5 but that this steady state also is globally and asymptotically stable. This statement evidently excludes the possibility of a limit cycle. The proof of this criterion has an analytic and a network topological part. In the analytic part one shows that the system of differential equations... 

In the network topological part of our proof we now convince ourselves that a loop-free network indeed satisfies the above listed conditions 1), 2) and 3). After our considerations in the preceding section and at the beginning of this section, this point almost needs no further argument. Condition 1) has been shown already in Section 7.5), condition 2) has essentially been utilized when proving the uniqueness property of Section 7.5 and condition 3) is a consequence of the fact that two capacities Xp X2 which are coupled by a flux of a reaction 2-port in a loop-free network are either on the same side or on opposite sides of this 2-port such that either 21 < 0 12 21 ... 

The analytic part of the above proof is evidently not restricted to loop-free networks but may be applied as a criterion of global and asymptotical stability to arbitrary networks as an individual test for the network with respect to conditions 1), 2) and 3). 

In general, the final network design should be achieved in the minimum number of units to keep down the capital cost (although this is not the only consideration to keep down the capital cost). To minimize the number of imits in Eq. (7.1), L should be zero and C should be a maximum. Assuming L to be zero in the final design is a reasonable assumption. However, what should be assumed about C Consider the network in Fig. 7.16, which has two components. For there to be two components, the heat duties for streams A and B must exactly balance the duties for streams E and F. Also, the heat duties for streams C and D must exactly balance the duties for streams G and H. Such balemces are likely to be unusual and not easy to predict. The safest assumption for C thus appears to be that there will be one component only, i.e., C = 1. This leads to an important special case when the network has a single component and is loop-free. In this case, ... 

This is a useful result, since if the network is assumed to be loop-free and has a single component, the minimum number of units can be predicted simply by knowing the number of streams. If the problem does not have a pinch, then Eq. (7.2) predicts the minimum number of units. If the problem has a pinch, then Eq. (7.2) is applied on each side of the pinch separately ... 

Raju, J. and Garcia-Luna-Aceves, J., A New Approach to On-demand Loop-free Multipath Routing, Proceedings of the 8th Annual IEEE International Conference on Computer Communications and Networks (ICCCN), pp. 522-527, Boston, MA, October 1999. 

Rangarajan, H., Garcia-Luna-Aceves, J. J. (2004). Using labeled paths for loop-free on-demand routing in ad hoc networks. InProceedings of the 5 > ACMMOBIHOC, (pp. 43-54). ACM Press. 

Networks obtained by anionic end-linking processes are not necessarily free of defects 106). There are always some dangling chains — which do not contribute to the elasticity of the network — and the formation of loops and of double connections cannot be excluded either. The probability of occurrence, of such defects decreases as the concentration of the reaction medium increases. Conversely, when the concentration is very high the network may contain entrapped entanglements which act as additional crosslinks. It remains that, upon reaction, the linear precursor chains (which are characterized independently) become elastically effective network chains, even though their number may be slightly lower than expected because of the defects. 

In the beginning there is a general loop to decide if more lot sizing procedures should be applied to the existing quant network to meet the constraint of the minimum batch sizes of products. Then the quant network is examined, free usable stocks and free quantities of quants are made available. The material balances of any quant are calculated and decisions are taken whether quants require further explosions of their BOM. Structures for a fast cycle checking, sorting of existing quants and quant links and forecast intervals are built up. A recalculation of the due dates for all quants - also the ones of orders - can be done if specified by the user. 

Free chain ends (unreacted functionalities) reduce the number of active network chains in a network compared with the same network without free ends. Disregarding possible presence of loops and entanglements, C — 1 C crosslinks are necessary according to Flory (55) to connect C chains into one giant macromolecule. Additional crosslinks will be elastically effective. Their number is given by... 

The argument leading to the conclusion that there can be no net circular reaction in a closed loop is based on the free energies of the members. As thermodynamic quantities, these are independent of whether or not the species involved also undergo other reactions. Accordingly, the rule is valid also for loops that are parts of larger networks, say, for ABC in... 

The populations of the I and Y free ends are independent of molecular ureight, as illustrated in Figure 7. On the other hand, it can be seen in Figure 6 that the occurrence of one-chain loops in the network agrees with the trend shown by the sol cyclics as described in the previous section. The observed increase in one-chain loop probabilities with shorter chain lengths is consistent with the Gaussian statistics assumed by the molecules. Network imperfections do not vanish at cong>lete conversion because of the loops. It is estimated by extrapolation that at 100% conversion, ca. 3% of the primary chains react to form loops for n>50. 

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