Equivalent size problem

In Chapters 3 and 4 we have shown that the vector of process variables can be partitioned into four different subsets (1) overmeasured, (2) just-measured, (3) determinable, and (4) indeterminable. It is clear from the previous developments that only the overmeasured (or overdetermined) process variables provide a spatial redundancy that can be exploited for the correction of their values. It was also shown that the general data reconciliation problem for the whole plant can be replaced by an equivalent two-problem formulation. This partitioning allows a significant reduction in the size of the constrained least squares problem. Accordingly, in order to identify the presence of gross (bias) errors in the measurements and to locate their sources, we need only to concentrate on the largely reduced set of balances... 

Reviewing again the fan-sizing problem of Chap. 5, the following equivalence of the terms in Eq. (5.52) is noted ... 

The above averages may be taken over actual dimensions or over dimensions of equivalent spheres of a certain type (see above). Different methods of calculation of equivalent sizes of irregularly shaped particles and of an average in an assembly of polydispersed particles can produce very different average particle sizes in the same assembly of particles. The problem of equivalent size of irregularly shaped... 

Our quest in ionophore and receptor design emerged from one of the seemingly intractable problems of contemporary biochemistry the selective recognition of the ammonium cation (NH/) [141], Much of the problem, as discussed in earlier section is due to the nearly equivalent sizes of NH/ and the potassium cation (K ) [67]. One also learns from our discussion of the cation-7i interaction, that receptors providing improved dispersion stabilization are more selective forNH/ [67,71,72]. 

The theory of cooperative capillary condensation in stochastic network of channels is developed. The corresponding mathematical problem is reduced to a three-component bond percolation problem. At a given relative pressure x in the network of channels three types of bonds are distinguished subundercritlcal channels of equivalent size pp (x)- Here functions p+(x) and p (x) determine the equivalent sizes of pores in which the capillary condensation and desorption are observed at relative pressure x ... 

Just as there are universal computers that, given a particular input, can simulate any other com-puter, there are NP-complete problems that, with the appropriate input, are effectively equivalent to any NP-hard problem of a given size. For example, Boolean satisfiability -i.e. the problem of determining truth values of the variable s of a Boolean expression so that the expression is true -is known to be an NP-complete problem. See section 12.3.5.2... 

The example problem was generated by picking the type and size at random, while ensuring that no size and type combination was repeated. This last requirement was imposed to avoid making the equivalence condition appear to perform better due to spurious equivalences. 

An interesting approach has been employed in paper [74] to find the distribution f(li, l2) of copolymer chains for numbers l and h of monomeric units Mi and M2. This distribution is evidently equivalent to the SCD, because the pair of numbers k and I2 unambiguously characterizes chemical size (l = h + l2) and composition ( 1 = l] //, 2 = h/l) of a macromolecule. The essence of this approach consists of invoking the Superposition Principle [81] that enables the problem of finding the Laplace transform G(pi,p2) of distribution f(li,k) to be reduced to the solution of two subsidiary problems. The first implies the derivation of the expression for the generating function [/(z1",z 2n ZjX,z ) of distribution P(ti, M2 mt, m2), and the second is concerned with finding the Laplace transforms g (pi,p2) and (pi,p2) of distributions (Eq. 91). With these two problems solved, it is possible to obtain the characteristic function G(pi,p2) of distribution f(li,h) using the Superposition Principle formula... 

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