Super-additive

SO, additive chemistry has been described previously in literature [3]. SO, reduction additives remove SO, from the regenerator flue gas and release the sulfur as H2S in the FCC reactor. In a full bum regenerator, the amount of SO2 removed is directly proportional to the amount of additive used. Normal additive levels in the catalyst inventory range from 1-10%, with up to 20% being used in some units. Typical SO, removal rates have historically been in the 20-60% range. With the introduction of new super additives, rates in excess of 95% are commonly being achieved [4]. 

Type 1 refers to ordinary thermodynamics. Type 4 is the classical analogue of the thermodynamics of the Scfiwarzschild black hole, it violates the property of concavity, however, it keeps the super additivity. 

With multiple items or multiattribute items the preference structure of agents can be exponentially large. For example, if there are n items and the agent has super-additive preferences then in general the agent could specify 2 bids. Multiattribute items with n binary attributes leads to similar informational complexity. Therefore an additional consideration is to provide a compact bid representation language that allows agents to implicitly specify their bid structure. 

Y = n/T = H and T = T w) (we assume T > 1). As it has already been shown in Section f. 6, establishing the existence of the free energy for periodic models is just a matter of exploiting the super-additive property of... 

What we have presented in Section 4.2 is a proof of the existence of the free energy by bare hands, based on super-additivity of numerical sequences and on the Law of Large Numbers. It is a useful exercise in manipulating the partition function, in particular the technique yields the result in Section 4.3. There are however quicker and less self-contained approaches. Here we discuss ... 

The approach by using Kingman Super-Additive Ergodic Theorem. 

As we have seen ElogZ AT is a super-additive sequence, cf. Theorem 4.2, and, by Proposition A.12, one has the following characterization... 

A natural and very important probabilistic generalization of the well-known converge result for super-additive deterministic sequences is due to J. F. C. Kingman [Kingman (1973)]. Let us first recall the deterministic result a... 

Proposition A.12 is a super-additive sequence, then the limit... 

It is worth recalling that, in spite of what (A.63) might suggest, in general neither n nor n/n are increasing. This is obvious, because linear sequences, choose e.g. an = —n, are super-additive. A less trivial example is given by a = — log(l - - n). 

Consider now the case s = sup a /n < - -oo it suffices to show that liminfn oo an/n > s. Note that for every > 0 one may find no G N such that > (s—e)no- By writing once again n as kno + m, the super-additive property immediately yields... 

Theorem A. 13 Let us assume that F is a super-additive process that is F is a 2-index process such that... 

Chapter 4 deals with the problem of the existence of the free energy for disordered models. An elementary proof is worked out in detail and some other proofs are sketched. The reason for emphasizing other is simply that I find it troublesome talking about different proofs when they are all based on super-additivity. 

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