Strong Law of Large Numbers

By the Strong Law of Large Numbers, from (4.13) we conclude that a.s. 

Let fn n be a sequence of Independent Identically Distributed (IID) random variables on the probability space Cl, Q, P). The following statement is the Strong Law of Large Numbers ... 

Since only a few molecules of the Y species exist in the MC state, a structural change to them strongly influences the catalytic activity of the protocell. On the other hand, a change to X molecules has a weaker influence, on average, since the deviation of the average catalytic activity caused by such a change is smaller, as can be deduced from the law of large numbers. Hence the MC state is important for a protocell to realize evolvability. 

Basic mechanisms involved in gas and vapor separation using ceramic membranes are schematized in Figure 6.14. In general, single gas permeation mechanisms in a porous ceramic membrane of thickness depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collisions. In membranes with large mesopores and macropores the separation selectivity is weak. The number of intermolecular collisions is strongly dominant and gas transport in the porosity is described as a viscous flow that can be quantified by a Hagen-Poiseuille type law ... 

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