TRANSMISSION OF VARIETY 8/12 



that has all its representative points at the same state can thus, under 

 the effect of T's variety, change to a set with its points scattered over 

 not more than Uj- states. There are «y such sets of C/'s, each capable 

 of being scattered over not more than rij states, so the total scattering 

 cannot, after one step, be greater than over rij-Uu states. If variety 

 is measured logarithmically, then the variety in U after one step 

 cannot exceed the sum of those initially in U and T. In other words, 

 the U's cannot gain in variety at one step by more than the variety 

 present in the T's. 



This is the fundamental law of the transmission of variety from 

 system to system. It will be used frequently in the rest of the book. 



Ex. 1: A system has states (t,ii) and transformation t' = 2t, u' = u + t, so t 

 dominates u. Eight such systems are started at the states (0,9), (2,5), 

 (0,5), (1,9), (1,5), (2,5), (0,9), (1,9) respectively. How much variety is in 

 the/'s? How much in the m's ? 



Ex. 2: (Continued.) Find the states at the next step. How much variety has 

 /now? Predict an upper limit to «'s variety. How much has » now ? 



Ex. 3 : In another system, T has two variables, ti and t2, and U has two, «i 

 and «2- The whole has states (ti, (2, iiu "2), and transformation ti' = tit2, 

 t2 = tu III = Ml + tiui, U2 = t\U2, so that T dominates U. Three replicas 

 are started from the initial states (0,0,0,1), (0,0,1,1) and (1,0,0,1). What is 

 r's variety ? What is t/'s ? 



Ex. 4: (Continued.) Find the three states one step later. What is f/'s variety 

 now? 



8/12. Transmission at second step. We have just seen that, at the 

 first step, U may gain in variety by an amount up to that in T; what 

 will happen at the second step? T may still have some variety: 

 will this too pass to U, increasing its variety still further? 



Take a simple example. Suppose that every member of the whole 

 set of replicates was at one of the six states {Ti,U^), (_Ti,Ui), (Ti,U^), 

 {Tj,Uk), (Tj,U,), {Tj,UJ, so that the T's were all at either T,- or Tj 

 and the U's were all at C/;^, Ui or U^. Now the system as a whole is 

 absolute; so all those at, say {7^,1},^, while they may change from 

 state to state, will all change similarly, visiting the various states 

 together. The same argument holds for those at each of the other 

 five states. It follows that the set's variety in state cannot exceed 

 six, however many replicates there may be in the set, or however 

 many states there may be in Tand U, or for however long the changes 

 may continue. From this it follows that the t/'s can never show 

 more variety than six t/-states. Thus, once U has increased in 

 variety by the amount in T, all further increase must cease. If 

 U receives the whole amount in one step (as above) then U receives 



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