

THE BLACK BOX 6/10 



The transformation P: 



y -^ 



is a shorthand way of describing the one-one transformation that 

 pairs off states in S and R thus: 



in S, (2,3) against (-3,2) in R 

 (1,0) „ (0,1) „ „ 



(4,5) „ (-5,4) „ „ 



„ „ (-3,0) „ (0,-3) „ „ 



i.e. „ „ {u,v) „ {—v,u) „ „ 



(Compare U of S.4/9.) Apply P to all the description of S; the 

 result is 



r y = -y + X 



I -A-' = -y - X 

 which is algebraically identical with R. So R and ^S' are isomorphic. 



Ex. 1 : What one-one transformation will show these absolute systems to be 

 isomorphic? 



y, a b c d e y.pqrst 



' ^ c c d d b ' ^ r q q p r 



(Hint: Try to identify some characteristic feature, such as a state of 

 equilibrium.) 



£.Y. 2: How many one-one transformations are there that will show these 

 absolute systems to be isomorphic? 



A:\l^ ' B:\P ^ ' 



* b c a * r p q 



*Ex. 3: Write the canonical equations of the two systems of Fig. 6/8/1 and show 

 that they are isomorphic. (Hint: How many variables are necessary if 

 the system is to be a machine with input?) 



^.v. 4: Find a re-labelling of variables that will show the absolute systems A 

 and B to be isomorphic. 



Cx' = -A-2 + y fu'= )V2 -f « 



A: {y' = -x2-y B: -{ v' = -v2+ w 



[_Z' = J'2 + Z lw'= -1-2- Vt' 



(Hint: On the right side of A one variable is mentioned only once; the 

 same is true of B. Also, in A, only one of the variables depends on itself 

 quadratically, i.e. if of the form a' = ± a2 . . . ; the same is true of B.) 



6/10. The previous section showed that two machines are iso- 

 morphic if one can be made identical to the other by simple re- 

 labelling. The "re-labelling", however, can have various degrees 

 of complexity, as we will now see. 



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