THE BLACK BOX 



6/11 



Ex. 1 : (Ex. 6/9/4 continued.) Compare the diagram of immediate etTects of 

 A and B. 



Ex. 2: Mark the following properties of an absolute system as changed or 

 unchanged by a re-labelling of its states: (i) The number of basins in its 

 phase-space; (ii) whether it is reducible; (iii) its number of states of equili- 

 brium; (iv) whether feedback is present; (v) the number of cycles in its 

 phase-space. 



Ex. 3 : (Continued.) How would they be affected by a re-labelling of variables? 



6/11. The subject of isomorphism is extensive, and only an intro- 

 duction to the subject can be given here. Before we leave it, however, 

 we should notice that transformations more complex than a simple 

 re-labelling of variables can change the diagram of immediate 

 effects. Thus the systems 



B: 



are isomorphic under the one-one transformation 



— u 



V -\- v^ 



P: 



Yet A's diagram is 



while B's diagram is 



i.e. two unconnected variables. 



The "method of normal co-ordinates", widely used in mathe- 

 matical physics, consists in applying just such a transformation as 

 will treat the system not in its obvious form but in an isomorphic 

 form that has all its variables independent. In this transformation 

 the diagram of immediate effects is altered grossly; what is retained 

 is the set of normal modes, i.e. its characteristic way of behaving. 



Such a transformation (as P above), that forms some function 

 of the variables (i.e. x — y) represents, to the experimenter, more 

 than a mere re-labelling of the x-, v-output dials. It means that 

 the Box's output of x and y must be put through some physical 

 apparatus that will take x and y as input and will emit x — y and 

 X + J as new outputs. This combining corresponds to a more 

 complex operation than was considered in S.6/10. 



Ex.: Show that A and B are isomorphic. (Hint: (x — y)' = x' — y': why 



101 



