DESIGN FOR A BRAIN 19/8 



in which the F's are single-valued functions of their arguments 

 but are otherwise quite unrestricted. Obviously, if the initial 

 state is at t = 0, we must have 



F,(«J, . . . , 4 1 0) = a$ (i = 1, . . . , n) 



19/8. Theorem : The lines of behaviour of a state-determined system 

 define a group. 



Let the initial state of the variables be x°, where the single 

 symbol represents all n, and let time t' elapse so that x° changes 

 to x' . With x' as initial state let time t" elapse so that x' changes 

 to x" . As the system is state-determined, the same total line of 

 behaviour will be followed if the system starts at x° and goes on 

 for time t' + t". So 



x[ = Ftih . . . , x n ; t") = F,(xl . . . , xl ; f + t") 



{i = 1, . . . , n) 

 But 



x\ = F,(4 . . . , x° n ; t') {i = 1,. . . , n) 



giving 



F t {F^; t'), . . . ,F n (x°; f); t") = F,(4 . . . ,x° n ; V + t") 



(i = 1, . . . . , n) 



for all values of x°, t\ and t" over some given region; and this is 

 one way of defining a one-parameter finite continuous group. 



The converse is not true. Thus x — (1 -f- t)x° defines a group 

 (with n = 1); but the times do not combine by addition, and the 

 system is not state-determined. 



Example : The system with lines of behaviour given by 

 x x = x\ + x? z t + t 2 

 x 2 = x% + 2i 

 is state-determined, but that with lines given by 



X-t ^ = X-t ~\~ X'jl \~ i 



x 2 = X?z + t 



is not. 



Canonical representation 



19/9. Theorem: That a system x v . . . , x n should be state- 

 determined it is necessary and sufficient that the x's, as functions oft, 

 should satisfy equations 



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