THE MACHINE WITH INPUT 4/2 



formation must tell what they are to be changed to. So all quan- 

 tities that appear on the right, but not on the left, must be parameters. 

 The examples below will clarify the facts. 



Ex. 1 : What are the three transformations obtained by giving parameter a the 

 values — 1, 0, or +1 in 7"^: 



jg' = {\ -a)g + {a- \)h 

 «•[/?'= 2^+ lah. 



Ex. 2: What are the two transformations given when the parameter a takes the 

 value or I in 5?: 



h' = (\ - a)J + log (1 + a + sin ah) 



j' = (1 + sin ay>(°-')/(. 



Ex. 3: The transducer n' = n + a^, in which a and n can take only positive 

 integral values, is started at « = 10. (i) At what value should a be kept if, 

 in spite of repeated transformations, n is to remain at 10? (ii) At what 

 value should a be kept if n is to advance in steps of 4 at a time (i.e. 10, 14, 

 18, . . .)? (iii) What values of a, chosen anew at each step, will make n 

 follow the series 10, 11, 15, 16, 20, 21, 25, 26, . . ., in which the differences 

 are alternately 1 and 4? (iv) What values of a will make n advance by unit 

 steps to 100 and then jump directly to 200? 



Ex. 4: If a transducer has n operands and also a parameter that can take n 

 values, the set shows a triunique correspondence between the values of oper- 

 and, transform, and parameter if (1) for given parameter value the trans- 

 formation is one-one, and (2) for given operand the correspondence between 

 parameter-value and transform is one-one. Such a set is 



Show that the transforms must form a Latin square, i.e. one in which 

 each row (and each column) contains each transform once and once only. 

 Ex. 5 : A certain system of one variable V behaves as 



- = ro(--?). 



where P is a parameter. Set P at some value Pi, e.g. 10, and find the limit 

 that V tends to as the transformation is repeated indefinitely often; call 

 this limit Vi. Then set P at another value P2, e.g. 3, and find the corres- 

 ponding limit V2. After several such pairs of values (of P and limit- F) 

 have been found, examine them to see if any law holds between them. Does 

 V behave like the volume of a gas when subjected to a pressure PI 

 Ex. 6: What transformation, with a parameter a, will give the three series of 

 values to «?: 



a=i: 0,^l,-> 2,-> 3,^ 4,... 



a = 2: 0,^4,^ 8,^12,^16,... 



fl = 3 : 0, -> 9, -^ 18, ^ 27, ^ 36, . . . 

 (Hint : Try some plausible expressions such as n' = n + a, n' = ahi, etc.) 

 Ex. 7: If n' = n -\- 3a, does the value given to a determine how large is n's 

 jump at each step? 



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