3/5 AN INTRODUCTION TO CYBERNETICS 



Ex. 9: In each decade a country's population diminishes by 10 per cent, but in 

 the same interval a million immigrants are added. Express the change from 

 decade to decade as a transformation, assuming that the changes occur 

 in finite steps. 



Ex. 10: (Continued.) If the country at one moment has twenty million in- 

 habitants, find what the population will be at the next three decades. 



Ex. 1 1 : (Continued.) Find, in any way you can, at what number the population 

 will remain stationary. (Hint: when the population is "stationary" what 

 relation exists between the numbers at the beginning and at the end of the 

 decade? — what relation between operand and transform?) 



Ex. 12: A growing tadpole increases in length each day by 1-2 mm. Express 

 this as a transformation. 



Ex. 13: Bacteria are growing in a culture by an assumed simple conversion of 

 food to bacterium; so if there was initially enough food for 10* bacteria 

 and the bacteria now number n, then the remaining food is proportional to 

 108 _ „. If the law of mass action holds, the bacteria will increase in each 

 interval by a number proportional to the product : (number of bacteria) x 

 (amount of remaining food). In this particular culture the bacteria are 

 increasing, in each hour, by 10 8/j (108 _ n). Express the changes from 

 hour to hour by a transformation. 



Ex. 14: (Continued.) If the culture now has 10,000,000 bacteria, find what the 

 number will be after 1, 2, . . ., 5 hours. 



Ex. 1 5 : (Continued.) Draw an ordinary graph with two axes showing how the 

 number of bacteria will change with time. 



VECTORS 



3/5. In the previous sections a machine's "state" has been regarded 

 as something that is known as a whole, not requiring more detailed 

 specification. States of this type are particularly common in 

 biological systems where, for instance, characteristic postures or 

 expressions or patterns can be recognised with confidence though 

 no analysis of their components has been made. The states des- 

 cribed by Tinbergen in S.3/1 are of this type. So are the types of 

 cloud recognised by the meteorologist. The earher sections of this 

 chapter will have made clear that a theory of such unanalysed states 

 can be rigorous. 



Nevertheless, systems often have states whose specification 

 demands (for whatever reason) further analysis. Thus suppose a 

 news item over the radio were to give us the "state", at a certain 

 hour, of a Marathon race now being run; it would proceed to give, 

 for each runner, his position on the road at that hour. These 

 positions, as a set, specify the "state" of the race. So the "state" of 

 the race as a whole is given by the various states (positions) of the 

 various runners, taken simultaneously. Such "compound" states 

 are extremely common, and the rest of the book will be much 



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