TH E D ET ERM IN A TE MA C HI N E
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 inhabit-
Ants, find what the population will be at the next three decades.
Ex. 11: (Continued.) Find, in any way you can, at what number the population
Will remain stationary. (Hint: when the population is “stationary” what rela-
Tion 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 108 bacteria and
The bacteria now number n, then the remaining food is proportional to 108–
N. 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 n (108–n). Express the changes from hour to hour by a transfor-
Mation.
Ex. 14: (Continued.) If the culture now has 10,000,000 bacteria, find what the
Number will be after 1, 2, . . ., 5 hours.
Ex. 15: (Continued.) Draw an ordinary graph with two axes showing how the
Number of bacteria will change with time.
Cerned with them. It should be noticed that we are now beginning
To consider the relation, most important in machinery that exists
Between the whole and the parts. Thus, it often happens that the
State of the whole is given by a list of the states taken, at that
Moment, by each of the parts.
Such a quantity is a vector, which is defined as a compound
Entity, having a definite number of components. It is conve-
Niently written thus: (a1, a2, . . ., an), which means that the first
Component has the particular value a1, the second the value a2,
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And so on.
A vector is essentially a sort of variable, but more complex than
The ordinary numerical variable met with in elementary mathe-
Matics. It is a natural generalisation of “variable”, and is of
Extreme importance, especially in the subjects considered in this
Book. The reader is advised to make himself as familiar as possi-
Ble with it, applying it incessantly in his everyday life, until it has
Become as ordinary and well understood as the idea of a variable.
It is not too much to say that his familiarity with vectors will
Largely determine his success with the rest of the book.
Here are some well-known examples.
A ship’s “position” at any moment cannot be described by a
Simple number; two numbers are necessary: its latitude and its
Longitude. “Position” is thus a vector with two components. One
ship s position might, for instance, be given by the vector (58 °N,
17 °W). In 24 hours, this position might undergo the transition
(58 °N, 17 °W) → (59 °N, 20 °W).
The weather at Kew” cannot be specified by a single num-
Ber, but it can be specified to any desired completeness by our tak-
Ing sufficient components. An approximation would be the
Vector: height of barometer, temperature, cloudiness, humidity),
and a particular state might be (998 mbars, 56.2 °F, 8, 72%). A
Weather prophet is accurate if he can predict correctly what state
This present a state will change to.
Most of the administrative “forms” that have to be filled in
Are really intended to define some vector. Thus the form that the
Motorist has to fill in:
Age of car: ......................
Horse-power: ..................
Colour: ............................
V EC TO RS
In the previous sections a machine’s “state” has been
Regarded as something that is known as a whole, not requiring
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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 confi-
Dence though no analysis of their components has been made. The
States described by Tinbergen in S.3/1 are of this type. So are the
Types of cloud recognised by the meteorologist. The earlier sec-
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