Point always moves either to some state at which it stops, or to
Some cycle around which it circulates indefinitely. Such a graph
Is like a map of a country’s water drainage, showing, if a drop of
Water or a representative point starts at any place, to what region
It will come eventually. These separate regions are the graph’s
Basins. These matters obviously have some relation to what is
Meant by “stability”, to which we shall come in Chapter 5.
Ex. 1: Draw the kinematic graphs of the transformations of A and B in Ex. 2/4/3.
Ex. 2: How can the graph of an identical transformation be recognised at a
Glance?
Ex. 3: Draw the graphs of some simple closed one-one transformations. What is
Their characteristic feature?
Ex. 4: Draw the graph of the transformation V in which n, is the third decimal
digit of log10(n + 20) and the operands are the ten digits 0, 1, . . ., 9.
Ex. 5: (Continued) From the graph of V read off at once what is V(8), V2(4),
V4(6), V84(5).
Ex. 6: If the transformation is one-one, can two arrows come to a single point?
Ex. 7: If the transformation is many-one, can two arrows come to a single point ?
Ex. 8: Form some closed single-valued transformations like T, draw their kine-
Matic graphs, and notice their characteristic features.
Ex. 9: If the transformation is single-valued, can one basin contain two cycles?
B → AD ← E ← C
And so on. These different shapes are not regarded as different
Graphs, provided the internal connexions are identical.
The elements that occur when C is transformed cumulatively by
U (the series C, E, D, D, …) and the states encountered by a point
In the kinematic graph that starts at C and moves over only one
Arrow at a step, always moving in the direction of the arrow, are
Obviously always in correspondence. Since we can often follow
The movement of a point along a line very much more easily than
We can compute U(C), U2(C), etc., especially if the transforma-
Tion is complicated, the graph is often a most convenient represen-
Tation of the transformation in pictorial form. The moving point
Will be called the representative point.
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23
TH E D ET ERM IN A TE MA C HI N E
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Chapter
3
T H E D ET ERM I N A T E M A C H I N E
Having now established a clear set of ideas about transforma-
Tions, we can turn to their first application: the establishment of an
Exact parallelism between the properties of transformations, as
Developed here, and the properties of machines and dynamic sys-
Tems, as found in the real world.
About the best definition of “machine” there could of course be
Much dispute. A determinate machine is defined as that which
Behaves in the same way as does a closed single-valued transfor-
Mation. The justification is simply that the definition works— that
It gives us what we want, and nowhere runs grossly counter to
What we feel intuitively to be reasonable. The real justification
Does not consist of what is said in this section, but of what follows
In the remainder of the book, and, perhaps, in further develop-
Ments.
It should be noticed that the definition refers to a way of behav-
Ing, not to a material thing. We are concerned in this book with
Those aspects of systems that are determinate— that follow regular
And reproducible courses. It is the determinateness that we shall
Study, not the material substance. (The matter has been referred to
Before in Chapter 1.)
Throughout Part I, we shall consider determinate machines, and
The transformations to be related to them will all be single-valued.
Not until S.9/2 shall we consider the more general type that is
Determinate only in a statistical sense.
As a second restriction, this Chapter will deal only with the
Machine in isolation— the machine to which nothing actively is
Being done.
As a simple and typical example of a determinate machine, con-
Sider a heavy iron frame that contains a number of heavy beads
Joined to each other and to the frame by springs. If the circum-
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Stances are constant, and the beads are repeatedly forced to some
Defined position and then released, the beads’ movements will on
Each occasion be the same, i.e. follow the same path. The whole
24
System, started at a given “state”, will thus repeatedly pass
Through the same succession of states
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