Is written as T2(n). The exercises are intended to make this nota-
Tion familiar, for the change is only one of notation.
Ex. 1: If f:
V is said to be the product or composition of T and U. It gives
Simply the result of T and U being applied in succession, in that
Order one step each.
If U is applied first, then U(b) is, in the example above, c, and
T(c) is a: so T(U(b)) is a, not the same as U(T(b)). The product,
When U and T are applied in the other order is
W: ↓
A b c d
B a b d
For convenience, V can be written as UT, and W as TU. It must
Always be remembered that a change of the order in the product
May change the transformation.
It will be noticed that V may be impossible, i.e. not exist, if
Some of T ’s transforms are not operands for U.)
Ex. 1: Write out in full the transformation U2T.
Ex. 2: Write out in full: UTU.
*Ex. 3: Represent T and U by matrices and then multiply these two matrices in
the usual way (rows into columns), letting the product and sum of +’s be +:
Call the resulting matrix M1. Represent V by a matrix, call it M2. Compare
M1 and M2.
↓ 3 1 2
1 2 3
What is f(3)? f(1)? f2(3)?
Ex. 2: Write out in full the transformation g on the operands, 6, 7, 8, if g(6) = 8,
g(7) = 7, g(8) = 8.
Ex. 3: Write out in full the transformation h on the operands α, β, χ, δ, if h( α) =
χ, h2( α) = β, h3( α) = δ , h4( α) = α.
Ex. 4: If A(n) is n + 2, what is A(15)?
Ex. 5: If f(n) is – n 2 + 4, what is f(2)?
Ex. 6: If T(n) is 3n, what is T2(n) ? (Hint: if uncertain, write out T in extenso.)
Ex. 7: If I is an identity transformation, and t one of its operands, what is I(t)?
Product. We have just seen that after a transformation T has
Been applied to an operand n, the transform T(n) can be treated as
An operand by T again, getting T(T(n)), which is written T2(n). In
Exactly the same way T(n) may perhaps become operand to a
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Kinematic graph. So far we have studied each transforma-
Tion chiefly by observing its effect, in a single action on all its pos-
Sible operands (e g. S.2/3). Another method (applicable only
When the transformation is closed) is to study its effect on a single
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Operand over many, repeated, applications. The method corre-
Sponds, in the study of a dynamic system, to setting it at some ini-
Tial state and then allowing it to go on, without further
Interference, through such a series of changes as its inner nature
Determines. Thus, in an automatic telephone system we might
Observe all the changes that follow the dialling of a number, or in
21
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
C H A NG E
An ants’ colony we might observe all the changes that follow the
Placing of a piece of meat near-by.
Suppose, for definiteness, we have the transformation
U: ↓
A B C D E
D A E D D
When the transformation becomes more complex an important
Feature begins to show. Thus suppose the transformation is
A B C DE F G H I J K L M N P QT: ↓ D H D I Q G Q H A E E N B A N E
Its kinematic graph is:
P
N → A → D
L
I
C
K
E
Q ← G ← F
M → B → H
If U is applied to C, then to U(C), then to U2(C), then to U3(C) and
So on, there results the series: C, E, D, D, D,... and so on, with D
Continuing for ever. If U is applied similarly to A there results the
Series A, D, D, D, . . . with D continuing again.
These results can be shown graphically, thereby displaying to the
Glance results that otherwise can be apprehended only after
Detailed study. To form the kinematic graph of a transformation,
The set of operands is written down, each in any convenient place,
And the elements joined by arrows with the rule that an arrow goes
From A to B if and only if A is transformed in one step to B. Thus
U gives the kinematic graph
C → E → D ← A ← B
(Whether D has a re-entrant arrow attached to itself is optional if
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No misunderstanding is likely to occur.)
If the graph consisted of buttons (the operands) tied together
With string (the transitions) it could, as a network, be pulled into
Different shapes:
C → E
D
Or:
B → A
J
By starting at any state and following the chain of arrows we can
Verify that, under repeated transformation, the representative
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