New idea is involved, only a new manipulation of symbols. (The
Reader is advised to work through all the exercises, since many
Important features appear, and they are not referred to elsewhere.)
Ex. 1: If the operands are of the form (a,b), and one of them is (1/2,2), find the
Vectors produced by repeated application to it of the transformation T:
a' = b
T: b' =– a
Hint: find T(1/2,2), T2(l,2), etc.)
Ex. 2: If the operands are vectors of the form (v,w,x,y,z) and U is
v' = w
w' = v
U: x' = x
y' = z
z' = y
find U(a), where a = (2,1,0,2,2).
Ex. 3: (Continued.) Draw the kinematic graph of U if its only operands are a,
U(a), U2(a), etc.
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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
TH E D ET ERM IN A TE MA C HI N E
Ex. 4: (Continued.) How would the graph alter if further operands were
Added ?
Ex. 5: Find the transform of (3, – 2,1) by A if the general form is (g,h,j) and the
Transformation is
g' = 2g – h
A: h' = h – j
j' = g + h
Ex. 6: Arthur and Bill agree to have a gamble. Each is to divide his money into
Two equal parts, and at the umpire’s signal each is to pass one part over to the
Other player. Each is then again to divide his new wealth into two equal parts
And at a signal to pass a half to the other; and so on. Arthur started with 8/-
And Bill with 4/-. Represent the initial operand by the vector (8,4). Find, in
Any way you can, all its subsequent transforms.
Ex. 7: (Continued.) Express the transformation by equations as in Ex. 5
Above.
Ex. 8: (Continued.) Charles and David decide to play a similar game except that
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Each will hand over a sum equal to a half of what the other possesses. If they
Start with 30/- and 34/- respectively, what will happen to these quantities ?
Ex. 9: (Continued.) Express the transformation by equations as in Ex. 5.
Ex. 10: If, in Ex. 8, other sums of money had been started with, who in general
Would be the winner?
Ex. 11 : In an aquarium two species of animalcule are prey and predator. In each
Day, each predator destroys one prey, and also divides to become two pred-
Ators. If today the aquarium has m prey and n predators, express their
Changes as a transformation.
Ex. 12: (Continued.) What is the operand of this transformation?
Ex. 13: (Continued.) If the state was initially (150,10), find how it changed over
The first four days.
Ex. 14: A certain pendulum swings approximately in accordance with the trans-
formation x' = 1/2(x– y), y' = 1/2(x + y), where x is its angular deviation from
the vertical and y is its angular velocity; x' and y' are their values one second
Later. It starts from the state (10,10); find how its angular deviation changes
from second to second over the first eight seconds. (Hint: find x', x", x"', etc.;
can they be found without calculating y', y", etc.?)
Ex. 15: (Continued.) Draw an ordinary graph (with axes for x and t) showing how
X’s value changed with time. Is the pendulum frictionless ?
Ex. 16: In a certain economic system a new law enacts that at each yearly read-
Justment the wages shall be raised by as many shillings as the price index
Exceeds 100 in points. The economic effect of wages on the price index is
Such that at the end of any year the price index has become equal to the wage
Rate at the beginning of the year. Express the changes of wage-level and
Price-index over the year as a transformation.
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Ex. 17: (Continued.) If this year starts with the wages at 110 and the price index
At 110, find what their values will be over the next ten years.
Ex. 18: (Continued.) Draw an ordinary graph to show how prices and wages will
Change. Is the law satisfactory?
Ex. 19: (Continued.) The system is next changed so that its transformation
becomes x' = 1/2(x + y), y = 1/2(x– y) + 100. It starts with wages and prices
Both at 110. Calculate what will happen over the next ten years.
Ex. 20: (Continued.) Draw an ordinary graph to show how prices and wages will
Change.
Ex. 21: Compare the graphs of Exs. 18 and 20. How would the distinction be
Described in the vocabulary of economics?
Ex. 22: If the system of Ex. 19 were suddenly disturbed so that wages fell to 80
And prices rose to 120, and then left undisturbed, what would happen over
The next ten years? (Hint: use (80,120) as operand.)
Ex. 23: (Continued.) Draw an ordinary graph to show how wages and prices
Would change after the disturbance.
Ex. 24: Is transformation T one-one between the vectors (x1, x2) and the vectors
(x1', x2') ?
x ' = 2x + x12T: 1
x2' = x1 + x2
(Hint: If (x1, x2) is given, is (x1', x2') uniquely determined ? And vice versa ?)
*Ex. 25: Draw the kinematic graph of the 9-state system whose components are
Residues:
x' = x + y
(Mod 3)
y' = y + 2
How many basins has it ?
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