Output) of the one machine into those of the other can convert the
One representation to the other.
Thus, in the example given, apply the one-one transformation P
The transformation P:
Uv
Y – x
Is a shorthand way of describing the one-one transformation that
Pairs off states in S and R thus:
In S,
,, ,,
,, ,,
,, ,,
I.e. ,, ,,
(Compare U of S.4/9.)
Result is
Against (– 3,2) in R
(1,0) ,,(0,1) ,, ,,
(4,5) ,,(– 5,4) ,, ,,
(– 3,0) ,,(0,– 3) ,, ,,
U,v) ,,(– v,u) ,, ,,
Apply P to all the description of S; the
δ ε g h j k
P: ↓ β α c a b d
To N’s table, applying it to the borders as well as to the body. The
Result is
↓
P: ↓
c
a
b
d
D b a c
D a c c
This is essentially the same as M. Thus, c and β in the border give
β
α
y' = – y + x
– x' = – y – x
Which is algebraically identical with R. So R and S are isomorphic.
Ex. 1: What one-one transformation will show these absolute systems to be iso-
Morphic?
a b c d ep q r s tY: ↓ c c d d bZ: ↓ r q q p r
Fig. 6/9/2
Hint: Try to identify some characteristic feature, such as a state of equilib-
Rium.)
Ex. 2: How many one-one transformations are there that will show these absolute
Systems to be isomorphic?
a b cB: ↓ p q rA: ↓ b c ar p q
*Ex. 3: Write the canonical equations of the two systems of Fig. 6/8/1 and show
That they are isomorphic. (Hint: How many variables are necessary if the sys-
Tem is to be a machine with input ?)
Ex. 4: Find a re-labelling of variables that will show the absolute systems A and
B to be isomorphic.
x' = – x2 + y u' = w2 + u
2A: y' = – x – yB: v' = – v2 + w
|
|
|
2 z' = y + z w' = – v2 – w
(Hint: On the right side of A one variable is mentioned only once; the same
Is true of B. Also, in A, only one of the variables depends on itself quadrat-
ically, i.e. if of the form a' = + a2 . . . ; the same is true of B.)
D in both. The isomorphism thus corresponds to the definition.
(The isomorphism can be seen more clearly if first the rows are
Interchanged, to
↓ c a b d
D a c c
D b a c
And then the columns interchanged, to
↓
α
β
a
b
c
d
A c d c
B a d c
But this re-arrangement is merely for visual convenience.)
When the states are defined by vectors the process is essentially
Unchanged. Suppose R and S are two absolute systems:
R: x' = x + yS: u' = – u – v
y' = x – y v' = – u + v
98
α
β
The previous section showed that two machines are isomor-
Phic if one can be made identical to the other by simple relabel-
Ling. The “re-labelling”, however, can have various degrees of
Complexity, as we will now see.
99
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 BL AC K B O X
The system that is specified only by states, as in the previous
Section, contains no direct reference either to parts or to variables.
In such a case, “re-labelling” can mean only “re-labelling the
States”. A system with parts or variables, however, can also be
Re-labelled at its variables— by no means the same thing. Relabel-
Ling the variables, in effect, re-labels the states but in a way sub-
Ject to considerable constraint (S.7/8), whereas the re-labelling of
States can be as arbitrary as we please. So a re-labelling of the
|
|
|
States is more general than a re-labelling of the variables.
Thus suppose a system has nine states; an arbitrary re-labelling
Of eight of the states does not restrict what label shall be given to
The ninth. Now suppose that the system has two variables, x and
Y, and that each can take three values: x1, x2, x3 and y1, y2, y3. Nine
States are possible, of which two are (x2,y3) and (x3,y1). Suppose
This system is re-labelled in its variables, thus
Ex. 1: (Ex. 6/9/4 continued.) Compare the diagram of immediate effects of A and
B.
Ex. 2: Mark the following properties of an absolute system as changed or
Unchanged by a re-labelling of its states: (i) The number of basins in its
Phase-space; (ii) whether it is reducible; (iii) its number of states of equilib-
Rium; (iv) whether feedback is present; (v) the number of cycles in its
Phase-space.
Ex. 3: (Continued.) How would they be affected by a re-labelling of variables?
↓
X y
ξ η
The subject of isomorphism is extensive, and only an intro-
Duction to the subject can be given here. Before we leave it, how-
Дата добавления: 2019-11-16; просмотров: 213; Мы поможем в написании вашей работы! |
Мы поможем в написании ваших работ!