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===Note 1===
===Note 1===
'''Linear Topics : The Differential Theory of Qualitative Equations'''
<blockquote>
<p>The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.</p>
<p>William Ross Ashby, ''An Introduction to Cybernetics'', Chapman and Hall, London, UK, 1956. Reprinted, Methuen and Company, London, UK, 1964.</p>
</blockquote>
This chapter is titled "Linear Topics" because that is the heading under which the derivatives and the differentials of any functions usually come up in mathematics, namely, in relation to the problem of computing "locally linear approximations" to the more arbitrary, unrestricted brands of functions that one finds in a given setting.
To denote lists of propositions and to detail their components, we use notations like:
: <math>\mathbf{a} = (a, b, c),\ \mathbf{p} = (p, q, r),\ \mathbf{x} = (x, y, z),\!</math>
or, in more complicated situations:
or, in more complicated situations:
: <math>x = (x_1, x_2, x_3),\ y = (y_1, y_2, y_3),\ z = (z_1, z_2, z_3).\!</math>
In a universe where some region is ruled by a proposition, it is natural to ask whether we can change the value of that proposition by changing the features of our current state.
Given a venn diagram with a shaded region and starting from any cell in that universe, what sequences of feature changes, what traverses of cell walls, will take us from shaded to unshaded areas, or the reverse?
In order to discuss questions of this type, it is useful to define several "operators" on functions. An operator is nothing more than a function between sets that happen to have functions as members.
A typical operator <math>\operatorname{F}</math> takes us from thinking about a given function <math>f\!</math> to thinking about another function <math>g\!</math>. To express the fact that <math>g\!</math> can be obtained by applying the operator <math>\operatorname{F}</math> to <math>f\!</math>, we write <math>g = \operatorname{F}f.</math>
to have functions as members.
<pre>
The first operator, E, associates with a function f : X -> Y
The first operator, E, associates with a function f : X -> Y
another function Ef, where Ef : X x X -> Y is defined by the
another function Ef, where Ef : X x X -> Y is defined by the
