From Osher Doctorow
I have often pointed out in threads in sci.physics that the Riccati
Differential Equation is the Fundamental equation of Probable
Influence/Causation (PI), where the former is:
1) dy/dt = A(t) + B(t)y + C(t)y^2
while the latter is:
2) P(A-->B) = 1 + y - x, 0 < = y < = x < = 1
The most important cases of the former are the exponential growth/
decay and the logistic differential equations, namely respectively:
3) dy/dt = ky
4) dy/dt = ky(1 - y) (for y normalized in the interval [0, 1])
Let's set k = 1 for simplicity in (4), and so we have:
5) dy/dt = y(1 - y) = y - y^2
Since P(A-->B) of (2) is the PI analog of dy/dt (the latter is
"Birkhoff Causation") except that PI arguably involves partial
derivatives rather than just ordinary derivatives, we want to know
whether the term y of (2) closely approximates y - y^2 of (5) and
where. The argument for x in (2) can be similar, or we can just
attribute x to the two-variable scenario.
We have:
Theorem. y closely approximates y^2 when y is very near to 0 ("Very
Rare events").
6) y - y^2 = y(1 - y) = (y - 0)(1 - y)
so y - y^2 has two roots, y = 0 and y = 1, at which y - y^2 is 0. By
continuity, in small enough neighborhoods respectively of y = 0 and of
y = 1, y - y^2 is very close to 0. Q.E.D.
For example, if y is 1/100, which is very near 0, then y - y^2 is
1/100 - 1/10,000 which is very near 0.
For the behavior of y - y^2 elsewhere in [0, 1], readers can examine
the first and second derivatives and show that there is a local
maximum at y = 1/2 and that y - y^2 increases from 0 to 1/4 in [0,
1/2] and decreases from 1/4 to 0 in [1/2, 1].
Osher Doctorow
.
|
