From Osher Doctorow Ph.D.
The simplest population growth equation is one that works for bacteria: 1) dx/dt = kx where k > 0. The simplest population decay equation is one that works for radioactive decay: 2) dx/dt = -kx (k > 0) The first has an exponential solution x(t) = x(0)exp(kt), the second a negative exponential solution x(t) = x(0)exp(-kt) I remarked some years ago that Logic-Based-Probability (LBP) and Rare Event Theory proceed somewhat differently from Category Theory, which latter is a branch of very advanced Algebra pioneered by Saunders MacLane and William Lawvere in which the "composition" operation is the main operation. For example, suppose that y = exp(t) and z = y^2, then y o z is the composition of y and z and is equal to exp(y^2), so y o z tells you to put z inside the parentheses of exp(t) instead of t (where y is exp(t)). In LBP and RET and Geometric Physics, we're mostly interested in much simpler operations. For example, we use y * z which is simply (exp(t))*(y^2) where * is just ordinary multiplication. Or we use y + z which is exp(t) + y^2. Since dx/dt is the instantaneous speed or velocity (rate of change of x as t changes), let's write it as an Operator Dt(x): 3) Dt(x) = dx/dt or more generally: 4) Dt(f(x)) = df(x)/dt Since dx/dt = kx is the simplest growth equation, our orientation toward multiplication rather than composition would suggest looking at: 5) Ut(x) = kx 6) Vt(x) = 1/(kx) because: 7) Ut(x) * Vt(x) = 1 so that Vt(x) is a multiplicative inverse of Ut(x). Now let's go one step further and define: 8) W(x, t) = 2x/t which has both a numerator and denominator variable of its arguments. So this leads us to: 9) dx/dt = 2x/t The solution of (9) is rather simple because dx/x = 2dt/t when integrated on both sides leads to log(x) = 2log(t) + c so taking exponentials to both sides leads to x = x(0)t^2. However, keep track of the inequality: 10) dx/dt < 2x/t while I remark about the Heisenberg Uncertainty Principles (HUPs) in the form: 11) U(x)U(p) > = h/(2pi) 12) U(E)U(t) > = h/(2pi) where x is position or distance/displacement (from the origin), p is momentum, E is energy, t is time, and this time U( ) means "Uncertainty of" or "Uncertainty in," so that U(x) means "uncertainty in x". If speeds are not too close to that of light and also potential or potential energy is small, we have approximately: 13) p = mv 14) E = (1/2)mv^2 where v is velocity or speed. In most applications in physics, for example to cosmology and quantum entanglement, a rather curious "oversight" is made - instead of using U( ) in (12), as in U(x), the origin variable is really used, that is to say x. For example, when physicists say that high energy forces act for short times or low energy forces act for long times or that the vacuum seethes with particle- antiparticle pairs coming into existence because short distances are associated with high momenta, they are using (for the latter) xp > h/(2pi) and for the former Et > h/(2pi) and not uncertainties. What happens if we replace U(E) by E, U(x) by x, U(v) by v, U(t) by t in equations (11) and (12)? We get: 15) x(mv) > h/(2pi) 16) (1/2)mv^2 t > h/(2pi) Now it's common to replace the inequalities in the last two equations by equalities and in that case we do have a choice of selecting slightly different quantities from h/(2pi) on the right hand sides. Let's call them constants k1 and k2 respectively: 17) xmv = k1 18) (1/2)mv^2 t = k2 Then we have, by subtracting (18) from (17) and defining k3 = k1 - k2: 19) xmv - (1/2)mv^2 t = k1 - k2 = k3 = mv(x - (1/2)vt) = k3 and therefore if (1/2)vt < x or v = dx/dt < 2x/t, then for nonnegative v the expression equal to k3 will be nonnegative. Readers will recognize this expression from the previous postings. Why would we want k3 to be nonnegative? Well, in RET and LBP and Geometric Physics, the probable influence of x on y can only be calculated if x - y is nonnegative, that is to say y < = x. So if k3 in (19) is nonnegative, then x influences t in x - (1/2)vt. I'll try to continue this shortly. Osher Doctorow