From Osher Doctorow Ph.D.

Following up further on Jerry Iuliano's email (see my previous posting), we find that cardioids and their more general relatives epicycloids relate strongly to optics and curve minimization (and thereby to calculus of variations which in turn relates to general relativity and quantum gravity and many other branches of physics) via such things as caustics, diacaustics, the brachistochrone problem (most rapid travel of a point acted upon only by gravity between two different fixed points), light sources at infinity (Huygens), nephroids, etc. I will let readers who are interested look these up as keywords on the internet and simply mention that when a circle of radius b rolls around a fixed circle of radius a (without slipping), a point on the first circle traces out an epicycloid, and when a = b it traces out a cycloid. How this relates to growth and expansion-contraction could well be regarded as a fascinating open problem. I want to spend the rest of this posting dealing with a rather curious "counterexample" that seems to go against the general idea of growth equations for Rare Events, namely the electromechanical analogy. Basically, a simple system of masses or weights with springs and friction acted upon by a force (simple problems in mechanical vibrations) is completely analogous to a simple electrical system (simple circuit) with a resistor and a capacitor and inductor and source of voltage, with mass ("weight" near the Earth) corresponding to inductance L, damping or friction corresponding to resistance R, spring constant ("springiness" vs "stiffness") corresponding to 1/capacitance = 1/C, impressed force F corresponding to source of voltage E(t) that depends on time t. The equation for the circuit, with Q being electrical charge, is: 1) LDtt(Q) + RDt(Q) + (1/C)Q = E(t) were Dtt(Q) is the acceleration of charge with respect to time, Dt(Q) is the velocity or speed of charge (called the current) with respect to time. Solutions of differential equations like (1), called linear differential equations of the second order, are sums of real and complex exponentials, or in real terms simple sums of exponentials and trigonometric sine and cosine functions with possibly some simple multiplication of exponential and sine or cosine factors. Those are familiar from growth expansion-contraction problems tat I have discussed except for the trigonometric functions. If we write Dt(Q) as I, the current, then Dtt(Q) is the change of current, which we can write as Dt(I), so that if the (1/C)Q term is absent we have an equation of growth in I type. Even if the LDtt(Q) term is absent, we have an equation of growth in Q. What makes equation (1) NOT a growth equation in general is the case where all the terms are not 0 except possibly for E(t). Yet the special cases where one term or two are zero other than E(t) indicate that growth is not entirely unrelated to mechanical vibrations and electrical circuits. What relates growth to mechanical vibrations and electrical circuits at a deep level is WAVES and their simulation. In mechanics, these are Vibrations. In electricity, these are Electromagnetic Waves. Waves tend to go outward from a source, although electrical or electromagnetic circuits tend to "capture" this behavior so that they travel instead circularly or rectangularly or perpendicular to these directions. When growth is "captured", it stops being similar to expansion-contraction as my previous postings have described the latter two and instead either becomes fixed (unchanging) or keeps repeating or oscillating (up and down, etc.) although sometimes the oscillations die down if the voltage source E(t) dies down for example or alternatively is 0 to begin with. This is somewhat similar to the case when planets orbit around stars or the Sun as opposed to comets or asteroids which pass by and never return (although some do return in larger orbits). I pointed out in an earlier posting that growth-expansion-contraction is typical or radiation from a source, biology, the expansion of the universe, development of galaxies (spherical to elliptical, etc.), condensation as from water to ice or gases (water vapor) to liquids (water), consciousness- perception-human memory. But we see that although the early Universe was expanding in growth, later on parts of it became attracted in cyclical and orbiting or oscillating ways to other parts, and we call this the Matter- Dominated Era as opposed to the earlier Radiation-Dominated Era. Radiation still exists to considerable extent in the Matter-Dominated Era, but some of it is "captured" by Matter and near Matter. We see that in Black Holes including photospheres also. We see that in NASA's recent finding of heat-light-to-sound- wave conversion in the Perseus cluster of galaxies which retains instead of diffuses heat outside the cluster. Rare Events are least "oscillating" or repeating - intuitively, if they repeat exactly at fixed intervals, their probabilities could no longer be less than .05 because they are perfectly certain to occur at those intervals or points in time. One could extend probability to include rarity of occurrence even if it is deterministically predictable, but I think that would be a different type of situation and I don't deal with that in my present work. Fairly Frequent Events do, however, begin to have a closer relationship to determinstic predictability because of processes similar to such repeats or oscillations that return to their original states or positions in some way. This is an explanation of why Bayesian (Conditional) Probability-Statistics or BCP, which involves division of probabilities to indicate dependencies, has been roughly fairly successful in Mainstream Fairly Frequent Event Probability- Statistics but not in applications to Rare Events usually. Very Frequent Events, such as random bombardment by molecules on dust particles, curiously enough have an intermediate status between Rare and Fairly Frequent Events/Processes, because there is so much contact between matter and radiation that there is "over-kill" - you can never tell when something will bombard you, so to speak, and these events are called Independent. With Fairly Frequent Events, there is typically only dependence on the immediately preceding Event so to speak - Markov Chains perfectly illustrate that, and have resemblance to Independent Events in that Markov Chains depend on only one preceding Event whereas Independent Events depend on NO preceding events. Very Frequent Events depend on either ALL or INFINITELY MANY preceding events, so that while more "random" in this sense than Independent Events, they also have some more predictability than Markov Chains since more past things are being simultaneously taken into consideration. It is sometimes claimed that Markov Chains do take more than the immediately past Event into consideration, but they do that by one-step-at-a-time paths like checkers or chess or event ants touching one other ant at a time in sequence rather than being simultaneously influenced by more than the immediately past Event. So when Markov Chains are used in Very Rare Event situations, they tend to go astray. Osher Doctorow Ph.D.