From Osher Doctorow Ph.D.
Readers may recall some of my contributions on Logic-Based Probability- Statistics (LBP) and Logical-Experimental Unification Theory (LEUT) and Rare Event Theory (RET) from some years back. My published paper in B. N. Kursunuglu et. al. Quantum Gravity, Generalized Theory of Gravitation, and Superstring Theory-Based Unification, Kluwer: N.Y. 2000, 89-97, is a good presentation of some of this from the viewpoint of LEUT. Some good recent work of mine is on www.researchmathematics@yahoogroups.com (a moderated forum), geometry-research@forum.swarthmore.edu which can also be accessed via www.mathforum.com, http://www.superstringtheory.com/forum in the String/M Theory - Duality subforum of their Forum section, Stardrive1@yahoogroups.com. Geometric Physics uses not only the 11 fundamental elements which I mentioned last time (Expansion, Contraction, Knowledge as Semantics, Information as Syntactics, Transmission, Rarity, Non-Rarity, inside, outside, potential, actual), but also two very powerful interdisciplinary relationships: A. The cross-discipline expression 1 + y - x for y < = x and usually x and y between 0 and 1 inclusive. In probability-statistics, this is the probable influence of x on y for Rare Events. In fuzzy multivalued logics (FMLs), this is implication (x-->y) such that x implies y for Lukaciewicz and Rational Pavelka FMLs which describe Rare Events. In proximity-geometry-topology, this is p1(x, y) or pn(x, y) for n = 2, 3, 4, ..., where pn(x, y) is the proximity or "nearness" of n-dimensional point or vector x to point y from the viewpoint of x (a one-sided "nearness") for Rare Events. B. The cross-discipline relationships between Expansion and Contraction. These cross the expansion of the Universe as a whole, biological growth and expansion- contraction, expansion-contraction of human (and perhaps animal) consciousness, expansion-contraction in fractals and chaos, self-similarity in dimensional analysis, expansion of radiation from a source, "implosion" or collapse of black holes, expansion of quantum entanglement, and even evolution of galaxies and organic memory materials including plastics. I want to begin here with a rather curious real-valued rather than complex- valued expression which comes from Geometric Physics including both quantum theory and general relativity: 1) dx/dt < 2x/t where dx/dt is the calculus derivative or instantaneous rate of change of x (say a spatial direction) with t (which we'll take as time). In order to x to exert probable influence on t in BOTH a certain quantum and a certain general relativisic (macroscopic) scenario, equation (1) must hold. For those who don't feel comfortable with their calculus or need refreshing in it, dx/dt is the speed of x for x a real variable representing position in space along a straight line or axis - not the average speed of elementary algebra but the instantaneous speed at time t, which is often indicated by writing x as x(t) to mean "x at time t exactly" (not x times t). Readers may find some interest in exploring (1) for the parabolas t = x^2, x = t^2, the ellipse (x^2/a^2) + (t^2/b^2) = 1, the exponential t = t(0)exp(kx), the exponential x = x(0)exp(kt). It turns out that equation (1) is best satisfied for t = x^2 with fewest restrictions, which from the equation t = x^2 brings to our attention the fact that dt/dx, which is 2x, is one way of measuring the SENSITIVITY of t to a change in x. When dt/dx is large, then dx'dt is small if it exists, which relates to (1). I'll try to continue shortly. Osher Doctorow Ph.D.