From Osher Doctorow Ph.D.

NASA at http://science.nasa.gov/headlines/y2003/09sep_blackholesounds.htm reports in "Black hole sound waves," Sept. 9, 2003 (the story can probably be accessed by the title as keywords if you have any problem with the http) that the following (in my words) seems to be roughly occurring in galaxy clusters to explain why they have so much hot gas and so little cool gas. Heat and light appears to be transformed into sound waves in vast cavities extending away from central black holes (in particular the Perseus cluster 250 million light years from Earth studied by the Institute of Astronomy in Cambridge, England, in coordination with NASA), and the sound waves then dissipate in the cluster gas to keep the gas warm. In fact, the deepest sound "note" ever detected in the Universe (57 octaves lower than middle-C in music, as compared with a typical piano that only contains about 7 octaves) has been detected by them coming from the Perseus cluster and its supermassive black hole. Readers who've been following my development of logistic growth equations here and on researchmathematics@yahoogroups.com (a moderated group) may be aware of my recent posting isolating the "momentum kernels" mv and v of the superluminal and subluminal logistic models respectively: 1) d(mv)/dt = kmv(1 - mv) 2) d(v)/dt = k1v(1 - v) Alternatively to (2), velocity or speed v could be replaced by kv where k1 is a constant usually having magnitude 1 but dimension that of mass which might be approximated by sqrt(1 - v1^2/c^) times a constant very close to the speed of light c and where v1 is a constant speed near but below the speed of light. In the second site above I isolated the superluminal/subluminal ratios: 3) mv/v 4) mv/(k1v) where the first has dimension of mass and the second is dimensionless since the mass and speed dimensions cancel. Now consider this equation, where speed v is written u and s is the speed of sound and m is mass: 5) (u/s)(s/c)(c/(mu))(mu/(uk1)) = 1/k1 This is a simple algebaic identity, but (u/s) is the well known Mach Number (Mach for short here) in dimensional analysis, mu/(k1u) is my superluminal/subluminal ratio which I abbreviate as SupSub here, and I will refer to s/c as PERSEUS because of the NASA results that relate sound to heat and light. So (5) becomes: 6) Mach * PERSEUS * (c/(mu))* SupSub = 1/k1 and where * is just multiplication of real numbers. Surely c/(mu) can't be important? On the contrary, it is the partial "reflection" of mu/u (ratio of superluminal momentum to subluminal "momentum" with constant mass) into the ratio of light speed ("light momentum" with the mass replaced by 1 since photons are supposed to have mass zero) to superluminal momentum. We could credit the NASA results (although not their theories, which didn't notice this) with Mach and PERSEUS dimensionless ratios, but logistic growth definitely gets credit for SupSub and c/(mu) (the first dimensionless, the second dimensional with dimension of mass to the negative 1 exponent). Equation (6) can be used by itself even though it is only an identity, but it also has a second use: it tells us the importance of the four ratios which can be used either separately or in combination in astrophysical and cosmological and various mathematical physics or theoretical physics problems. Since growth via the logistic equations and fractals and chaos are so interrelated, we seem to have entered a new "dimension" of applications of fractals and chaos. Osher Doctorow Ph.D.