Science > Physics > MOND, Negative & "Generalized Probabilities" Via PI 2: Zhao and Famaey and Riccati
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
01 Apr 2006 06:05:10 PM |
| Object: |
MOND, Negative & "Generalized Probabilities" Via PI 2: Zhao and Famaey and Riccati |
From Osher Doctorow
H. S. Zhao of U. St. Andrews, Scotland U.K., and B. Famaey of U. Libre
de Bruxelles, Belgium in "Refining MOND interpolating function and
TeVeS Langrangian, astro-ph/0512425 v3 5 Jan 2006, point out that the
Famaey and Binney 2005 interpolating function for the "constant" G of F
= Gm1m2/r^2 (that is, G is replaced by the interpolating function more
or less) is:
1) u~(x) = x/(1 + x)
This, which they don't mention, "obeys" in its key expression the
Riccati Differential Equation with t replaced by x:
2) dy/dt = A(t) + B(t)y + C(t)y^2
since if y = x/(1 + x), then by long division:
3) x/(1 + x) = 1 - 1/(x + 1)
Now set y equal to 1/(x + 1) instead of x/(1 + x):
4) y = 1/(x + 1)
where x is not -1, and therefore:
5) dy/dx = -1/(x + 1)^2
Let's rewrite (3) as:
6) x/(1 + x) = 1 - y
Then:
7) d(x/(1 + x))/dx = - dy/dx = + 1/(x + 1)^2 from (5) = y^2
Substituting from (6) into (7) yields:
8) d(1 - y)/dx = y^2
and therefore:
9) 0 - dy/dx = y^2
which can be rewritten:
1) dy/dx = - y^2
which is the Riccati Differential Equation (2) with A(t) = 0 and B(t) =
0 and C(t) = C(x) = -1.
The Riccati Differential Equation is fundamental to Probable
Influence/Causation (PI) as I have argued for years on sci.physics and
elsewhere. For example, it embodies expansion-contraction both
exponentially via exponential growth/decay (dy/dt = ky) and logistic or
supply-limited growth (dy/dt = ky(1 - y) when y is
normalized/standardized).
Osher Doctorow
.
|
|
|
Related Articles |
MOND, Negative & "Generalized Probabilities" Via PI
MOND, Negative & "Generalized Probabilities" Via PI 3: The Dark Matter Alternative
Quantum Gravity 156.0: Generalized P and P ' PI Via Generalized-Modified Uniform Distribution Weighting
Quantum Gravity 211.7: F = k1r - k2/r^2 Via PI, Phase Theory,Generalized Hyperbolic Function Theory, Newton-Hooke-Laplace
Quantum Gravity 156.1: More Re Generalized P and P ' PI Via Generalized-Modified Uniform Distribution Weighting
Quantum Gravity 119.5: Bosonic Fields Via Generalized Lie Derivatives and Supersymmetry Algebra
Quantum Gravity 118.2: Phase Change Via v/c --> c/v in Generalized SR
Quantum Gravity Via Expansion-Contraction 2.1: Generalized W Lambda Function
| Osher Doctorow's Reply To Tony Scott (Re Quantum Gravity Via Expansion-Contraction 2.0: The Generalized Lambert W Function Relates QM and GR (Harvard, Waterloo, Aachen)
Coulomb Forces Generalized Via Newton-PI
Quantum Gravity Via Expansion-Contraction 2.0: The Generalized Lambert W Function Relates QM and GR (Harvard, Waterloo, Aachen)
Independent/Dependent Phases 14.1: 1, 2, sqrt 3 in Generalized Isoperimetric
Independent/Dependent Phases 14: Generalized Isoperimetric Inequalities
Derivative Products of Form (df/dx)(dg/dx) in Physics 14: Leibniz and Generalized Leibniz Rules
Quantum Gravity 171.3: Dirac and Pauli Matrices As Generalized Alternating Sign Matrices
|
|
|
