Science > Physics > Quantum Gravity Via Expansion-Contraction 17.0: The Quadratic Asymmetric Nonlinear Nature of Causation Via Riccati
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
27 Aug 2006 01:46:54 AM |
| Object: |
Quantum Gravity Via Expansion-Contraction 17.0: The Quadratic Asymmetric Nonlinear Nature of Causation Via Riccati |
From Osher Doctorow
The Riccati Differential equation:
1) dy/dt = A(t) + B(t)y + C(t)y^2
can better be understood by generalizing it to:
2) Dt(y(x, t)) = A(t) + B(t)y + C(t)y^2, Dt = partial derivative with
respect to t
In the scenarios discussed in recent posts, we have:
3) Dt(y(x,t)) = A(t) -/B(t)/y - /C(t)/y^2, A(t) nonnegative
The asymmetry here between the A, B, and C terms is rather obvious when
y is in [0, 1], but there is also a geometric similarity in the general
parabola using x instead of y because of convention:
4) x = ay^2 + by + c
Parabolas have asymmetric x and y powers, and in the (y, Dt(y)) plane
we can regard (3) as a time-varying parabola. This differs from
ellipses (including circles) and hyperbolas, and there are
corresponding results for 3-dimensional generalizations (quadric
surfaces) including ellipsoids, paraboloids, etc. Ellipses and
ellipsoids are symmetric in x and y powers or at least degrees, which
parabolas and paraboloids aren't. Hyperbolas and hyperboloids are a
bit more complicated because of the negative sign always.
The origin of this asymmetry can be found in Probable
Influence/Causation (PI), although even an intuitive causal argument
begins to detect it. In physics, it is a "great fad" to claim that
something is causal if it doesn't violate light speed constraints in
terms of accessibility/motion and the light cone. This is like saying
that "what part of causal don't you understand" refers to the "c" in
causal, not to over-do a pun. In fact, however, physics intuition
recognizes causation in terms of priority in time or else simultaneity
and in terms of change of the prior thing resulting in change of the
subsequent thing and analogously for simultaneity. I'll discussion PI
below.
PI (Probable Influence/Causation) has an "underlying" asymmetry which
sometimes changes to symmetry in a "reflexive" way, that is to say
comparably to a < = b and b < = a yielding a = b so that the
description of a, b changes from inequality to equality although the
latter is in a sense implicit in the former. So P(A-->B) with A the
causing and B the (probably) caused set/event and P(B-->A) with these
reversed "changes' to P(A<-->B) in simultaneity and in fact:
5) (A<-->B) = (A-->B)(B-->A) (adjacent parentheses indicate
intersection)
by definition (an analogous situation occurs in logic, where a<-->b =
(a-->b) ^ (b-->a) for a, b propositions and ^ conjunction ("and")).
Although the asymmetry only becomes super-obvious in Probability and
Causation and (mathematical) logic, it is also found in nonlinear
differential equations including the Riccati Differential equation. It
is also found in Nonsmooth Analysis (look this up as keywords on the
internet or arXiv/Front for the Mathematics ArXiv) where inequality and
set inclusion theorems almost entirely replace equalities.
Osher Doctorow
.
|
|
| User: "OsherD" |
|
| Title: Re: Quantum Gravity Via Expansion-Contraction 17.0: The Quadratic Asymmetric Nonlinear Nature of Causation Via Riccati |
27 Aug 2006 02:08:23 AM |
|
|
From Osher Doctorow
There are two ways to view the results indicated in the previous post:
a "strong" and a "weak" way. The "weak" way says essentially that the
reason for the Riccati Differential equation embodying Quantum Gravity
and/or much if not all of physics is the nature of Probable
Influence/Causation, period - end of argument. The "strong" way says
that physics should not be based on symmetric equations except as
special cases of usually asymmetric equations which change to symmetric
by combining two opposite directional asymmeties such as a < = b and b
< = a.
If the "strong" view of the previous results is correct, then neither
mainstream correlations nor Probable Correlation P(A<-->B) should not
be regarded as the basis of physics and much of the structure of
statistical mechanics and statistical thermodynamics and Quantum Theory
is questionable if not undermined. There may also be a question as to
whether invariance principles and symmetry groups that embody them are
really an ultimate foundation for physics rather than merely a part-way
milestone in physics.
But what do we make of the Riccati Differential equation generalized
to:
1) Dt(y(x, t)) = A(t) + B(t)y + C(t)y^2
in terms of the lack of specification of x in this equation? Wouldn't
it be more aesthetically pleasing to have a system of equations rather
than one equation (1), where the second equation of the system gives
Dx(y(x,t))?
There is implicitly a second expression that complements equation (1),
namely either one or both of the following:
2) 0 < = y < = x < = k (k usually but not always 1)
3) 0 < = y < = t < = k (ditto for k)
I would coin the name "System of Eq-Inequations" for (1) and either (2)
or (3), or something analogous (perhaps less tongue-tying). The point
is that x, y, and t are not being one-sidedly ignored by (1), but
rather are all involved in (1), (2), and/or (3).
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Quantum Gravity Via Expansion-Contraction 17.0: The Quadratic Asymmetric Nonlinear Nature of Causation Via Riccati |
27 Aug 2006 02:13:02 AM |
|
|
From Osher Doctorow
The second paragraph of the previous posting has a double negative or
triple negative typo, "neither...nor...should not...," which should
eliminate the last "not".
Osher Doctorow
.
|
|
|
|
|
|
Related Articles |
|
|
