Riccati Differential Equation Relates Probability, Geometry, Physics, Engineering, Cosmology
OsherD
COPYRIGHT NOTICE
Riccati Differential Equation Relates Probability, Geometry, Physics,
Engineering, Cosmology
Copyright By Owner Osher Doctorow Ph.D.
First Published 2006
The Riccati Differential Equation relates probability, geometry,
physics, engineering. The equation is:
1) dy/dt = A(t) + B(t)y + C(t)y^2
and can involve real scalar functions, vectors, matrices, and can be
even generalized to the partial derivative Dt(y) replacing dy/dt.
In engineering, kalman filter-predictors and optimal control theory
(e.g., dynamic programming) centrally involve the Riccati Differential
and Algebraic (setting dy/dt = 0 in the latter) equations.
In cosmology and other branches of physics, simple exponential
growth/decay as special cases of the Riccati Differential Equation
describe both "explosive" expansion and contraction of the Universe and
various subpopulations of the Universe, while the Logistic Differential
Equation as a special case of Riccati describes supply-limited growth.
(to be continued hopefully)
Osher Doctorow
OsherD
The exponential and rational exponential (fraction containing
exponentials in numerator and denominator) solutions of the Riccati
Differential Equation explain explosive growth/decay.
Now hold that thought and take a look at the Logistic Differential
Equation:
1) dy/dt = ky(1 - y)
Although 1 on the right hand side could be replaced by another
constant, say k2, the equation is especially valuable when used with 1
in which case y is a proper fraction or proportion. But that almost
instantly reminds probability-statistics people of probability itself
and cumulative distribution functions (cdfs) and most probability
density (pdf) and all probability mass functions which are between 0
and 1. Sometimes a pdf jumps above 1 for some families or subfamilies
of pdfs which get intensely concentrated about a point (although only
in a small neighborhood of that point usually).
So the Riccati Differential Equation in Logistic form suggests
probabilistic solutions. Since a remarkable number of pdfs and cdfs
involve exponentials and power functions and their products, the
connection between geometry and probability is actually coming about
via such things as exponential functions and power functions or their
products, as well as the ubiquitous "uniform" distributions which are
constant on their interval of support:
2) fX(x) = 1/(b - a), x in (a, b}
and linear in their cdfs:
3) FX(x) = (x - a)/(b - a) on (a, b)
where FX(x) is the cdf of random variable X and fX(x) = dFX(x)/dx is
the pdf of X and the latter is 0 outside the interval in question.
Notice that dFX(x)/dx = constant satisfies the Riccati Differential
Equation with A(t) = constant and B(t) = C(t) = 0.
OsherD
Now we notice something curious:
1) exp(x) = exp(-x) iff x = 0, in which case exp(x) = 1 = exp(-x)
Exponential functions have a curious connection with 1 and 0 and
thereby with the interval [0, 1]. Logarithmic functions also do, but
rather disasterously when x = 0 in terms of the undefined log(0) and
the blow-up very near x = 0 from above. This still leaves log(1) = 0,
but log(0) doesn't quite make it.
Arguably, this is what relates cosmological geometry to probability.
"Acceleration" or "explosion" at high levels have a remarkable
relationship to deceleration or contraction at high levels and to
constancy or even linearity. They are much less noticeable in
tangential or one-direction-at-a-time motion which is not
expansive-contractive, and physics has been mostly preoccupied with the
latter.
These "little details" suggest a level of analysis far beyond the
interrelationships of category theory and algebraic geometry and
algebraic topology. For example, the above paragraphs indicate
cosmology-geometry-probability-combinatorics interrelations. This
involves 4 major fields, not two as in algebraic geometry and algebraic
topology. The hint seems to emerge that "little details involving
many disciplines" are being missed in the search for
interdisciplinarity via algebra and its "logical" applications (by the
latter I mean its theorems and proliferated definitions that allegedly
push it into geometry or topology).
Of course, once probability is in the picture, so is logic via fuzzy
multivalued logics (FMLs).
Osher Doctorow
OsherD
Do I mean to say that orientation toward 0 and 1 are "combinatoric"
orientations and that algebraic orientation tends to exclude this?
Yes!
The algebraic orientation can be summarized by Saunders MacLane's and
S. Eilenberg's category theory: compose functions and morphisms,
emphasize morphisms (isomorphisms, homomorphisms, etc.), "go beyond
sets".
Let me say this more "graphically". If an algebraist were asked to
build a maze in a sort of rat type experiment, he/she would keep
composing functions, go beyond sets, morph ad infinitum. An algebraist
is happy when he/she looks at something like this:
1) sin(exp(2x - y) + z^5 (x - sinh(log(z)))^3)
But that's only the start of it. (1) would be the start of the maze,
which would eventually stretch to hundreds of multiplications and
divisions and composite functions ("parentheses within parentheses but
not only multiplication").
If a probability-cosmology-geometry-combinatorics-FML(logic) person
were to be run through a similar experiment, he/she would probably come
up with 0, 1, or [0, 1].
>From the viewpoint of simplicity, I think that Einstein, Paul Dirac,
Steven Weinberg, Pierre de Fermat would be very happy with the
probability-cosmology-geometry-combinatorics-FML(logic) person. I
think that the algebraist would be sent back to a lower year of school
:>)
Osher Doctorow
tadchem
OsherD wrote:
> >From Osher Doctorow mdoc...@comcast.net
>
> COPYRIGHT NOTICE
> Riccati Differential Equation Relates Probability, Geometry, Physics,
> Engineering, Cosmology
> Copyright By Owner Osher Doctorow Ph.D.
> First Published 2006
>
> The Riccati Differential Equation relates probability, geometry,
> physics, engineering. The equation is:
>
> 1) dy/dt = A(t) + B(t)y + C(t)y^2
The Ricatti equation with constant coefficients
y' + Ay^2 + By + C = 0
has several solutions depending on the value of the discriminant B^2 -
4AC
Let D = (B^2-4AC) and k an arbitrary scalar constant
When B^2 - 4AC > 0:
y = +/- sqrt(D)/2A*tan(-/+sqrt(D)x + k) - B/2A
When B^2-4AC < 0:
y = +/- sqrt(-D)/2A*tanh(-sqrt(-D)x + k) - B/2A
When B^2-4AC < 0:
y = +1/(Ax + k) - B/2A
When A, B, or C are *variables* the situation becomes quite complex and
the form of the answer depends on the value of D, which itself will
vary.
Tom Davidson
Richmond, VA