From Osher Doctorow

COPYRIGHT NOTICE
Riccati Equation Generalized For Universe Influencing Intersections
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005

The Riccati Differential equation is:

1) dy/dt = A(t) + B(t)y + C(t)y^2

where A(t), B(t), C(t) are real-valued functions, although there are
matrix generalization of (1) called Riccati Differential equations
also. There are also generalizations with dy/dt replaced by the
partial derivative Dt(y) of y = y(x, t) with respect to t holding x
constant, although these are less common.

For a "far out" generalization, let O(y) be some kind of operator or
operator-like expression involving y, and consider:

2) O(y) = A(t) + B(t)y + C(t)y^2

For example, take a look at this:

3) P{(A U B) -- AB} = P(A--B) = P(AB) + P(A' B' )

which I've proved earlier. Looking at the far left hand side of
equation (3), let's define:

4) O(AB) = P{(A U B) -- AB}

or in general:

5) O(C) = P{(A U B) -- C}

which we could indicate in functional argument-like form:

6) O(.) = P{(A U B) -- .)

The (generalized) operator O should probably have a subscript A U B,
but that's rather awkward here, so let's just assume that.

Now rewrite (3) as:

7) O(AB) = P(AB) + P(A' B' )

This begins to look similar to (2) which is our "generalized Riccati
Differential equation". Moreover, by Birkhoff Causation, differential
equations and hence derivatives involve Causation, and since P(D -- E)
for any set/events D, E in a probability space expresses Probable
Causation/Influence, O(AB) can be regarded as a generalization or part
of a generalizatio of derivatives. (Read my earlier postings for an
explanation of Garrett Birkhoff's views on differential equations
embodying causation).

We know that P(AB) and P(A' B' ) are in general almost arbitrary
probabilities for any set/events A, B, except for the slight constraint
that AB and A' B' are mutually exclusive or disjoint sets which still
doesn't itself have much restriction other than requiring that the sum
P(AB) + P(A' B' ) can't exceed 1 or be less than 0. So if we write:

8) A(t) = (definition) P(A' B' ), B(t) = 1, and P(AB) = y, then we can
rewrite (7) as:

9) O(AB) = A(t) + B(t)y

which has the right hand side as two of the terms of the Riccati
Differential equation and the left hand side a generalized derivative
as indicated. Since y = P(AB), we could put:

10) Q(y) = (definition) Q(P(AB)) = O(AB)

for some expression Q. There is a difficulty in doing this in the
sense that a direct correspondence between O(AB) and P(AB) is difficult
to establish if not impossible in general, but we have an idea as to
what is formally meant, and with this notation the probability of
intersections would follow a generalized Riccati Differential equation.


I'll call (9) an "almost-Riccati Generalized Differential Equation for
probabilities of intersections".

Osher Doctorow

From Osher Doctorow

It can be shown that for fixed A, we have:

1) P(A--C) = 1 + P(AC) - k, any C, k = P(A)

and since P(AC) is monotone increasing in AC (probabilities are
monotone increasing on the non-null parts of their set/event arguments
where sets are ordered by inclusion), P(A--C) is monotone increasing
in P(AC). Note that in stochastic processes and random set theory,
sets would be time-indexed by time subscripts, but for simplicity I
drop time-subscripts and assume that the "same" (in a sense) set
changes in time. Those people who prefer to use A_t instead of A (A_t
at time t) will find that the equations only change in requiring extra
subscripts.

Osher Doctorow

From Osher Doctorow

Let's take a look at this:

1) P_Z ( ( . ) -- C) = P(C) + f(t)

By putting Z = A U B into ( . ) and f(t) = (definition) P(A' B' ), and
C = AB we get the equation which I proved:

2) P{(A U B) -- AB} = P(AB) + P(A' B' )

So the equation (1) is related in an interesting way to (2). We
didn't need f(t) in a sense, but it's a mnemonic symbol that reminds us
of A(t) (not the A above) in the Riccati Differential equation dy/dt =
A(t) + B(t)y + C(t)y^2 where B(t) isn't related to B of (2). It also
helps "decouple" our thinking from P(AB) when we look at P(A' B' ),
which is useful in realizing that AB and A' B' don't constrain each
other much and similarly for their probabilities.

The Bayesians, who use conditional probability almost as a "way of
life" in mathematical probability-statistics and AI (artificial
intelligence), often use a subscript in a slightly similar way:

3) P_A(B) = P(B|A) = (definition) P(AB)/P(A) if P(A) is not 0
(undefined otherwise)

The slash / on the far right side of this equation represents division,
while the vertical symbol | in the middle expression is interpreted as
"given" or "holding constant". This is sometimes called a relative or
conditional probability measure.

Osher Doctorow

From Osher Doctorow

We could actually write:

1) P_( . ) ( ( . ) -- C) = P(C) + f(t)

where Z from last time is not needed. Compare this with:

2) P(A--B) = 1 + P(AB) - P(A)

which when replacing AB by ( . )B and A by ( . ) becomes:

3) P( ( . ) -- B) = 1 + P( ( . )B ) - P( . )

The conditional probability definition:

4) P(B|A) = P(AB)/P(A)

could also be expressed as:

5) P(B| ( . ) ) = P( ( . )B)/P( . )

The reason for stating these equations in this form is to notice that
they are "functional equations" (not functional analysis equations)
which have rather deep foundations and many open questions in
mathematics. Aczel and Renyi and D'Hombres of Canada, Hungary, France
pioneered in the study of such equations, although they didn't get to
the particular ones above. Among equations that are of key importance
are the functional equations of logarithms and exponentials which are
variants of Cauchy equations:

6) f(x + y) = f(x)f(y) (exponentials)
7) f(xy) = f(x) + f(y) (logarithms)

as well as for example the equation of the normal/Gaussian distribution
and many others.

Osher Doctorow

OsherD wrote:
for simplicity I
drop time-subscripts and assume that the "same" (in a sense) set
changes in time. Those people who prefer to use A_t instead of A (A_t
at time t) will find that the equations only change in requiring extra
subscripts.

I'm not sure I'm quite with you on this. As I explained to you
earlier:

My own favorite theory of cosmology, for which the existence of the
universe is the only evidence, predicts that the universe will expand
forever, eventually becoming empty and eventless. The gravitational
field throughout becomes increasingly small, but while it continues to
exist, particles and their anti-particles spontaneously come into
existence. The particles can be thought of as quantum tunnelling
relative to each other, which is equivalent to saying that one particle
is in the past relative to the other (as though it had travelled faster
than the speed of light). This quantum-tunnelling into the past is
more likely to be a very tiny amount rather than a large distance, but
as the vast eons of the future progress, larger and larger quantum
tunnellings will occasionally occur. After a while, particles in the
far-distant future will have tunnelled appreciable amounts into the
past, and will start appearing at the beginning of the universe.
Eventually, all the matter in the future will have tunnelled back to
the beginning of the universe, and the future will be empty of matter
and energy.

According to my theory, all the matter in the universe does not exist
until it has finished quantum-tunnelling back to the beginning of time,
at which stage (call this time T) the matter then continues to not
exist forever. In the meantime, at every single instant that occurs,
the universe, viewed as the set A_t in your description, vanishes and
is replaced by A_t+1.

The existence of the universe as I described above would also imply
that an eventless eternity of empty space has 'already' existed beyond
time T in the future. In your notation, A_t = {} for all t = T. The
existence of the beginning of the universe (at t = 0) is an event which
we know, by observation using standard astronomical equipment, has
'already' happened. According to my theory, the universe (including
the Big Bang at t = 0) is being continuously created for 0 t T.

The last part of my theory is that the universe exists as described
because of human conscious minds interacting with reality. Of course,
the whole theory could be proved wrong if the universe ceased to exist
before time T - the last part may eventually be falsified by aliens
existing elsewhere in the universe, if not by God.

OsherD wrote:
for simplicity I
drop time-subscripts and assume that the "same" (in a sense) set
changes in time. Those people who prefer to use A_t instead of A (A_t
at time t) will find that the equations only change in requiring extra
subscripts.

I'm not sure I'm quite with you on that. As I explained earlier:

My own favorite theory of cosmology, for which the existence of the
universe is the only evidence, predicts that the universe will expand
forever, eventually becoming empty and eventless. The gravitational
field throughout becomes increasingly small, but while it continues to
exist, particles and their anti-particles spontaneously come into
existence. The particles can be thought of as quantum tunnelling
relative to each other, which is equivalent to saying that one particle
is in the past relative to the other (as though it had travelled faster
than the speed of light). This quantum-tunnelling into the past is
more likely to be a very tiny amount rather than a large distance, but
as the vast eons of the future progress, larger and larger quantum
tunnellings will occasionally occur. After a while, particles in the
far-distant future will have tunnelled appreciable amounts into the
past, and will start appearing at the beginning of the universe.
Eventually, all the matter in the future will have tunnelled back to
the beginning of the universe, and the future will be empty of matter
and energy.

According to this theory, all the matter in the universe does not exist
until it has finished quantum-tunnelling back to the beginning of time,
at which stage (call this time T) the matter then continues to not
exist forever. In the meantime, at every single instant that occurs,
the universe, viewed as the set A_t in your description, vanishes and
is replaced by A_t+1.

The existence of the universe as described above implies that an
eventless eternity of empty space has 'already' existed beyond time T
in the future. In your notation, A_t = {} for all t = T. The
existence of the beginning of the universe (at t = 0) is an event which
we know, by observation using standard astronomical equipment, has
'already' happened. According to my theory, the universe (including
the Big Bang at t = 0) is being continuously created for 0 t T.

The last part of the theory is that the universe exists as described
because of human conscious minds interacting with reality. Of course,
the whole theory could be proved wrong if the universe ceased to exist
before time T - the last part may eventually be falsified by aliens
existing elsewhere in the universe, if not by God.

OsherD wrote:

for simplicity I
drop time-subscripts and assume that the "same" (in a sense) set
changes in time. Those people who prefer to use A_t instead of A (A_t
at time t) will find that the equations only change in requiring extra
subscripts.

I'm not sure I'm quite with you on that. As I explained earlier:

My own favorite theory of cosmology, for which the existence of the
universe is the only evidence, predicts that the universe will expand
forever, eventually becoming empty and eventless. The gravitational
field throughout becomes increasingly small, but while it continues to
exist, particles and their anti-particles spontaneously come into
existence. The particles can be thought of as quantum tunnelling
relative to each other, which is equivalent to saying that one particle
is in the past relative to the other (as though it had travelled faster
than the speed of light). This quantum-tunnelling into the past is
more likely to be a very tiny amount rather than a large distance, but
as the vast eons of the future progress, larger and larger quantum
tunnellings will occasionally occur. After a while, particles in the
far-distant future will have tunnelled appreciable amounts into the
past, and will start appearing at the beginning of the universe.
Eventually, all the matter in the future will have tunnelled back to
the beginning of the universe, and the future will be empty of matter
and energy.

According to this theory, all the matter in the universe does not exist
until it has finished quantum-tunnelling back to the beginning of time,
at which stage (call this time T) the matter then continues to not
exist forever. In the meantime, at every single instant that occurs,
the universe, viewed as the set A_t in your description, vanishes and
is replaced by A_t+1.

The existence of the universe as described above implies that an
eventless eternity of empty space has 'already' existed beyond time T
in the future. In your notation, A_t = {} for all t = T. The
existence of the beginning of the universe (at t = 0) is an event which
we know, by observation using standard astronomical equipment, has
'already' happened. According to my theory, the universe (including
the Big Bang at t = 0) is being continuously created for 0 t T.

The last part of the theory is that the universe exists as described
because of conscious human minds interacting with reality. Of course,
the whole theory could be proved wrong if the universe ceased to exist
before time T - the last part may eventually be falsified by aliens
existing elsewhere in the universe, if not by God.