Re: Probability Transform And Applications

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 02:54:32 +0000 (UTC)

On 27 Dec 04 23:23:48 -0500 (EST), Osher Doctorow wrote:
>Both Max Jammer and T. Y. Cao, in their 1974 and 1997 respective
>volumes which analyzed quantum mechanics and quantum theory from
>the beginning through 1997, noted all types of anomalies and vague
>concepts in quantum theory. (See my previous postings for referen-

It is of considerable interest for the development of probability-
statistics models to examine the equations of physics from the view-
point of the ranges of the PTs:

1) PT(y/x) = x - y
2) PT(xy) = x + y

The first PT's range generates determinants and even more generally
alternating series (series with alternating + and - terms), while
the second PT's range generates "whole is the sum of the parts"
equations which apply to vectors as well as scalars and so on. Let's
examine each type from a research viewpoint.

A. Determinant or alternating series

We already know from statistical multivariate research design that
tensor products or Kronecker products play an important role there.
Tensors found their main applications in general relativity in
physics, although they were discovered by mathematicians. Since a
tensor has a representation as a matrix (possibly multidimensional)
in a particular coordinate system, and square matrices each have
determinants (non-square ones have "sub-determinants"), we have a
hint from the PTs that determinants may underlie the known fund-
amental property of tensors in physics. This is especially the case
since in PI, the sign of a term distinguishes between cause and
effect or influencing and influenced variable. Matrices of course
extend beyond multivariate experimental research design in
probability-statistics since multivariate distributions often have
matrix representations or forms.

B. Whole is the sum of parts expressions/equations

Two of the most common equations in physics are:

3) E (total) = E1 + E2 + ... + En
4) F ("resultant") = F1 + F2 + ... + Fn

where E is energy, F is force, Ei and Fi are particular energies or
forces respectively, i = 1 to n, and in the case of forces the Fi
and F are taken as vectors usually. Yet there is typically no
mention of this in mathematical probability-statistics research
courses. Both (3) and (4) are automatically dimensionally homo-
geneous (each term has the same dimensions when reduced algebraic-
ally), which is usually stated as a key requirement of physical
equations and should be considered in other sciences. Some sciences
outside the "hard" sciences have introduced dimensional homogeneity
and dimensionless ratios which are part of dimensional analysis
(look up the terminology on the internet under these keywords),
including economics (De Jong), biology (Schepartz), and even
population research (Bruno, Doctorow, Kappner), but progress has
been slow largely because they are not part of an overall program or
"agenda" or organized Knowledge base or theoretical structure.

I'll just conclude this post by mentioning that the Schrodinger
equation of quantum theory has the form dimensionally E (total) =
E1 + E2 and arguably Schrodinger discovered the equation from such
considerations since according to W. J. Fushchych 2004 (look him
up on the internet) Schrodinger did not derive his equation from
any (equivalent of) elementary principles in the usual way.

Osher Doctorow

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