[Axiom-developer] ideas on algebra of tainted points

[Tim Daly]

Bill,

He was looking for your email address...

Tim

------- Start of forwarded message -------
Date: Thu, 10 Jun 2004 06:28:00 +1200
To: Tim Daly <address@hidden>
From: Craig Carey <address@hidden>
Subject: Can't subscribe to mailing list; ideas on algebra of tainted
  points
Cc: Bertfried Fauser <address@hidden>

CCed to Mr Fauser under a special axiomatic principle I am not
disclosing (maybe the axiom is wrong).


I attempted to subscribe to 2 Axiom mailing lists. My attempts
failed with this error message:

- ----------------------------------------
Axiom-developer Subscription results
You must supply a valid email address. 
- ----------------------------------------
 From URL: http://lists.nongnu.org/mailman/subscribe/axiom-developer

The only information I had entered into the form was my e-mail
address.

I see Mr Fauser wrote. E.g. wrote this:

"iii) u _| ( v _| w) = (u /\ v) _| w  (left action on the module V^)"
   http://lists.gnu.org/archive/html/axiom-developer/2004-06/msg00006.html

The _| is like a projection onto a flat except that the projected part
is rotated by 90 degrees:
(A _| B) = (Sum r,s)(A[r].B[s])[s-r].(r<=s)
  where x[r] is the part of x that is the product of r vectors.
  i.j = -j.i;  i.j = +1. I say so here: http://www.ijs.co.nz/polytopes.htm

Mr Fauser might have to do some coding himself. E.g. Clifford algebra
does not have polytopes to rotate since lacking a "<" less-than operator,
yet being antisymmetric, it is mainly about rotating. In logic,
rotations don't occur but shearing might.

It could be a special case of logic, i.e. extended this:
 1  polytopes
 2  polytopes with real-values. This seems too difficult since Exists
    creates polynomials and roots are solved since Exists of And's, is
    Min-ing triangle functions.
    Booleans are replaced with non-negative reals, and:
      x And  y = Min(x,y), x Or y = Max(x,y). 'Not' has to be deleted.
    Could define "<" onto polytopes.
 3  next, polytopes contain vectors.
 4  next, the extension is have Clifford/etc. numbers inside of
     polytopes.

That is a bit general. I need not look beyond the software problems
involved in implementing the algebra is to Hopf algebras, what
polytopes are to points (joke). Still no evidence that Mr Grassmann's
failure was due to the lack of computers, as far a I can assess.

Mathematica has got pattern matching.

I don't have Mr Page's e-mail address.

- --

My Tope solver now includes the O.R. Double-description/Chernikova
algorithm made in France by LeVerge of irisa.fr a decade ago. I merely
make it look like my solver is all symbolic but the breakthrough was
to:
 * impose restrictions on the user (no parameterized polytopes for the
   shadow operator. (Bagnali calls it the "image" operator:
     http://www.cs.unipr.it/ppl/ )

 * copy in a Operations Research algorithm and give it a fully
   symbolic front-end.

 * not needed but present is no support for quadratics, e.g.
     "Exists x)(0<x*x)".

- --

I just uploaded my project to tope.tigris.org.

- --

I mention this for Mr Fauser who might have missed this when reviewing
all existing systems. Notes:

(1) The offline "Bear" Dirac algebra program of Tkachov and for (CERN, etc.)
physicists:
   http://www.inr.ac.ru/~ftkachov/projects/bear/dirac.htm
 
(2) On teaching Russia (i.e. their physics students) to prefer the
fast failing Oberon language (actually the narrowly free Blackbox
GUI compiler, it seemed):
   http://www.inr.ac.ru/~info21/info/fvtjmlc2003.htm
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