Re: [Axiom-mail] Abstract Vector Algebra

[u1204]

>> This would be a first example of a special symbolic domain that would
>> create "arbitrary integers" rather than actual integers.
>>
>> This is my approach to the Maple "assume" facility. I want to say that
>> the uderlying domain is an "indeterminate integer" rather than
>> "assume x is an integer". This idea seems more in-line with Axiom's
>> approach to computational mathematics.
>
>OpenAxiom's syntax is " assume x : T" where T can be any domain.

The idea of an "assume" facility seems antithetical to the design
of Axiom. It feels more natural to me to create categories and
domains which remain symbolic (e.g. symbolic matrices). If necessary
they could be coerced to definite values at the appropriate time.

I want

  t2:POLY(IndefinteInteger(x)) := x^2+2*x+3
  t3:POLY(IndefinitePositiveInteger(x)) := x^2+2*x+3
  t4:POLY(Complex(IndefiniteInteger(x))) := x^2+2*x+3

and I can choose the appropriate manipulation algorithms based on
the given types following the same rules we now use.

I'd like to be able to say something like:

  A:SymbolicMatrix := [[a b],[c,d]]
  B:SymbolicMatrix := [[e,f],[g,h]]
  t1 := (A*B)^T
                  B^T A^T
  Type:                       Expression(SymbolicMatrix)

and coerce B^T to a matrix domain if I need to see the entries.

Or I can subtype it to SymbolicMatrix(DoubleFloat), do the
manipulations symbolically and then coerce the final result
to Matrix(DoubleFloat).

Done this way, the indefinites (or symbolics) fit naturally into
the mathematical structure of Axiom. There is probably even some
category theory support for lifting operations (arrows) to 
symbolic form.

Tim Daly