Re: [Axiom-developer] Is anybody actually doing mathematics with Axiom?

[Alasdair McAndrew]

Hi Martin,

Here is my ztrans.input file in its current diabolical form - all the cleverness is yours and Themos's; the extra hack work is my own.  It does forward transforms pretty well (of powers, exponential functions and mixtures thereof), but inverse transforms are less well handled.  For example,

ex:=ztrans(n^5,n,z)

produces a formidable polynomial quotient; to invert it you need to apply partial fractions to ex/z, and apply the inverse transform to each individual fractional term multiplied by z.  This is not happening right now.

As well, I need to implement shifts:

ztrans(f(n-k),n,z) == z^(-b)*ztrans(f(n),n,z)

and for f(n+k).

My own thoughts for difference equations were much more modest than yours: linear equations with constant coefficients, and equations which can be transformed into linear equations.  I can't imagine it would be too hard?  Of course, if I get the z-ztransforms working, that can be used....

cheers,
Alasdair

On 05 Jun 2007 11:53:24 +0200, Martin Rubey <address@hidden > wrote:

"Alasdair McAndrew" <address@hidden > writes:

> but does anybody actually use Axiom for mathematics: teaching, research, or
> even fun?

I am.  For example, I wrote a tiny script to spit out all set partitions in
(Coxeter-) Type B and D and counted the first few terms, then entered them into
Sloane's database, to find the reference Ruedi Suter.

I also used Axiom to check a conjecture on Jeu de Taquin.  To this end I
implemented growth diagrams by Fomin, but I am unable to pursue this project on
my own.

Last year I used the (in my opinion) formidable graphics to show students how a
saddle point and a tangential plane looks like.  Gorgeous!

> Is anybody out there adding further mathematical functionality to Axiom -
> numerical routines, difference equation solvers, discrete mathematics etc
> etc?

I added the guessing package (currently state of the art), and together with
Ralf a project to deal with combinatorial species.

Concerning difference equations, you might know that my favorite would-be
project is a hierarchy covering

  functions satisfying algebraic diffential equations,
  holonomic functions,
  algebraic functions,
  rational functions
  polynomials

on one hand and

  "admissible" recurrence relations
  D-finite recurrence relations
  recurrence relations with constant coefficients

on the other hand.  But I could not find a collaborator so far --
unfortunately, Antoine Hersen gave up.  (One needs to know a bit about
Ore-algebras, AKA skew-polynomial rings.)

> FWIW, as Martin well knows, I've been banging my head over pattern matching
> and z-transforms lately. I have a file which works to about 75%; the last
> bit, which involves partial fractions, is still giving me gyp.

Try to formulate a question, then I can try to answer it.

> And on the meta-mathematical side, has anybody got an emacs mode to work
> with *.input files?

Francois Maltey sent me some files, but I am unable to incorporate them into my
emacs mode (which I use myself meanwhile), for lack of time.

Martin

ztrans.input
Description: Binary data