Axiom/FriCAS - Additional Code I have Written

Here is various Axiom/FriCAS code that I have written for my own use and in case they may be of use to others:

Further Work

New for Feb 2016 - I have made some changes and additions to some of my code. Would you be interested in including this in the FriCAS library?

The changed/new files are:

These files are available here.

These 4 files are interdependent so it would be really good, from my point of view, if you could include all of them.

Further details of the changes are:

logic.spad

This is new code. Includes various logic related structures including partial order, lattice structures

I have also moved intuitionistic logic here from computation.spad

Since I last mentioned this on the forum I was kindly sent some unpublished code (related to Moebius function) by Franz Lehner with permission to merge this with my own poset related code. This is now all included in the code on Github.

These are my notes related to the code written by Franz here.

I have included Logic category, this is the same as existing Logic
category in Boolean.spad as it is intended to replace it. Since this
category is more general than just Boolean I thought it more appropriate
to put it in logic.spad.

For further information see the documentation here:
https://www.euclideanspace.com/prog/scratchpad/mycode/discrete/logic/
https://www.euclideanspace.com/prog/scratchpad/mycode/discrete/logic/moebius/

graph.spad

There are some updates to existing code, this includes:

For further information see the documentation here.

algebraictopology.spad

This is new code.

This includes simplicial complexes. At the moment only abstract simplicial complexes (purely combinitorial) are operational. The plan is to add geometric complexes later.

My aim is also to export to homotopy and homotopy groups.

For further information see the documentation here.

computation.spad

There are some updates to existing code, this includes:

For further information see the documentation here.

TODO

There is a lot more work to be done on this code but I think it is in a state where it can be of some use to others. If it is included in the FriCAS library then there is always a chance that someone else may be able to help. I would welcome as much help as I can get.

Some improvements I would like to make at some time in the future are:

  1. I would like to fully implement the geometric version of simplicial complex. One reason is that I would like to make it the standard way to represent 2D, 3D and n-dimensional shapes in code such as scene.spad. So the shapes can be transformed and output in various ways however they would have better mathematical properties than the structures usually used to represent shapes in computer science. This would involve writing code to test for overlapping simplexes and so on.
  2. The code that Franz Lehner sent to me also included isomorphism testing and drawing (using algorithms from Freese's book on "Free Lattices"). I would like to merge this with my code. (it may not all be generalisable to graphs/complexes but parts of it may be and the remainder can still be used for posets).
  3. Homology - Can we use simplicial complex to generate the Smith normal form which already exists in FriCAS?
  4. Homotopy - Although a lot of stuff in homotopy does not have a general algorithmic solution, I would still like to be able to generate groups from complexes. This would require implementing groups as presentations (free groups). I realise there are limitations on what can be done with these but, as an example GAP can do some simplifications of such groups. Franz tells me he has implemented code for free products of groups so it would be interesting to see if this could be merged in.

I have a long list of changes that I would like to see in FriCAS which I have listed on the page here.


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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Geometric Algebra for Physicists - This is intended for physicists so it soon gets onto relativity, spacetime, electrodynamcs, quantum theory, etc. However the introduction to Geometric Algebra and classical mechanics is useful.
flag flag flag flag flag flag Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry. This book stresses the Geometry in Geometric Algebra, although it is still very mathematically orientated. Programmers using this book will need to have a lot of mathematical knowledge. Its good to have a Geometric Algebra book aimed at computer scientists rather than physicists. There is more information about this book here.
flag flag flag flag flag flagClifford Algebras: Applications to Mathematics, Physics and Engineering (Progress in Mathematical Physics) (Hardcover) also see arXiv:math-ph/0212032 Clifford and Grassmann Hopf algebras via the BIGEBRA package for Maple
Rafal Ablamowicz, B. Fauser: Comp. Physics Comm. 170, 2005:115--130 available from the arXiv

 

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