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From: Bill Page
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Tim,
On 2002/10/31 Thu PM 11:23:33 EST you wrote:
>
> Actually, I have no estimate. The key to the game is to
> get the lisp running as this is the only part of the
> system that is new to me. Axiom used to be hosted on
> AKCL, now GCL, and Camm and I have discussed rehosting
> it there. I expect it to run on both as they each have
> advantages. In any case, though, the game is to get it
> to run anywhere and I'm working on that at the moment.
> There really isn't any point to posting the sources as
> the build process is very complex and not documented (as
> yet. it will be). I've built the first version of the
> lisp and now am working on building the "image" file.
> Unfortunately there isn't any obvious way to share this
> task.
What is Camm? You mentions Codemist Lisp earlier. If I
remember installing REDUCE correctly, after the initial
installation there was a step involving creating
(compiling?) an image file that is loaded when Lisp starts
up. Is that what you are creating for AXIOM?
What operating system are you using? What platform should
I set up for this work? Linux? Any particular flavor?
(I'm a RedHat kind of Linux person.)
>
> The new system build uses noweb (search for noweb Ramsey
> in google) which is a tool to support literate
> programming. If Axiom has any chance to survive it has
> got to be documented so anyone who is willing to put out
> the effort can learn how to build, modify and maintain
> it. I suggest you look at noweb and I can send you an
> example file or two to bring you up to speed on how I'm
> using it.
Ok, I'll do that. I agree about the need for documentation
and a structured approach. Literate programming to me means
writing code in manner intended specifically to make it
easy to read. I am not so sure that is so easy or even
possible in older languages like Lisp but I am
willing to give it a try.
>
> Once the lisp build works locally I can upload the lisp
> portion of the system, you can try to build it, and we
> can work on correcting the problems with the build.
>
Ok.
> What is your background? Programmer? Mathematician?
> What area of Axiom strikes your interest?
>
Programmer, physicist, mathematician in order of decreasing
expertise. See previous email.
What is of most interest to me in Axiom is the programming
language. But I want it embedded in a user interface that
is somewhat more powerful than emacs. I am quite strongly
motivated to develop code for symbolic manipulation of
tensors (as in general relativity and differential
geometry). I am also interested in applied (physics)
mathematical research in Clifford algebra and in category
theory.
Regards,
Bill Page.
From bill.page1@sympatico.ca Fri Nov 01 09:43:32 2002
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From: Bill Page
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Subject: Re: Re: [Axiom-mail] Status
Date: Fri, 1 Nov 2002 9:43:27 -0500
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Tim,
On 2002/10/31 Thu PM 11:23:33 EST you wrote:
> ...
> The new system build uses noweb (search for noweb Ramsey
> in google) which is a tool to support literate
> programming. If Axiom has any chance to survive it has
> got to be documented so anyone who is willing to put out
> the effort can learn how to build, modify and maintain
> it. I suggest you look at noweb and I can send you an
> example file or two to bring you up to speed on how
> I'm using it.
>
Ok. I've read Ramsey's article and I now understand that
noweb is a stripdown version of Knuth's WEB. No problem.
A little "old fashioned" but that's ok.
I suppose almost everyone has their own "literate
programming" philosophy. For me it usually just involves
writing code in a style and format (within the constraints
of a particular language) that is intended to be read and
understood by others as well as compiled. When the language
is sufficiently rich, I also prefer to move as much of the
documentation as possible into the compilable code itself
rather than depending on a parallel textual description to
be accurate - too often the comments are right but the code
is still wrong. Of course you can't do that very well if
the code already exists or if the language is too arcane
to be easily read by a human. So WEB (noweb) is ok.
Are you using LaTex or some other output format?
Ramsey (and WEB in general) does not emphasize the use of
diagrammatic techniques, but I am rather much oriented that
way. Did you have any particular graphical documentation
tools in mind?
I am setting up the MSDOS/Windows/Perl version of noweb
in order to play. Feel free to send a few sample files
when you get a chance.
Bill Page.
From daly@idsi.net Fri Nov 01 13:27:54 2002
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Subject: [Axiom-mail] [axiom] Finitely Presented Groups and Axiom
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Dr Sims,
I've recently purchased your book "Computation with Finitely
Presented Groups".
My particular interest in the book is related to Axiom. Axiom
is a computer algebra system that was developed at IBM, sold
to the Numerical Algorithms Group (NAG), and released as
free source code in September. I'm leading an effort to bring
the Axiom system back to life as free code. You can find it at:
http://savannah.nongnu.org/projects/axiom if you are interested.
Axiom is weak in group theory and really has nothing related to
infinite group theory. I've looked at the algorithms in your book
briefly with the hope of building up strength in infinite group
theory.
The key item for me is the question of the representation of these
groups and the representation of a presentation. The examples in the
book use a matrix but the groups are really infinite objects. It is
unclear what a good computational representation would be for them.
Do you have any guidance you can provide? I've done work with the
Magnus group (Baumslag) but that code is all C++ and the group
representations are not easily expressed.
I'd like to develop a small set of representations in Axiom that
will work across your set of algorithms.
Tim Daly
daly@idsi.net
From daly@idsi.net Fri Nov 01 22:18:38 2002
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All,
Be sure you've subscribed to the axiom-developer mailing list as I'll only
be copying the list from now on so we can journal the discussions. I urge
you to actually try these instructions because it will make sure that we
share the same ability to create, use and view pamphlets. If these
instructions don't work for you let me know because we need to make
sure you've got a proper environment set up.
I've uploaded 3 examples of literate programming to illustrate styles.
An Algebra Example:
The first is called dhmatrix.pamphlet. The instructions for use are:
1) wget http://home.earthlink.net/~jgg964/dhmatrix.pamphlet
2) notangle dhmatrix.pamphlet >dhmatrix.spad
3) noweave dhmatrix.pamphlet >dhmatrix.tex
4) latex dhmatrix.tex
5) latex dhmatrix.tex
6) xdvi dhmatrix.dvi
step (1) will fetch the file. (wget is a useful utility for fetching
files given by a url. how you actually get the file is up to you)
step (2) will read the pamphlet file and create a spad file.
spad is the algebra language of Axiom. This file is extracted from
the pamphlet file and when diff'ed against the original source
shows no difference (an important point since we don't want to
introduce errors by wrapping the original source files into a
literate style).
step (3) will extract the tex output from the pamphlet file. notice
that the dhmatrix.tex file has commands like \documentclass
prepended automatically by noweb. This is, in general, not what
we want so normally we will use the -delay parameter to noweave.
However for this particular example I am experimenting with the
ability to collect many pamphlets into a booklet and I deliberately
left off the \documentclass, etc. The default behavior of noweb
works in this case.
step (4) and (5) read the dhmatrix.tex file and create dhmatrix.dvi
We need to latex it twice to get the cross-references right. I
don't believe there are any cross-references in this file yet but
I do this by habit.
step (6) will let us view the dhmatrix.dvi file. You could also use
other utilities like dvips to create a postscript file or dvipdf
to create a pdf file. I tend to work directly from the dvi file.
The dhmatrix.pamphlet file is an example of documenting the algebra
code. The original dhmatrix.spad file (which I wrote) was taken from
Richard Paul's Ph.D. thesis which became a book (Robot Manipulators).
Richard gave me permission to quote from his thesis for documenting
the domain. There is much more work to be done but this pamphlet was
written as a first experiment.
In general I hope to search out primary sources for the algebra
that lives in Axiom and either get permission to directly quote
the relevant paper or study the paper and write a new pamphlet.
There is a large piece of work to track down the original works.
The next piece of algebra documentation is to use Barry Trager's
Ph.D. thesis to document the integration code. I have his thesis
and permission to use it for documentation purposes. There is a
long leap from the thesis to the code so I have a fair bit of
background research I need to do before I can write up the
pamphlet. The original thesis is 85 pages.
A MAKEFILE Example:
The second file I've uploaded is actually from the sources we'll
be using. It is a Makefile in the Codemist Common Lisp (CCL) subtree.
The instructions are:
1) wget http://home.earthlink.net/~jgg964/Makefile.pamphlet
2) notangle -t8 Makefile.pamphlet >Makefile
3) noweave -delay Makefile.pamphlet >Makefile.tex
4) latex Makefile.tex
5) latex Makefile.tex
6) xdvi -expert -s 3 Makefile.dvi
step (1) will fetch the file.
step (2) will read the pamphlet file and create the Makefile. Notice
the -t8 (tabs every 8 spaces) parameter to notangle. In general we
will need this parameter and it doesn't hurt to have it if there are
no tabs so we will use it all the time.
step (3) noweave will read the pamphlet file and create the tex output.
Notice the -delay option to noweave. We have included the \documentstyle
and other header information in the pamphlet file already. The -delay
option allows us to do this. Unless the file is going to be included
as part of a larger document (like the algebra file above) we will
generally write our own document headers.
steps (4) and (5) are latex->dvi done twice to get reference right.
You are certain to get a complaint about a missing noweb.sty file
because this Makefile was ripped out of the real source tree.
For demo purposes you can modify the line in the original pamphlet
file from:
\usepackage{/home/axiomgnu/new/mnt/linux/bin/tex/noweb}
to:
\usepackage{noweb}
and rerun the command. I would ask you to resist the urge to
change the tex file directly as you need to think of the pamphlet
file as the source and everything else as machine-generated files.
step (6) invokes xdvi with -expert (which eliminates the buttons as
I know the keyboard commands) and -s 3 (which gives sufficient
magnification so I can read it).
This makefile shows a documentation style for Makefiles in general.
Makefiles are generally very verbose in their stanzas because we will
not be doing compiles in the same directories as the source files nor
will we be depending on default stanzas to do compiles.
We need to document the various stanzas and any special options that
we might add to the compile commands. Look for the TPD string which
I use to ifdef any changes I made to the original C sources. This
shows up as a -DTPD on the compile line for that particular stanza.
Also note that the end of the Makefile.tex contains references to
other pamphlets. The reason for these to support a future plan. We'd
like to be able to accept new code, particularly algebra code, that
we can add to the system in a reasonably automated fashion. The
references will give us this connection. I expect to expand the
format later. These are only placeholders.
The other reason for the references is that the commands are not
actually documented in the Makefile. The Makefile only documents
build information and other special instructions. For instance,
in another Makefile (not shown here) you need to build a special
Axiom library before you build the C code. While the Makefiles
know this it would be easy for a human to overlook. Instructions
on how to use a particular command as well as documentation on
the C code that implements the command do not belong in the Makefile
but reside in the pamphlet file for the command itself.
Makefiles will also contain information about what a particular
directory contains and why it exists in the source tree.
A Non-Axiom Example:
This is an example using Java to show how to build a stand-alone
command and its Makefile
1) wget http://home.earthlink.net/~jgg964/Magman.pamphlet
2) notangle -t8 Magman.pamphlet >Makefile
3) make
This illustrates a couple of useful points. First, the default
output of running notangle is the stanza marked <<*>>= in the
original source file. In Magman.pamphlet this is the Makefile.
If you read the dvi file that gets generated (or read the
generated Makefile if you must (sigh)) you will see that we
use the -R feature of notangle. The -R flag will extract particular
tags from the pamphlet file. So, if this were C code we could embed
the .h file, the .c file, and the associated Makefile into the pamphlet
and send them as one file.
Another point is that this pamphlet file has real mathematics and
real references in it. We will need this to explain sections of Axiom.
In the long term plan we want people to submit papers to an Axiom
Journal that includes the mathematics and executable code. That way
reviewers can test the code, readers can understand the code, the
system can import the code and maintainers can update the code.
This code is not related to Axiom but is here to illustrate the
embedded Makefile and mathematics points. If you don't have Java
this will fail but that is not important. Note that we mix Java
C and Makefile in the same pamphlet file.
Tim
From daly@idsi.net Fri Nov 08 15:10:55 2002
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From: root
To: Bertfried.Fauser@uni-konstanz.de
CC: bill.page1@sympatico.ca, miked@nag.co.uk, daly@idsi.net,
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Subject: [Axiom-mail] Re: aldor
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Dear Prof Bertfried,
Axiom was one of "the big three" (Maple and Mathematica were the other
two) general purpose computer algebra systems.
A small bit of history is needed here to clarify the discussion.
Axiom started at IBM Research as "Scratchpad" in 1971. It has a
language called "Spad". In the 1990s the compiler language was
formalized and a new compiler implemented. One design decision was to
make the new compiler, now called Aldor, a stand-alone program. The
Spad compiler (written in Lisp) and the Aldor compiler (written in C)
are compatible within certain limits dictated by their respective
roles. Aldor code runs inside of the Axiom system.
In the early 90s Axiom was boxed up and sold to the Numerical
Algorithms Group (NAG), a respected scientific software house in
England. They sold Axiom commercially and recently withdrew it from
the market. They have released the source code under a BSD style
license.
The Texexplorer interface was a separate product by one of the
IBM group members. It was added to the Axiom product from NAG.
It is now sold as a product by IBM and was not released as part
of the Axiom source code per IBM's request. It only ran on the
Windows version of the system as I understand it.
I'm currently the lead person on the Axiom project whose primary
goal is to document, maintain, upgrade and support Axiom for the
next 30 years. (http://savannah.nongnu.org/projects/axiom). A
list of tasks, both present and proposed, is available at:
(http://www.nongnu.org/axiom).
The good Professor Page has volunteered to help with this effort.
Now, back to your comments:
> Bertfried,
>
> On Friday, November 08, 2002 1:03 PM you wrote:
> >
> > Dear Prof Page!
> >
>
> It is a very pleasant European honorific title, but
> one to which I am not entitled since I am not currently
> employed as a teacher...
>
> > thank you once more for your mail about aldor/axiom. I
> > have downloaded the compiler and tested it for my purpose.
>
> I hope that you do not mind that I have copied this reply
> to your message two other people with whom I am currently
> associated in the development of a new open source version
> of AXIOM. They in turn may be aware of some other people
> that share some of your concerns and ambitions.
>
> As you now know, Aldor is a high level compiler for the
> language that is really a part of AXIOM. AXIOM used to
> be available from NAG with a graphical front end called
> Techexplorer (Techexplorer is now a separate product
> available from IBM but without AXIOM). In this packaging
> Axiom was a competitor to Maple, but for various reasons
> (most of which I am not aware) NAG decided against
> continuing the development.
>
> The open source version of AXIOM is intended to preserve
> and then continue the enhancement of AXIOM as a fullfledged
> computer algebra system. The intial version of OpenAxiom
> that will be available will have a basic character style
> interface rather like command line Maple.
There is work in progress in cooperation with TeXmacs to make a
tex-like user interface. As TeXmacs works with many other computer
algebra systems I don't anticipate any difficulty making it work
with Axiom. I've sent them parts of the source code so they can
optimize the connection.
>
> At this point it is not entirely clear to me what the
> long term relationship between Aldor and OpenAxiom will
> be. Perhaps Mike or Tim could comment...
Stephen Watt (www.aldor.org) was the compiler lead on the
Scratchpad project. He is the most qualified to maintain
and extend it. Stephen and I have both stated our goal of
keeping both projects compatible. Aldor code in the system
is in files with the .as suffix, Spad code with the .spad
suffix. This should be transparent to you if you use the
)compile command.
>
> > However, I have several problems:
> >
> > -- Indeed I like very much the categorial approach, but
> > its implementation is beyond the ability to use it as a
> > student teaching tool. (OK, but good for research)
> >
>
> My understanding is that it is the intention of some of
> the people who are currently interested in Axiom to use
> it as a teaching tool for abstract algebra.
There are a large number of people in Europe who have ambitions
to use Axiom as a teaching tool in various courses. The design
of the system and the open source nature of the code will make
it possible to reshape the system to their teaching needs.
Contact Paul Zimmerman
for more information on the people interested in a teaching use.
Axiom's great strength has been in research so it is no
surprise that this is where people find it easiest to use.
>
> > -- We use in Clifford/Bigebra at several very important
> > steps the equation solver of maple. That's tremendously
> > convenient! In aldor itself there is nothing comparable
> > and in the Algebra library (which I am on the way to
> > install) it looks from the documentation as if only very
> > basic solving mechanisms are incorporated. This means
> > that you have to implement your own solver and thats heavy
> > work to do...
> >
>
> I think you are right, however most of the algorithms for
> more general solution methods (Grobner bases etc.) are well
> known and well documented. I expect that a lot of work will
> be done in this area in a short time once AXIOM becomes
> available.
We are hoping that people will use Axiom to develop their
algorithms and contribute the results back to the project.
This has been true historically.
>
> > -- This situation was similar to the state of the art with
> > GINAC, which I had a look on some time ago.
> >
> > -- I will see what kind of features are provided by
> > MACSIMA, which went open source too. Colleagues who used
> > MACSYMA reported that it was poor in graphics (no problem
> > for me) but very good in performance.
> >
>
> I think that you will find MACSYMA rather like Maple (that's
> both good and bad) - much easier to convert what you have
> already written in Maple, but it lacks the very high level
> object orientation and rigorous type system that is a
> trademark of Axiom. I think that a better, more expressive
> and more "mathematically" or better said, "algebraically"
> compatible language is critical to advanced research
> applications such as yours. I am rather amazed that you
> have accomplished as much as you have with Maple. Some
> times it takes a lot of dedication and hard work to see
> the mathematics in Maple code!
MACSYMA no longer exists as the source code was bought and
has never been heard from again. MAXIMA is a project (originally
done by Bill Schelter) to port the original Department of Energy
code to Common Lisp. (maxima.sourceforge.net)
>
> > However, I have learned already now many things from
> > aldor/AXIOM and the overall sped of an compiled aldor
> > program is amazing and makes me hungry about more :)
> > Perhaps it would be a good idea to see a sort of maple
> > compiler?
>
> >From Maple 7 on there has been the capability to call
> external programs (written in C or other high level
> language) and to return results to Maple. Maple also
> has the ability to convert Maple procedures to "C"
> code which can then be compiled and called externally.
> Unfortunately this feature is seriously limited in
> the case of algebraically intense applications by the
> limited nature of the native data structures available
> in most other high level languages.
>
> > I am curious about the maple aldor interface which is
> > announced on the web-page.
> >
>
> As far as I know this is only in the research stage. At
> one time a few years ago there was also a project to link
> Maple and Axiom and even very optimistic statements to
> the effect that Axiom would likely be the successor to
> Maple. The problems to solve involve the conversion
> of data structures between these two systems. But it
> should be possible in principle to call an Aldor routine
> from Maple.
Aldor, in stand-alone mode, can call foreign programs. There is an
effort underway to make Axiom's algebra available in this stand-alone
form. (Contact www.aldor.org for further information). Of course,
given that Lisp exists I can't imagine why anyone would write in
C. All the device drivers I need already exist :-)
>
> Kind Regards,
> Bill Page.
Ultimately, the difference between the Maple, Mathematica, and MAXIMA
computer algebra systems and the Axiom system lies in the design.
Maple, Mathematica, and MAXIMA are all "engineering" systems. They do
whatever it takes to make the answer work. (Thus subtracting two equal
matrices will give an "integer" 0 result). Axiom is a theory-based
system. (Thus subtracting two matrices will give a zero matrix). I
believe that for correctness and scalability you need to have a
guiding theory. These systems are much too complex to check the
details so you have to trust the results. Without theory there is no
guidance about correctness. There has been much talk recently, by the
way, about proving Axiom's algorithms correct. Let me know if you are
interested in such work and I can put you in touch with the people.
As for teaching purposes Axiom has a much higher learning curve
than the other systems. There is also an effort underway to try
to match Axiom with a "zero-learning curve" front end which should
make it much easier to use for students. Again, let me know if you are
interested in such work and I can put you in touch with the people.
Feel free to send me any questions you have. You might want to
visit the Axiom home page and sign up for the Axiom mailing list
(In spite of the shameless plug).
Tim Daly
daly@idsi.net
From fauser@spock.physik.uni-konstanz.de Fri Nov 08 18:46:57 2002
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From: Bertfried Fauser
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,
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Dear Tim!
thank you for the offer to ask questions. But let me beforehand
explain what my background is.
By the way you should drop 'Professor', which is untrue (hopefully
not for ever). I am working in quantum field theory and need urgently
abstract algebra, however I am not canonically trained in this field. The
last three years showed me that I need urgently things like categories,
functors, but also Groebner bases for non-commutative algebras (that's not
yet solved on my side).
My interest in computer algebra started when I used the Clifford
package of Rafal Ablamowicz (TN Tech Univ. Cookevile) for my work in 1996.
First I asked for procedures, then I was beta tester, then I wrote an own
package on Clifford Hopf algebras and since last September I am coauthor
of the Clifford/Bigebra package. This was done in maple, since Rafal had
by chance a maple around when he started his first hack.
The package is awkward, since it grew up from a quick and dirty
hack and we had to do much nasty tricks to speed it up and make it usable
for a general purpose. However, you might acknowledge that this was also
due my very basic programming skills.
For me, research has priority and I would not spend at the current
time to much efforts in using CAS systems in teaching, since in a physics
department (I am member of an astrophysics group, hahaha ..) I cannot say
the word category loudly (even if some guests which I invited did so).
Things are different with Rafal, he uses the software for teaching
mathematics and engineering students. Some of our algorithms are very fast
an my be useful in projective geometry and then in Robotics, such people
do not have any idea about categories (nor on Hopf algebras, but one can
just explaing the algorithm without saying 'Hopf' :).
Indeed I was pressed several time by maple to do quite nasty
things to keep the type of objects, e.g. to write &t(0) for zero just to
be able to typecheck it being a tensor! Such things would not happen with
AXIOM ! My experience with multiliner algebra, Hopf algebras and quantum
groups is that a serious type system is 75% of the thing to do, so one
more point for Aldor/AXIOM.
On the other hand, I spend several weeks and looked at the
Internet and asked lots of people at the ACA 2002 in Volos, Greece about
Groebner basis methods for non-commutative algebras (I would also be
interested in the theory of generating functions in non-commutative
variables which is related) but I was not able yet to implement an
algorithm in maple (and that's not maples fault :).
I could imagine to look closer to Aldor, I have not yet any hint
how to get AXIOM or to use it, but I think together with the Algebra
library Aldor/Algebra = AXIOM, isn't it? For research projects I do not
need any graphical interface.
I will sign up the mailing list you told me. Bill Page has send me
the Aldor, AXIOM etc web pages which I have visited. (There I got
Aldor/Algebra). However, since my main activity is in research, I cannot
guarantee that I even have enough time to do some reasonable things there.
As you have confessed the learning curve is really high!
I am trying to do a base free package for Grassmann and Clifford
Hopf algebras now for several times, but it seems as if the mathematics is
not yet fully understood. That is a further problem with such a high
level approach as is issued by AXIOM: You need from the very beginning a
very precise and theoretical idea about the objects (better morphisms) you
want to deal with.
Rafal is a person for a quick and dirty hack. He wrote several
procedures which I would not have succeeded to write because he went back
to some maple internal abilities which I didn't want to use since it shows
that one has not understand the case at hand. However it worked pretty
fine! I am not sure if I really will be able to convince him to use
something like AXIOM or Aldor (I was not even able to convince him to use
text based code in maple, but he prefers these awkward worksheets which
cannot be transfered from one maple version to the other). However, I
know that in Rafal's bookshelf is a volume on AXIOM which he got some time
ago when he went with Prof. Lounesto (who died this year unfortunately)
to some conference on computer algebra. I was already then fascinated by
the clear theoretical concept of this system.
Compared to GINAC, Aldor has a concept. GINAC is just a collection
of C++ library routines which can be used in any C++ program. However I
miss there a theoretical background, the user has to think there to much
on his own, this was the reason for me not to join that thread.
I spend quite a bit of time on the maple external c interface.
However, as every time with WMI, it was badly documented and the wrappers
generated by maple took all efficiecy away by data translation into
external C compliant code. Maple does not want you to get access to
internal data strcuturs but that would be necessary fro algebraically
operating functions. Hence I was not able to go beyond some numerical
examples.
All in all, I cannot promise you to develop anything, but I will
certainly keep track of the further development of Aldor (Where to get
AXIOM?) I will also try to do some private experiments, and let's see what
happens...
Thank you for you kind mail, and hope to stay loosely in contact.
best
BF.
% Bertfried Fauser Fachbereich Physik Fach M 678
% Universit"at Konstanz 78457 Konstanz Germany
% Phone : +49 7531 883786 FAX : +49 7531 88-4864 or 4266
% E-mail: Bertfried.Fauser@uni-konstanz.de
% Web : http://clifford.physik.uni-konstanz.de/~fauser
From daly@idsi.net Mon Nov 11 22:19:44 2002
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From: root
To: Bertfried.Fauser@uni-konstanz.de
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Subject: [Axiom-mail] Hopf Algebras
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Bertfried,
When Axiom comes out we should have a further discussion about
possible contributions. I'm unfamiliar with Hopf Algebras but
that is no surprise. Is there a paper or reference on the subject
you can recommend?
Re: GINAC. Be aware that the library-only approach to system building
leads to a complex tangle that gets progressively more difficult to
use. System like Axiom help you to coercions and simplify results.
Tim
From fauser@spock.physik.uni-konstanz.de Tue Nov 12 05:14:25 2002
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From: Bertfried Fauser
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To: root
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Subject: [Axiom-mail] Re: Hopf Algebras
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On Mon, 11 Nov 2002, root wrote:
> When Axiom comes out we should have a further discussion about
> possible contributions.
Dear Tim,
please keep me informed, but I guess it will be posted on the axiom-mail
group?
> I'm unfamiliar with Hopf Algebras but
> that is no surprise. Is there a paper or reference on the subject
> you can recommend?
The idea of a Hopf algebra came out of topology in the early 40ies, the
name was given by Milnor and Moore (see refs below) and refers to Heinz
Hopf`s pioneering work. However, they survived in the theory of
combinatorics for some decades. In physics, Hopf algebras became famous in
a special flavor called quantum groups in the late 80ies. You can look at
a Hopf algebra as a sort of generalized group (yes, group not algebra)
which has a pseudo inverse called antipode and allows to transport much
reasoning of group theory to the a general setting.
Technically there is one new operation:
-- Algebra =3D=3D Module + morphism A \otimes A --> A ++ as common
-- Coalgebra =3D=3D coModule + morphism A --> A \otimes A ++ new
In computer science, you have already coproducts, if you think of anything
which generates a tree of possibilities. It is possible to build a
coalgebra theory on an equal footing with an algebra theory, there are
comodules, corepresentation, coideals, etc which however have sometimes
different properties than modules, ideals, representations of algebras.
-- Hopf algebra =3D=3D Algebra + Coalgebra + compatibility laws -> Antipode
A Hopf algebra is hence in the same time an algebra and coalgebra, where
the algebra product is a coalgebra homomorphism and the coalgebra
coproduct is an algebra homomorphism and the existence of a map
S : module -> module is required, which is called antipode and is
a generalized inverse.
-- Convolution
Today it seems to be most promising to start with a convolution. That is
with a product say m : V \otimes V -> V and a coproduct say \Delta : U
--> U \otimes U. Given morphisms f,g : U ---> V, you can define an
-- convolution algebra with product * on morphisms
via the formula
f * g(x) :=3D m =B0 ( f \otimes g) =B0 \Delta(x) +++ You can drop the x sa=
vely!
One can prove that if product and coproduct are (co)associative and
(co)unital and the identity morphism has a convolutive inverse, than this
inverse is the antipode called S and the convolution is already a Hopf
algebra, i.e. Id^(-1) (w.r.t. *) =3D S
-- Quantum groups
in physics are Hopf algebras having special `nice` features, called
(quasi) triangular structures, which can be recast in categorial terms to
be based on naturalness and coherence of certain categorial maps. (In
physicist's terms, they have a sort of integrability condition which
guarantee that if you reach a result by different way's of reasoning in a
category then the result does not depend on the way you got it, which is
convenient and that you can identify several tensor modules)
-- Example: A group is a (trivial) Hopf algebra
let G be a group, * the group multiplication and define \Delta to be the
`diagonal map` \Delta : G --> G \otimes G, \Delta(g) =3D g \otimes g (just
doubling the element). One can show that the antipode S is an
anti-homomorphisms and is given in the group case by the inverse
S : G --> G, S(g) =3D g^(-1). A group action may be given as (using a switc=
h
sw : G \otimes G --> G \otimes G, sw(g \otimes h ) =3D h \otimes g)
g > h =3D m =B0 (m \otimes Id) =B0 (Id\otimes S) =B0 \Delta(g) \otimes h
=3D g * h * g^(-1)
--> generalization. \Delta(g) =3D \sum g_[1] \otimes g_[2] which is no
longer diagonal.
-- Example: Universal enveloping Lie algebras are Hopf algebras
(Its generalizations are called sometimes 'matrix pseudo groups',
you end up with matrices which have non-commuting entries)
The product is the matrix product in a suitable representation (fundamental
representation) and the coproduct is given on generators x as
\Delta(x) =3D x \otimes 1 + 1 \otimes x and extended to the whole (graded)
module via the homomorphism properties. The antipode is given on
generators by S(x)=3D-x and also extended in a recursive way.
The extension of Lie symmetries was the main motivation in physics to study
this type of objects.
Given this short explanation you might judge yourself if you really need
to look for such mathematics. Some references are:
-- Classical texts
[1] Milnor, Moore, "On the structure of Hopf algebras", Ann. of Math.
81:211---264, 1965
[2] M.E. Sweedler, "Hopf algebras", W.A. Benjamin, INC, New York, 1996
--
if you are interested in combinatorics, you have to seek for texts of
Gian-Calro Rota and collaborators or on Umbral calculus.
-- more recent
[3] S. Montgomery, "Hopf Algebras and their Actions on Rings", AMS
conference series CBMS, No. 82, 1993
[4] my habilitation, see xxx.lanl.gov math.QA/0202059 (partly physics)
(gives also refs to geometry and Rota's work)
-- physics
[5] Shahn Majid, "Foundations of quantum group theory", Cambridge Univ.
Press, Cambridge, 1995
this is a subjective selection, but you might scan references of these
refs. to find more suitable reading.
greetings
BF.
% Bertfried Fauser Fachbereich Physik Fach M 678
% Universit"at Konstanz 78457 Konstanz Germany
% Phone : +49 7531 883786 FAX : +49 7531 88-4864 or 4266
% E-mail: Bertfried.Fauser@uni-konstanz.de
% Web : http://clifford.physik.uni-konstanz.de/~fauser
From daly@idsi.net Sat Nov 23 12:21:42 2002
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From: root
To: "'Joris van der Hoeven'"
Cc: , , ,
"Bill Page" , Norman Ramsey ,
axiom-mail@nongnu.org, Barry Trager ,
Manuel Bronstein ,
William Sit
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Subject: [Axiom-mail] noweb, pamphlets, and TeXmacs
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All,
I've written some tutorial notes on the pamphlet idea to try to get
everyone at the same level of discussion. Essentially what Bill
has been pursuing is a way to integrate noweb and TeXmacs so that
we can support pamphlet file documents. As there is some confusion
about what each part is I've decided to write it out in full.
Feel free to complain about mistakes.
Bill's ideas are essentially correct. His note is attached.
=================
RE: NOWEB
=================
Knuth and Dijkstra advocated literate programming to try to solve
the problem of marrying the knowledge behind a program with the
text of the program itself. Knuth wrote Web which was designed to
work with Pascal thus:
.web formatted document
| |
| ------> tangle ----> pascal code ----> compile ---> execute
----------> weave ----> tex format ----> latex ---> read
As this was Pascal-specific various other language-specific versions
were generated, e.g. CWeb for C.
Norman's innovation is that we don't need to be language specific.
With just a few additional tags above TeX we gain great power.
Since Axiom uses many forms of code (Makefiles, C, lisp, boot, spad, etc)
this is a key idea. We need to be able to embed many things transparently.
If we remove the language-specific options and simplify things we
can reduce the problem to this:
.noweb formatted document
| |
| ------> notangle ----> any code ----> compile ---> execute
----------> noweave ----> tex format ----> latex ---> read
Norman's implementation is called noweb. In essence, a noweb document
consists of alternations of code chunks and text blocks. A code
block is marked by:
<<(some string)>>=
code
@
Code chunks continue until encountering an @ in column 1 or another
chunk marker (the <<(some string)>>= tag).
The trailing equal sign marks this as a "definition" of the (some
string) block. Lack of a trailing equal sign marks this as a "use"
of the (some string) block. Uses are expanded by notangle.
Another important idea is that multiple occurences of the definition
string are concatenated into one definition thus:
<>=
code 1
@
....
<>=
code 2
@
....
<> ==> expands into:
code 1
code 2
We use this idea extensively in the documentation of code.
Text blocks are all that are not code chunks. Text blocks are tex
formatted document blocks.
=================
RE: TEXMACS
=================
TeXmacs is neither emacs not TeX but is an interesting cross-product
of the two ideas. Joris set out to make a useful front-end to a
computer algebra system and ended up with a generally useful tool.
It communicates with many computer algebra systems and is able to
properly format the math output in TeX style yet retain it as a
live object that can be handed back to the underlying system. In
addition, TeXmacs is able to properly format a large subset of
Tex and Latex documents.
TeXmacs, as Bill has been pointing out, is an excellent target for
an Axiom front-end. It already can talk directly to Axiom's interpreter
and embed the output into the TeXmacs buffer. It can already display
the .tex output from noweb.
Support for native noweb format would be most useful. The subtle
distinction that Bill was mentioning is that currently we can take
the "tex format" output and display it in TeXmacs. However, we would
like to fully support noweb as a standard format. This implies a couple
changes.
As mentioned above noweb does:
.noweb formatted document
| |
| ------> notangle ----> any code ----> compile ---> execute
----------> noweave ----> tex format ----> latex ---> read
If TeXmacs understood the noweb format fully it would need to have
the following features:
0) The ability to recognize and format a code chunk.
1) The ability to recognize the <>=, concatenation, and <