(Revised May 23, 1996)


'CLIFFORD - A Maple V (R. 3) Package for Clifford Algebra Computations'
                (version 2, February 15, 1996)
                          
                            by
                      
                      Rafal Ablamowicz
                 Department of Mathematics
                     Gannon University
                       Erie, PA 16541
                ablamowicz@cluster.gannon.edu
                   phone: USA(814)871-7592
                     fax: USA(814)871-7382

Here is a list and a brief decription of files available at 

        http://www.gannon.edu/service/dept/mathdept.  

These files (also stored here as part of clioct.zip) contain a Maple
package called 'Clifford' (version 2) (Copyright Rafal Ablamowicz, Gannon
University, 1995, 1996) and examples how to use it.  A new package for
computations with octonions called 'Octonion' (version 1) has also been
added. It supplements the 'Clifford' package and must be used with it.

NOTES:
1. Due to some changes in Maple's programming language from release 3
   to release 4, these two packages don't work yet under the newest
   release 4. Plans are under way to acommodate these changes.

2. These two packages do not run under the Student Version of Maple V R. 3
   since some functions available only in the academic (full) version of
   Maple are used.  A Student Version of these packages is possible if
   there is enough demand for it.  Please send me an e-mail if you would
   be interested in having such a version.

Permission is given to download these files as long as they are for
your individual use and not for distribution of any kind including
posting these files on any other server but this one.

These files will be modified on a continuous basis as a need may arise
and additional files may be added at a later time.
              
                Type    Size    Description

readme.txt   -  ASCII   8,074   This readme.txt file.

clifford.m   -  binary 39,008   Maple V R. 3 .m file containing the package 
                                'Clifford' version 2
                                
cliffor4.ms  -  ASCII 125,452   Maple V R. 3 worksheet created under 
                                Windows 95 which contains listings of 
                                procedures in 'Clifford' (ver. 2); needed to 
                                create file clifford.m

cliffor4.txt -  ASCII  83,860   .txt version of cliffor4.ms

octonion.m   -  binary  8,088   Maple V R. 3 .m file containing the package
                                'Octonion' ver. 1 (supplement to 'Clifford')

octon1.ms    -  ASCII  18,686   Maple V R. 3 worksheet created under
                                Windows 95 which contains listings of
                                procedures in 'Octonion' (ver. 1); needed to 
                                create file octonion.m  
                                
octon1.txt   -  ASCII  11,496   .txt version of octon1.ms 

cliproc2.ms  -  ASCII 147,520   Maple V R. 3 worksheet created under
                                Windows 95 which contains examples how
                                to use procedures in 'Clifford' (ver. 2) and
                                in 'Octonion' (ver. 1)

cliproc2.txt -  ASCII  51,369   .txt version of cliproc2.ms

cliff2.tex   -  ASCII 122,826   LaTeX source code of an article/guide
                                "Clifford algebra computations with
                                Maple," 40 pages, where all procedures
                                from version 1 are described and examples 
                                are given (needs climacro.tex and mapleenv.sty
                                described below for processing)

NOTE: to see additional features of version 2, in addition to cliff2.tex 
      load cliproc2.txt or cliproc2.ms.

climacro.tex -  ASCII   5,666   LaTeX macros needed for cliff2.tex

mapleenv.sty -  ASCII  14,510   Maple style file with LaTeX macros
                                needed to process cliff2.tex

clioct.zip  -  binary 177,104   contains all files mentioned above


THE SIMPLEST WAY TO DOWNLOAD FILES

The simplest way to download these files is to download the .zip file
clioct.zip and then unzip it using pkunzip.  After that, pick and choose
files you want to use (either the .ms or .txt files).  For more information
about these files and how to install them, read the rest of this file.

FOR WINDOWS USERS

If you run Maple V R. 3 under Windows 3.1, 3.11, or Windows 95, first
transfer cliffor4.ms with the 'Clifford' package, and octon1.ms with the
'Octonion' package (or transfer and unzip clioct.zip).

Then, execute cliffor4.ms and octon1.ms worksheets after loading them
into Maple.  

After you have run cliffor4.ms and octon1.ms, you will have created file
clifford.m  with the 'Clifford' package and file octonion.m  with the
'Octonion' package. 

Finally, transfer, load and execute (or view) file cliproc2.ms 
(see also information about a LaTeX file cliff2.tex below).


FOR OTHER OPERATING SYSTEMS

If you run Maple V R. 3 under DOS or some other operating system, first 
transfer cliffor4.txt for the 'Clifford' package, and octon1.txt 
for the 'Octonion' package (or transfer and unzip clioct.zip).  

Then, execute cliffor4.txt and octon1.txt worksheets after loading them
into Maple.  

After you have run cliffor4.txt and octon1.txt, you will have created file
clifford.m  with the 'Clifford' package and file octonion.m  with the
'Octonion' package. 

Finally, transfer, load and execute (or view) file cliproc2.txt 
(see also information about a LaTeX file cliff2.tex below).


LATEX FILES

I have added a LaTeX 2.09 file "cliff2.tex" (along with supporting
documents "climacro.tex" and "mapleenv.sty") which contains a 40 page
article showing how to use 'Clifford' package (version 1).  There are 
examples and descriptions of each procedure along with new mathematics 
created with the package.

Load and see cliproc2.ms or cliproc2.txt for information about ALL features
of version 2 including new ones such as:

- rmulm  - multiplication of matrices with Clifford algebra entries

- cmulQ and `&cQ` - Clifford multiplication in the orthogonal Clifford
                    algebra Cl(Q) (may be used in addition to cmul and `&c`)

- LCQ    - left contraction in Cl(Q) 

- isVahlenmatrix - procedure that checks if the given 2 x 2 Clifford matrix
                   is a Vahlen matrix (not all grades are allowed yet)

- isproduct - checks if the given Clifford polynomial of grade no higher than 
              2 is a product of 1-vectors

- omultable - displays octonion multiplication table

For more information and explanation of the new features please see also the
preamble to cliffor4.ms or cliffor4.txt.


LOADING PACKAGES

To load the packages, while in Maple session, type

> restart: with(Clifford); with(Octonion);

Please note that you can load and use the 'Clifford' package without 
loading the 'Octonion' package; however, the 'Octonion' package requires
that the 'Clifford' package has been loaded first.


MORE INFORMATION

More information about the use of this package as well as the use of
other software packages written in Mathematica, Reduce, Matlab, C++,
Fortran, Clical, and Maple for the Clifford algebra research, see the
newest book and associated with it electronic files located at the
Birkhauser Web site: www.birkhauser.com.

Title: ``Clifford Algebras with Numeric and Symbolic Computations''
Editors: Rafa\l Ab\lamowicz, Pertti Lounesto, Josep M. Parra
ISBN: 0-8176-3907-1
PUBLISHER: Birkh\"{a}user, Boston
YEAR: 1996


Please direct all questions, comments, and reports on bugs in 'Clifford'
and 'Octonion' to:

Rafal Ablamowicz
Department of Mathematics
Gannon University
Erie, PA 16541
ablamowicz@cluster.gannon.edu
phone: USA(814)871-7592
  fax: USA(814)871-7382

Erie, May 22, 1996
