From: mckay@cs.concordia.ca (MCKAY john)
Newsgroups: sci.math.symbolic
Subject: simplify torture - some flesh!
Date: 19 Jun 1998 13:17:08 GMT

The coefficients of a theta fn. count vectors and so are integers >=0.

Conway & Sloane,[Sphere Packings, Lattices and Groups, Chap 4 Eq.56
page 110 2nd edition] give a formula for the theta function of a lattice
of type A_n.

Expanding this in Maple as a q-series for A_8, the coefficients were
not "obvious" rational integers. The coefficient of q is 2c1 where 

c1 :=2*cos(4/9*Pi)-2*cos(1/9*Pi)+2*cos(2/9*Pi);

Note that this expression has a well-defined value.

Attempts at simplifying the series of the theta fn. for A_8 started all
this! Simplifying an initial segment of the q-series for theta of A_n
for n up to,say, 30 might be made part of a test suite for simplification.
Note that simplify is not idempotent in maple. 

=============================================================================
	TTY Iris, Maple V, Release 5, DEC ALPHA UNIX, Jan 11 1998


               c := 2 cos(4/9 Pi) - 2 cos(1/9 Pi) + 2 cos(2/9 Pi)

convert(c,exp)
                           1                               1
x := exp(4/9 I Pi) + ------------- - exp(1/9 I Pi) - -------------
                     exp(4/9 I Pi)                   exp(1/9 I Pi)

                             1
     + exp(2/9 I Pi) + -------------
                       exp(2/9 I Pi)


Watch closely...

simplify(x):
                              4/9       1/9       7/9
                          (-1)    - (-1)    - (-1)

rationalize(simplify(x)):
                            1/9       1/3           2/3
                       -(-1)    (-(-1)    + 1 + (-1)   )

simplify(rationalize(simplify(x))):
                                       0


Keep watching...

expand(x):
               4/9       5/9       1/9       8/9       2/9       7/9
           (-1)    - (-1)    - (-1)    + (-1)    + (-1)    - (-1)

rationalize(expand(x)):
             1/9      1/3       4/9           7/9       1/9       2/3
         (-1)    ((-1)    - (-1)    - 1 + (-1)    + (-1)    - (-1)   )

simplify(rationalize(expand(x))):
                            2/9       1/3           2/3
                        (-1)    (-(-1)    + 1 + (-1)   )

simplify(simplify(rationalize(expand(x)))):
                                       0


More interesting effects...

simplify(simplify(x)*simplify(x)):
                                       0

simplify(x*x):
                                                         2
       (-exp(- 7/9 I Pi) + exp(8/9 I Pi) - exp(5/9 I Pi))  exp(4/9 I Pi)

simplify(simplify(x*x)):
                                                         2
       (-exp(- 7/9 I Pi) + exp(8/9 I Pi) - exp(5/9 I Pi))  exp(4/9 I Pi)

expand(x*x):
          2/9         8/9         5/9         4/9         1/9         7/9
6 + 3 (-1)    + 3 (-1)    - 3 (-1)    + 3 (-1)    - 3 (-1)    - 3 (-1)

             1/3         2/3
     - 6 (-1)    + 6 (-1)

simplify(expand(x*x)):
           2/9         8/9         5/9         4/9         1/9         7/9
     3 (-1)    + 3 (-1)    - 3 (-1)    + 3 (-1)    - 3 (-1)    - 3 (-1)

expand(simplify(expand(x*x))):
           2/9         8/9         5/9         4/9         1/9         7/9
     3 (-1)    + 3 (-1)    - 3 (-1)    + 3 (-1)    - 3 (-1)    - 3 (-1)

expand(simplify(x*x)):
                             8/9         5/9         2/9
                       3 (-1)    - 3 (-1)    + 3 (-1)

simplify(expand(simplify(x*x))):
                             8/9         5/9         2/9
                       3 (-1)    - 3 (-1)    + 3 (-1)

radsimp(simplify(x)):
                                       3

=============================================================================
It occurs to me (now!) that a neater presentation of this material is to
give the effect of each non-trivial operator together with all the relevant
operators under which an expression is invariant.

JM

-- 
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.

==============================================================================

From: mckay@cs.concordia.ca (MCKAY john)
Newsgroups: sci.math.symbolic
Subject: Re: simplify torture - a test program
Date: 25 Jun 1998 17:45:25 GMT

A test maple program is available by anonymous ftp at 

URL ftp://ftp.cs.concordia.ca/pub/mckay/thetas

JM
 
-- 
But leave the wise to wrangle, and with me
the quarrel of the universe let be;
and, in some corner of the hubbub couched,
make game of that which makes as much of thee.