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.