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Re: Solve question
*To*: mathgroup at smc.vnet.net
*Subject*: [mg48934] Re: Solve question
*From*: Paul Abbott <paul at physics.uwa.edu.au>
*Date*: Thu, 24 Jun 2004 05:35:52 -0400 (EDT)
*Organization*: The University of Western Australia
*References*: <cb0u6a$r36$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
In article <cb0u6a$r36$1 at smc.vnet.net>, Arturas Acus <acus at itpa.lt>
wrote:
> Dear group,
>
> Suppose we have a system of linear equations eqns, with more variables,
> than can be solved. The question is. Do the number of solved variables
> can depend on the order of how these variables are listed in second
> Solve argument?
>
> Suppose not.
>
>
> The original problem, why I need to know this, is an attempt
> to eliminate dummy indices in formal sum involving Clebsch Gordan
> coefficients:
> CG[{j1,m1},{j2,m2},{j3,m3}]*CG[{j4,m4},{j3,m3},{j6,m6}]*...
>
> To calculate the sum explicitly. Dummy indices inside CG coefficient
> satisfy well known relations:
> m1+m2==m3, m4+m3==m6,...
>
> Not all indices are dummy, some of them are fixed or numbers. Sum may
> involve other objects, so generally there can be more or, rarely, less
> variables than can be solved. It is of primary
> interest then how Solve will act in these cases, and most important, how
> the solution it provides depends on the variables ordering.
I would avoid using Solve for such problems and instead suggest using
pattern-matching. That is, write the orthogonality conditions or the
contraction summations (e.g. leading to 6-j coefficients), etc., with
general coefficients, as a rule. For example, here is a way to code a
summation identity as a replacement rule:
s1[a_, b_] =
Sum[(-1)^(2k_) (2k_ + 1) SixJSymbol[{a, b, k_}, {a, b, f_}],
{k_, Abs[a - b], a + b}] -> 1
(we want k and f to be generic). Now, to reduce such a sum appearing in
an expression, we apply this rule:
Sum[(-1)^(2l) (2l + 1) SixJSymbol[{c, d, l}, {c, d, g}],
{l, Abs[c - d], c + d}] /. s1[a_, b_]
and get the desired reduction. Note the use of a_ and b_ in the
replacement rule.
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
School of Physics, M013 Fax: +61 8 9380 1014
The University of Western Australia (CRICOS Provider No 00126G)
35 Stirling Highway
Crawley WA 6009 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://physics.uwa.edu.au/~paul
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