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MathGroup Archive 2004

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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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