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RE: simplifying operator experessions with Dot, Times a nd Plus

• To: mathgroup at smc.vnet.net
• Subject: [mg35506] RE: [mg35481] simplifying operator experessions with Dot, Times a nd Plus
• From: "Wolf, Hartmut" <Hartmut.Wolf at t-systems.com>
• Date: Wed, 17 Jul 2002 02:08:52 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

> -----Original Message-----
> From: JL [mailto:jl at aol.com]
To: mathgroup at smc.vnet.net
> Sent: Tuesday, July 16, 2002 10:50 AM
> Subject: [mg35506] [mg35481] simplifying operator experessions with 
> Dot, Times and
> Plus
> 
> 
> I am trying to expand a large expression that has terms that look as
> follows:
> 
> -ex[0,1].(ex[0,1].ex[1,2]-ex[1,2].ex[0,1])
> 
> where ex[i,j] are unevaluated expressions.  I would like 
> Mathematica to
> simplify this and analogous expressions so that they read:
> 
> ex[0,1].ex[1,2].ex[0,1]-ex[0,1].ex[0,1].ex[1,2]
> 
> However, I cannot seem to find anything that will work.  The 
> problem is that
> I need to keep track of the order of the expressions ex[i,j]. 
>  If Dot were
> replaced by Times, there would be no problem whatsoever.
> 
> If anyone knows how to help me with this problem, I would 
> greatly appreciate
> it.
> 
> Thanks,
> 
> Jeremy Levy
> jlevy at pitt.edu
> 
> 
> 
> 
> 

Jeremy,

perhaps this is all humbug, ...

In[74]:=
dotExpandRules = 
  {a_ .(b_ + c__) :> a.b + a.Plus[c], 
   (a_ + b__). c_ :> a.c + Plus[b].c, 
   Dot[a___, \[Beta]_?NumericQ*Dot[b__], c___] :> \[Beta]*Dot[a, b, c]};


In[77]:=
-ex[0, 1].(ex[0, 1].(2 ex[1, 2]) + 3 ex[2, 2] - 
          ex[1, 2].ex[0, 1]).(5 ex[3, 0]) //. dotExpandRules
Out[77]=
-15 ex[0, 1].ex[2, 2].ex[3, 0] - 10 ex[0, 1].ex[0, 1].ex[1, 2].ex[3, 0] + 
  5 ex[0, 1].ex[1, 2].ex[0, 1].ex[3, 0]

...but regard this as an idea to try, test it and check for correctness. A
crucial point here is to separate scalar objects from (symbols/expressions
to become) tensors. There might be better ways to do that. (Or simpler, if
your problem is restricted, e.g. all your vectors/tensors have Head "ex" in
there symbolic form.) Make sure to set up the parentheses right in mixed
expressions with Times and Dot when scalars are involved. Be cautious, -1
quite often creeps in!

--
Hartmut