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MathGroup Archive 2006

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Re: Re: Re: Simplifying algebraic expressions


Andrzej Kozlowski wrote:
> Here is, I think, an optimized version of the "simplification" I sent  
> earlier:
> 
> 
> rule1 = {2x -> u, 3y -> v}; rule2 = Map[Reverse, rule1];
> 
> Simplify[TrigFactor[Simplify[ExpToTrig[
>       Simplify[ExpToTrig[(-1)^(2*x + 3*y) /. rule1],
>         Mod[u, 2] == 0] /. rule2], y â?? Integers]],
>    y â?? Integers]
> 
> 
> (-1)^y
> 
> "Optimized" means that I can't see any obvious way to make this simpler.
> 
> Unlike other answers to the original post, this works in Mathematica  
> 5.1 and 5.2, and involves only reversible operations. In other words,  
> it constitutes a proof. On the other hand, obviously, it would be  
> ridiculous to use this approach in practice: there must be a simple  
> transformation rule (or rules) missing from Simplify, which  
> apparently has already been added in the development version.
> 

Andrzej,

It seems to me that your simplification approach isn't as general as the 
replacement method I advocated (a variant of which is given below), as 
rule1 and rule2 are specific to the input. Also, my replacement method 
is based on the fact that:

In[30]:= Simplify[((-1)^a)^b == (-1)^(a b), Element[{a, b}, Integers]]

Out[30]= True

So, I think my approach is generally valid.

> Note also the following problem, which I suspect is related:
> 
> 
> 
> Simplify[(-1)^(u + v), (u | v) â?? Integers &&
>     Mod[u, 2] == 0 && Mod[v, 2] == 1]
> 
> -1
> 
> 
> Simplify[(-1)^(u + v), (u | v) â?? Integers &&
>     Mod[u, 2] == 0 && Mod[v, 2] == 0]
> 
> 1
> 
> But
> 
> Simplify[(-1)^(u + v), (u | v) â?? Integers &&
>     Mod[u, 2] == 0]
> 
> (-1)^(u + v)
> 
> 
> 
> I am sure this also gives (-1)^v in the development version.
> 

To obtain the desired simplification in 5.2, we can modify my previous 
approach as follows:

power[e_, a_Plus] := Times @@ (Simplify /@ (power[e, #1]&) /@ List @@ a)
power[e_, a_Integer b_?(Refine[Element[#,Integers]]&)] := (e^a)^b
power[e_, a_] := e^a

simp[expr_, assum_] := Block[{$Assumptions = assum},
   Simplify[expr /. Power -> power]
]

In the above I rely on the following two equivalences, verified by 
Mathematica:

In[50]:=
Simplify[e^a e^b == e^(a+b)]

Out[50]=
True

In[51]:=
Simplify[(e^a)^b == e^(a b), Element[{a,b},Integers]]

Out[51]=
True

The tricky part is that Mathematica automatically converts e^a e^b --> 
e^(a+b), and as you point out above Simplify[(-1)^(u+v),...] doesn't 
work. However, Simplify[(-1)^u, ...] does work, which is why the 
Simplify appears in the definition of power[e_,a_Plus] before Times is 
applied.

Here are some examples:

In[56]:=
simp[(-1)^(u+v),Mod[u,2]==0]

Out[56]=
     v
(-1)

In[57]:=
simp[(-2)^(u+v),Mod[u,2]==0]

Out[57]=
     v  u + v
(-1)  2

In[58]:=
simp[(-1)^(2x+3y),Element[{x,y},Integers]]

Out[58]=
     y
(-1)

In[59]:=
simp[(-2)^(2x+3y),Element[{x,y},Integers]]

Out[59]=
     y  2 x + 3 y
(-1)  2

Carl Woll
Wolfram Research

> 
> 
> Andrzej Kozlowski
> 
> 
> On 2 Jun 2006, at 17:09, Andrzej Kozlowski wrote:
> 
> 
>>On 1 Jun 2006, at 19:54, Amitabha Roy wrote:
>>
>>
>>>Hello:
>>>
>>>I would like Mathematica to be able to take an expression, say,
>>>
>>>(-1)^{2 x  + 3 y} and be able to simplify to (-1)^y.
>>>
>>>Is there a way one can do this ?
>>>
>>>Thanks
>>>
>>
>>Note that you are using {} instead of (). Different kind of brackets
>>have completely different meaning in Mathematica.
>>I have found it amazingly hard to force Mathematica to perform this
>>simplification. The best I could do is this. We need two rules. rule1
>>will replace 2x by u and 3y by v. rule 2 does the opposite: it
>>replaces u by 2x and v by 2y.
>>
>>rule1 = {2x -> u, 3y -> v};rule2 = Map[Reverse, rule1];
>>
>>Now:
>>
>>
>>Simplify[TrigFactor[
>>    FullSimplify[ExpToTrig[
>>      FullSimplify[
>>        ComplexExpand[
>>         (-1)^(2*x + 3*y) /.
>>          rule1], Mod[u, 2] ==
>>         0] /. rule2],
>>     y â?? Integers]],
>>   y â?? Integers]
>>
>>
>>(-1)^y
>>
>>Uff... Surely, this ought to be easier...
>>
>>Andrzej Kozlowski
>>Tokyo, Japan
>>


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