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Re: Re: Weird result in Mathematica 6

• To: mathgroup at smc.vnet.net
• Subject: [mg76554] Re: [mg76432] Re: [mg76393] Weird result in Mathematica 6
• From: Adam Strzebonski <adams at wolfram.com>
• Date: Thu, 24 May 2007 05:52:51 -0400 (EDT)
• References: <200705211001.GAA10071@smc.vnet.net> <EB6D3224-597F-4DD6-B05D-08B9F6A05D2D@mimuw.edu.pl> <200705220648.CAA19836@smc.vnet.net> <63B2BBD7-455D-42F6-AFB2-63F7D37D62D3@mimuw.edu.pl> <42849280-2256-40D7-826A-2626E1B6411D@mimuw.edu.pl>
• Reply-to: adams at wolfram.com

Andrzej Kozlowski wrote:
> 
> On 22 May 2007, at 23:44, Andrzej Kozlowski wrote:
> 
>> *This message was transferred with a trial version of CommuniGate(tm) 
>> Pro*
>>
>> On 22 May 2007, at 15:48, Adam Strzebonski wrote:
>>
>>> Andrzej Kozlowski wrote:
>>>>
>>>> On 21 May 2007, at 19:01, Sebastian Meznaric wrote:
>>>>
>>>>> I was playing around with Mathematica 6 a bit and ran this command to
>>>>> solve for the inverse of the Moebius transformation
>>>>>
>>>>> FullSimplify[
>>>>>  Reduce[(z - a)/(1 - a\[Conjugate] z) == w && a a\[Conjugate] < 1 &&
>>>>>    w w\[Conjugate] < 1, z]]
>>>>>
>>>>> This is what I got as a result:
>>>>> -1 < w < 1 && -1 < a < 1 && z == (a + w)/(1 + w Conjugate[a])
>>>>>
>>>>> Why is Mathematica assuming a and w are real? The Moebius
>>>>> transformation is invertible in the unit disc regardless of whether a
>>>>> and w are real or not. Any thoughts?
>>>>>
>>>>>
>>>>
>>>>
>>>> Reduce and FullSimplify will usually deduce form the presence of
>>>> inequalities in an expression like the above that the variables
>>>> involved in the inequalites are real. In your case it "sees"
>>>> a*Conjugate[a]<1 and "deduces" that you wanted a to be real. This was
>>>> of coruse not your intention but you can get the correct behaviour by
>>>> using:
>>>>
>>>>  FullSimplify[
>>>>  Reduce[(z - a)/(1 - Conjugate[a]*z) == w && Abs[a]^2 < 1 && Abs[w] 
>>>> ^2 <
>>>> 1, z]]
>>>>
>>>>
>>>>  -1 < Re[w] < 1 && -Sqrt[1 - Re[w]^2] < Im[w] < Sqrt[1 - Re[w]^2] 
>>>> &&  -1 <
>>>>   Re[a] < 1 &&
>>>>    -Sqrt[1 - Re[a]^2] < Im[a] < Sqrt[1 - Re[a]^2] &&
>>>>  z == (a + w)/(w*Conjugate[a] + 1)
>>>>
>>>> Mathematica knows that the fact that an inequality involves Abs[a]  
>>>> does
>>>> not imply that a is real but it does not "know" the same thing  about
>>>> a*Conjugate[a]. This is clearly dictated by considerations of
>>>> performance than a straight forward bug.
>>>> Andrzej Kozlowski
>>>>
>>>
>>> By default, Reduce assumes that all algebraic level variables appearing
>>> in inequalities are real. You can specify domain Complexes, to make
>>> Reduce assume that all variables are complex and inequalities
>>>
>>> expr1 < expr2
>>>
>>> should be interpretted as
>>>
>>> Im[expr1]==0 && Im[expr2]==0 && Re[expr1]<Re[expr2]
>>>
>>> For more info look at
>>>
>>> http://reference.wolfram.com/mathematica/ref/Reduce.html
>>> http://reference.wolfram.com/mathematica/tutorial/RealReduce.html
>>> http://reference.wolfram.com/mathematica/tutorial/ComplexPolynomialSystems.html 
>>>
>>>
>>> In your example we get
>>>
>>> In[2]:= Reduce[(z - a)/(1 - a\[Conjugate] z) == w && a a\[Conjugate] 
>>> < 1 &&
>>>     w w\[Conjugate] < 1, z, Complexes]
>>>
>>>                                           2                          2
>>> Out[2]= -1 < Re[w] < 1 && -Sqrt[1 - Re[w] ] < Im[w] < Sqrt[1 - Re[w] 
>>> ] &&
>>>
>>>                                        2                          2
>>>>    -1 < Re[a] < 1 && -Sqrt[1 - Re[a] ] < Im[a] < Sqrt[1 - Re[a] ] &&
>>>
>>>                  a + w
>>>>    z == ------------------
>>>            1 + w Conjugate[a]
>>>
>>>
>>> Evaluate
>>>
>>> Reduce[x^2+y^2<=1, {x, y}, Complexes]
>>>
>>> to see why I think that assuming that variables appearing
>>> in inequalities are real is a reasonable default behaviour.
>>>
>>> Best Regards,
>>>
>>> Adam Strzebonski
>>> Wolfram Research
>>>
>>
>> Still, it seems to me that there is a certain problem with this, not 
>> very important but still, a "logical difficulty". It concerns not 
>> Reduce, where you can specify the domain to be Reals or Complexes etc, 
>> but Simplify, where you can't. So for example:
>>
>> Simplify[Re[x], x*Conjugate[x] > 1]
>> x
>>
>> folowing the principle also used by reduce, Simplify assumed that x is 
>> real. On the other hand:
>>
>> Simplify[Re[x], Abs[x] > 1]
>>  Re(x)
>>
>> which also agrees with the principle, sicne Abs in non-algebraic. But, 
>> unlike in the case of Reduce, there seems to be no way to make 
>> Simplify treat the first assumption as taking place over the Complexes 
>> as in the Reduce example:
>>
>>  Simplify[Re[x], x*Conjugate[x] > 1 && Elment[x, Complexes]]
>> x
>>
>> Simplify[Re[x] && Element[x, Complexes], x*Conjugate[x] > 1]
>> x
>>
>> In other words, it seems that when using Simplify one really needs to 
>> use Abs in inequalities, if one does not want to force the assumption 
>> that a variable is real. (?)
>>
>> Andrzej Kozlowski
> 
> Furthermore:
> 
> Simplify[Re[x] && ! Element[x, Reals], x*Conjugate[x] > 1]
> False
> 
> Andrzej Kozlowski
> 

Since x is assumed to be real, !Element[x, Reals] simplifies
to False. Then we have

In[2]:= ArbitraryExpression[any, variables] && False
Out[2]= False

Best Regards,

Adam Strzebonski
Wolfram Research

• References:
  • Weird result in Mathematica 6
    • From: Sebastian Meznaric <meznaric@gmail.com>
  • Re: Weird result in Mathematica 6
    • From: Adam Strzebonski <adams@wolfram.com>