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>

**Re: Re: Weird result in Mathematica 6**

**Re: Mathematica 6.0 easier for me ... (small review)**

**Re: Re: Weird result in Mathematica 6**

**Re: Re: Weird result in Mathematica 6**