Re: Re: parameter restrictions

*To*: mathgroup at smc.vnet.net*Subject*: [mg32460] Re: [mg32442] Re: [mg32428] parameter restrictions*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>*Date*: Tue, 22 Jan 2002 03:19:25 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

The way to do this sort of thing is completely different. The usage of parameters for pattern matching has nothing to do with deduction of one condition form another. The latter is done with Simplify and FullSimplify. Unfortunately your conclusion involves symbolic exponents the methods used by Simplify and FullSimplify do not work at present with symbolic exponents. In other words Mathematica can't answer this: In[1]:= Simplify[x^a>0,{x>0,a>0}] Out[1]= x^a > 0 So this will not work: In[2]:= f[x_,a_,b_]:=-x^(a+b) In[3]:= Simplify[D[f[x,a,b],{x,2}]>0,{x>0,a>0,b>0, (a+b)<1}] Out[3]= 0 > (-1 + a + b)*(a + b)*x^(-2 + a + b) One way to deal with this is to replace the symbolic power of x by some other variable, say t, and tell Mathematica that t is >0: In[4]:= Simplify[(D[f[x,a,b],{x,2}]/.x^a_:>t)>0,{(a+b)<1,a>0,b>0,z>0,t>0}] Out[4]= True Of course if the exponent of x is a number Mathematica has no problem with this sort of thing: In[5]:= Simplify[0>(-1+a+b)*(a+b)*x^53,{x>0,a>0,b>0, (a+b)<1}] Out[5]= True Andrzej Kozlowski Toyama International University JAPAN http://platon.c.u-tokyo.ac.jp/andrzej/ On Monday, January 21, 2002, at 04:54 PM, soso lala wrote: > Hi > > Thanks for your answer. Unfortunately, it does not work like this. > Consider > the following example: > > Clear[f,x,a,b] > f[x_,a_,b_]:= -x^(a+b) > > The second derivation with respect to x is: > D[f[x,a,b],{x,2}] > -(-1+a+b)(a+b)x^(-2+a+b) > > Given that x>0,a>0,b>0 and (a+b)<1 this derivation is unambiguously > positive. I have tried to show this with Mathematica in the manner > proposed: > > Clear[f,x,a,b] > f[x_?Positive,a_?Positive,b_?Positive]:= -x^(a+b) /; (a+b)<1 > > Now, if I write > > D[f[x,a,b],{x,2}]>0 > > I would expect the output > > TRUE > > but I receive > > f^(2,0,0)[x,a,b]>0 > > Can somebody tell me where the mistake is or how I must define parameter > restrictions? > > Thanks for your efforts > > Jack > > >> From: BobHanlon at aol.com To: mathgroup at smc.vnet.net > To: mathgroup at smc.vnet.net >> Subject: [mg32460] [mg32442] Re: [mg32428] parameter restrictions >> Date: Sat, 19 Jan 2002 20:47:10 EST >> >> >> In a message dated 1/19/02 8:05:36 PM, sosolala at hotmail.com writes: >> >>> I am using Mathematica 4.0 and have a question about parameter >> restrictions. >>> >>> How can I define the range of values of a parameter, e.g. that alpha >>> must >>> be >>> positive or that (alpha + beta) must be less than unity? >>> >>> Although it may be a very simple question (and/or answer) I would be >>> delighted if I get an answer. >>> >> >> f[x_, a_?Positive] := a*x; >> >> f[x_, a_?Positive, b_?NonNegative] := a^x*b /; (a+b) < 1; >> >> >> Bob Hanlon >> Chantilly, VA USA > > > > > _________________________________________________________________ > Downloaden Sie MSN Explorer kostenlos unter > http://explorer.msn.de/intl.asp. > > > >