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Re: Re: System of NonLinear Inequalities

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  • Subject: [mg48597] Re: [mg48564] Re: [mg48511] System of NonLinear Inequalities
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Mon, 7 Jun 2004 05:33:20 -0400 (EDT)
  • References: <200406040849.EAA23604@smc.vnet.net> <200406051119.HAA11747@smc.vnet.net> <6.0.1.1.1.20040605133437.01f906b8@pop.hfx.eastlink.ca>
  • Sender: owner-wri-mathgroup at wolfram.com
Indeed, but one should note that a numerical solution both of an 
equation and a non-strict inequality is a rather different thing from 
an exact solution. Of course for many purposes it may be just as good 
or better but for some problems the existence of a strict and not just 
an approximate solution is crucial. For example, when using Morse 
theory (the subject of my talk at IMS 2001) to study the topology of a 
surface the existence of an 'approximate' critical point is 
insufficient. The same is true of a few other areas of mathemtics.
Mathematica has powerful algebraic tools for strict solutions of both 
equations and inequalities (the main being Collins's 
CylindricalAlgebraicDecomposition) but naturally they are theoretically 
limited to algebraic equations and practically to algebraic equations 
with a small number of variables.

Andrzej Kozlowski



On 6 Jun 2004, at 01:45, Janos D. Pinter wrote:

>
> Colleagues,
>
> systems of nonlinear equations and inequalities - under general 
> analytical conditions - can be transformed into global optimization 
> problems that can be solved numerically. (See e.g. my book 'Global 
> Optimization in Action', Ch. 4.1.) Of course, such systems may have no 
> solution, infinitely many solns, and 'anything in between'.
>
> If you are interested in a single particular soln, then try to express 
> the quality of that soln by an 'objective function' and then you can 
> solve a std math programming (optimization) problem. For example, you 
> may search for the soln with minimal least squares error in an 
> inconsistent system, or you may like to find the soln that is closest 
> to the origin, etc.
>
> (The MathOptimizer Professional User Guide includes an example with 
> multiple solutions to a system of nonlinear equations, and how to 
> handle them numerically.)
>
> Regards,
> Janos Pinter
> _________________________________________________
>
> Janos D. Pinter, PhD, DSc
> President & Research Scientist, PCS Inc.
> Adjunct Professor, Dalhousie University
> 129 Glenforest Drive, Halifax, NS, Canada B3M 1J2
> Telephone:      +1-(902)-443-5910
> Fax:            +1-(902)-431-5100; +1-(902)-443-5910
> E-mail: jdpinter at hfx.eastlink.ca
> Web:            www.dal.ca/~jdpinter
>                 www.pinterconsulting.com
>
>
>
>
>
>
>
>
> At 08:19 AM 6/5/2004, Andrzej Kozlowski wrote:
>> First of all, your inequalities are not written using Mathematica
>> syntax (you can't use square brackets in this way). But looking at 
>> them
>> I see it does not matter whether you use proper syntax or not: no
>> computer program will ever solve a system of inequalites involving
>> someting like x^(2/(2 - x)). Your only chance is a human brain and 
>> some
>> fantastic stroke of luck.
>> Sorry for being so unhelpful.
>>
>> Andrzej
>>
>> On 4 Jun 2004, at 17:49, maurizio lisciandra wrote:
>>
>> > Dear Friends,
>> >
>> > I tried to solve the following system of nonlinear inequalities with
>> > Mathematica 5.0:
>> >
>> > F < (1/2)*[a^(x/(2 - x))]*[x^(2/(2 - x))] &&
>> > (2*a*B)^(x/2) + F - B > (2*a*F)^(x/2) &&
>> > F > B > 0 &&
>> > 0 <= a <= 1 &&
>> > 0 < x < 2.
>> >
>> > I tried with Reduce, SolveInequality, SemiAlgebraicComponent,
>> > FindInstance,
>> > but all these function do not solve it. I tried to substitute x for
>> > some
>> > fixed value, but again I cannot solve it. The only value that I can
>> > substitute for x is 1, and in this case the solution is an empty
>> > space. I
>> > may be happy if I find a value that solves the system, although I
>> > really
>> > need for which inetervals in the variable this system is not an 
>> empty
>> > space.
>> >
>> > Hope some nice Mathematica expert can help me.
>> >
>> > Cheers,
>> >
>> > Maurizio Lisciandra
>> > Trinity College
>> > Cambridge (UK)
>> >
>> > _________________________________________________________________
>> > Ricerche online più semplici e veloci con MSN Toolbar!
>> > http://toolbar.msn.it/
>> >
>> >
>
>
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