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Re: Re: Re: System of NonLinear Inequalities

Of course, I agree with Andrzej's comments. In certain cases, one needs 
exact solutions (which can be obtained only under special conditions). In 
many other cases, one needs to rely on numerical techniques: within the 
latter category, global scope search/optimization tools will be very useful.

Janos Pinter


At 06:33 AM 6/7/2004, you wrote:
>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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