Re: Determining continuity of regions/curves from inequalities
- To: mathgroup at smc.vnet.net
- Subject: [mg67271] Re: [mg67216] Determining continuity of regions/curves from inequalities
- From: "Carl K. Woll" <carlw at wolfram.com>
- Date: Thu, 15 Jun 2006 03:25:54 -0400 (EDT)
- References: <200606130506.BAA23751@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Bonny Banerjee wrote: > Is there an easy way in Mathematica to determine whether the region or curve > formed by a system of inequalities is continuous or not? > > For example, the output of some function (e.g. Reduce) might be as follows: > > x>2 && y>0 > > which forms a continuous region. Again, the following output > > (x<2 && y<0) || (x>2 && y>0) > > is not continuous. Similarly, for curves. > > Given such a system of inequalities, how to determine whether the > region/curve it forms is continuous or not? Or in other words, if I pick any > two random points, say P1 and P2, lying on the output curve/region, does > there exist a continuous path lying entirely within the output curve/region > from P1 to P2? > > Any help will be appreciated. > > Thanks, > Bonny. > The function SemialgebraicComponents in the package Algebra`AlgebraicInequalities may help. From the help browser: "The package provides a function for solving systems of strong polynomial inequalities in one or more unknowns. To be precise, SemialgebraicComponents[ineqs, vars] gives a finite set of solutions of the system of inequalities. That is, within the set of solutions, any solution can be connected by a continuous path to a solution in the finite set. The variable ineqs is a list of strong inequalities, where both sides of each inequality are polynomials in variables vars with rational coefficients. In other words, SemialgebraicComponents[ineqs, vars] gives at least one point in each connected component of the open semialgebraic set defined by inequalities ineqs. " Needs["Algebra`AlgebraicInequalities`"] First example: x>2 && y>0 SemialgebraicComponents accepts a list of inequalities: In[6]:= SemialgebraicComponents[{x>2,y>0},{x,y}] Out[6]= {{3,1}} Only one point is returned, so the region is connected. Second example: (x<2 && y<0) || (x>2 && y>0) This example is not a list of inequalities due to the Or. However, in this case it is easy to construct an equivalent inequality that encompasses the same region: In[5]:= SemialgebraicComponents[(x-2)y>0,{x,y}] Out[5]= {{0,-1},{3,1}} There are two components, so the region is not connected. Carl Woll Wolfram Research
- References:
- Determining continuity of regions/curves from inequalities
- From: "Bonny Banerjee" <banerjee@cse.ohio-state.edu>
- Determining continuity of regions/curves from inequalities