From: mailbox@dmitry-kazakov.de   
      
   On 3 Oct 2006 04:18:45 -0700, Bill Silvert wrote:   
      
   > I guess that you could put this more succinctly as pointing out that mu   
   > lies in the open interval (0,1) and not the closed interval [0,1]. I   
   > call such sets "strictly fuzzy" and have defined the operation   
   > symmetric summation on them (which is in Zimmerman's book).   
   >   
   > The symmetric sum deals with both intersection and union.   
      
   Hmm, an intersection of a left and a right shoulders cannot be again a   
   shoulder, semantically it cannot. You could define an operation that would,   
   but then it violated intuitively obvious axioms of sets intersection.   
      
   > Alhough this might be of concern to a mathematician, for environmental   
   > variables the difference betwen 0.00001 and 0 is not meaningful.   
      
   I.e. you consider it rather numerically, as an approximation of some mu,   
   which is not bound to any class of analytical functions. In that case, as I   
   said, there could be better approximations. You can drop smoothness or even   
   continuity, if it is just an approximation in C-norm.   
      
   > As for using splines, I don't think that any model with more than two   
   > parameters is useful in environmental applicaitons.   
      
   It could be same two. You can use the following spline approximation:   
      
   ]-oo, x0[ = 0   
   [x0, x1] = S(x) = ax^3 + bx^2 + cx + d   
   [x1, +oo[ = 1   
      
   The parameters here are x0 and x1. The coefficients a,b,c,d are determined   
   by the boundary conditions:   
      
   S(x0) = 0   
   S(x1) = 1   
   S'(x0) = S'(x1) = 0   
      
   Denoting: t=(x-x0)/(x1-x0)   
      
   S(x)=t^2 (3 - 2t)   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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