Newsgroup sci.math.research 6207

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Subject: Re: polynomial question 2
From: rgep@can.pmms.cam.ac.uk (Richard Pinch)
Date: 9 Nov 1996 09:45:26 GMT

In article <5604i8$7a3@vixen.cso.uiuc.edu> cotrell@ajpiris.me.uiuc.edu writes:
>I have a system of N polynomial equations (f_i=0:1<=i<=N)
>in the N variables (x_1,...,x_N), and would like
>to find a method (numerical or analytical) that is
>able tell if the system has a finite number
>of solutions or that the system has an infinite
>number of solutions.
You can do this is Maple with the Grobner basis package:
> ?grobner[finite]
FUNCTION: grobner[finite] - decide if a given algebraic system has (at most)
                            finitely many solutions
CALLING SEQUENCE:
   finite(F)
   finite(F, X)
PARAMETERS:
   F - set or list of polynomials
   X - set or list of indeterminates (not including parameters; default is
       indets(F))
SYNOPSIS:   
- The command finite(F, X) decides, by using the total degree Grobner basis and
  a criterion of Buchberger, if a set/list of polynomials F with respect to the
  indeterminates X has (at most) finitely many solutions.
- If X is a list, the given permutation of variables is used in all associated
  Grobner basis computations.
- If X is omitted, the set indets(F) is used as a default.
- This function is part of the grobner package, and so can be used in the form
  finite(..) only after performing the command with(grobner) or
  with(grobner,finite).  The function can always be accessed in the long form
  grobner[finite](..).
EXAMPLES:   
> with(grobner):
> F := [x^2 - 2*x*z + 5, x*y^2 + y*z^3, 3*y^2 - 8*z^3]:
> finite(F);
                                      true
> finite([F[1],F[2]]);
                                     false
SEE ALSO:  grobner[solvable], grobner[finduni], grobner[gsolve]

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Subject: measure
From: Biblioteca matematica
Date: Sat, 09 Nov 1996 11:19:33 -0800

Let C be Cantor set.
can you find a subset D of R that is omeomorph with C, and such that 
Lebesgue measure of D isn' t zero?

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Subject: Uniform Limits of Monotone Functions with Summable Discontinuities
From: lones@lones.mit.edu (Lones A Smith)
Date: 9 Nov 1996 21:12:42 GMT

Let U(x,y) be a bounded function on [0,1]X[0,1], with U(x,0)=0 for all x,
and U(x,y) monotone nondecreasing in all y>=0. Suppose that the sum of 
all discontinuities of U in x, the sum taken over all (x,y) in [0,1]X[0,1],
is at most 1. I presume this means that there are countably many such 
discontinuities (x_1,y_1),(x_2,y_2),... with the left or right jump in x 
at (x_i,y_i) equal to a_i, with |a_1|+|a_2|+... <= 1. 
CLAIM: For all epsilon>0 there exists delta>0 s.t. for almost all x, 
U(x,y)
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Subject: Nearby Continuous Supermodular Functions Don't Often Differ in Sign
From: lones@lones.mit.edu (Lones A Smith)
Date: 9 Nov 1996 17:23:08 GMT

Hi,
Let f be a supermodular (f_{xy}>0) and Lipschitz function on [0,1]X[0,1].
Let h be a Lipschitz function on [0,1], and put g(x,y) = f(x,y) + h(x) + h(y).
Define the indicator function I(x,y) = 1 if f(x,y)>=0 and g(x,y)<0 or 
f(x,y)<0 and g(x,y)>=0, and zero otherwise. 
CLAIM       For all epsilon>0, there exists delta>0 such that 
sup(|h(z)|)  ess-sup_x \int I(x,y) dy < epsilon
The motivating idea is that for x10. But I find it hard to deduce the above 
of continuity. (If it helps, observe that g_{xy}=f_{xy}.) 
Any help with the claim would be greatly appreciated.    -- Lones
  .-.     .-.     .-.     .-.     .-.     .-.    
 / L \ O / N \ E / S \   / S \ M / I \ T / H \   
/     `-'     `-'     `-'     `-'     `-'     ` 
 Lones Smith, Economics Department, M.I.T., E52-252C, Cambridge MA 02139
 (617)-253-0914 (work)  253-6915 (fax)   lones@lones.mit.edu 

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