Re: Sampling Without Replacement


"JoeS" <jhs@xxxxxxxxxxxxxx> wrote in message
news:1137027886.123605.92340@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
>
> JoeS wrote:
> > Jim Dars wrote:
> > > "quasi" <quasi@xxxxxxxx> wrote in message
> > > news:dru9s150ol251q0d3j64c5df0tlshut446@xxxxxxxxxx
> > > > On Tue, 10 Jan 2006 11:16:44 -0800, "Jim Dars"
<jim-dars@xxxxxxxxxxx>
> > > > wrote:
> > > >
> > > > >Hi All,
> > > > >
> > > > >Consider
> > > > >
> > > > > (x+1)(x)(x-1)/[(y+1)(y)(y-1)] = R/S
> > > > >
> > > > > y > x > 1; S > R >1 ; x, y, R, and S are integers.
> > > > >
> > > > >Question 1)
> > > > >
> > > > > Let x = 4 and y = 7 Thus R = 5 and S =23
> > > > >
> > > > >For R/S = 5/23 is there another solution(s)?
> > > > >
> > > > >Question 2)
> > > > >
> > > > > For R/S = 7/11 How do you go about finding x and y ?
> > > > >
> > > > >Question 3)
> > > > >
> > > > >For given R and S does a solution always exist?
> > > >
> > > > I don't see what this has to do with "Sampling Without Replacement".
> > > >
> > > > What you have here is a diophantine equation.
> > > >
> > > > It doesn't look easy.
> > > >
> > > > Hiw did the problem arise? Did it come from a book or a course? If
so,
> > > > which book or what course?
> > > >
> > > > quasi
> > >
> > > Hi quasi,
> > >
> > > Most everything you've said it true (see comment referencing title
below.).
> > > And thank you for at least contributing to the thread. I was afraid
I'd be
> > > completely ignored.
> > >
> > > The problem began many years ago. Some "innocent" (claiming to be
away from
> > > math for some years) posted the question: If I start with an urn
containing
> > > R red balls and N total balls what should R and N be so that the
chance of
> > > drawing two straight red balls, without replacement, is 1/6. After a
little
> > > thought you'd probably reply 2 red balls in a total number of 4.
> > > (2/4)*(1/3) = 1/6. But how, he asked, do you solve this problem in
general?
> > > (Thus the title.)
> > >
> > > Well, the problem proved damned difficult. Without getting into it,
the
> > > problem usually has an infinite number of solutions. It was a
friendly NG
> > > and it took a while, and a lot of interesting discussion, to work out
the
> > > problem. During the course of the discussion, as the solution was
becoming
> > > apparent, someone commented that if the number of samples were three
the
> > > problem looked impossible. This set me off, but after 3 years I
decided;
> > > Jim, best you let some others have a whack at this.
> > >
> > > Best wishes, Jim
> >
> > Hi Jim
> >
> > If I understand correctly, you're fixing nonzero integers R and S and
> > asking for nontrivial integer solutions to the cubic Diophantine
> > equation
> >
> > S(x^3 - x) = R(y^3 - y).
> >
> > A _nonsingular_ cubic equation has only finitely many integer solutions
> > by a famous theorem of Siegel on integral points on elliptic curves. So
> > let's see if your equation is singular. Taking the partial derivatives
> > and setting them equal to 0 gives
> > S(3x^2 - 1) = 0 and R(3y^2 - 1) = 0.
> > So the potential singular points have x^2=1/3 and y^2=1/3. But plugging
> > into the original equation, these points are not on the curve unless
> > S=R or S=-R. So if you assume that |S| does not equal |R|, then there
> > are no singular points. (I'm cheating a bit, you also need to check the
> > point(s) at infinity, but I did check and the three points are infinity
> > are always nonsingular.)
> >
> > Conclusion: If S and R are nonzero integers and |S| != |R|, then there
> > are only finitely many integer solutions to the equation S(x^3 - x) =
> > R(y^3 - y).
> >
> > So maybe I'm missing something, because you indicated the problem tends
> > to have infinitely many solutions.
> >
> > For particular values of S and R, one may be able to determine all of
> > the solutions using one or more of the "standard" methods for finding
> > integer points on elliptic curves, but the process tends to be fairly
> > difficult. There are lots of papers in the literature (and presumably
> > material on the web) giving examples of considerable complexity.
> > And in principle one can use the theory of linear forms in logarithms
> > to write down an explicit upper bound for the largest solution in terms
> > of R and S, although the bound would be quite large.
> >
> > As for your Question 3: "For given R and S does a solution always
> > exist?", do you have some reason to suspect that there should always be
> > a solution? The general situation for families of elliptic curves seems
> > to be that sometimes there are integer solutions and sometimes there
> > are none. (This is just an observation, not a theorem.) Indeed, aside
> > from the trivial solutions to your equation, I'd suspect that there are
> > infinitely many integer pairs (R,S) with R > S for which the equation
> > doesn't even have any rational solutions other than the few obvious
> > ones.
> >
> > One final comment. Your curves have a lot of "trivial" solutions, more
> > precisely, there are nine of them, all of the points (a,b) where a and
> > b can be any one of -1,0,1. My guess is that this means that the group
> > of rational points on the elliptic curve includes a subgroup of the
> > form Z/3Z x Z/3Z.
>
> Sorry, just realized your comment about "infinitely many solutions"
> referred to the case of taking 2 samples, not 3. In that case, you're
> right, there may well be infinitely many solutions, since
> S(x^2-x)=R(y^2-y) is more-or-less a Pell equation. So I hope the
> observation that there are only finitely many solutions for the "choose
> 3" case is of interest to you.
>
Hi JoeS,

Thank you! Not surprisingly I have a few questions. First a comment or
two, mainly so you can know what your dealing with.

1) Yes, your dead center right on about the two sample problem solution
revolving about the Pell equation.
2) While not a mathematician I have a great deal of practical math
training; over 60 semester credits. Alas, acquired over 50 years ago at an
engineering college. To my shame, I am not familiar with elliptical
equations, though I've often seen them referred to.
3) If I ask a question which you feel requires too much of your time, I
would appreciate your saying so --- trust me, I'll understand. But just
don't ignore the question because then I'll wonder if you over looked it.

With the foregoing in mind could you pick your own NUMERICAL examples (to
simplify the situation) and illustrate 2 situations:

a) An (R,S) pair that has no solution.

b) An (R,S) pair where you can delineate all the solutions.

Let me close with the obvious, you can't dumb this down too much.

Best wishes, Jim


.

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    ... >> Jim, best you let some others have a whack at this. ... > A _nonsingular_ cubic equation has only finitely many integer solutions ... > by a famous theorem of Siegel on integral points on elliptic curves. ... > pointat infinity, but I did check and the three points are infinity ...
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