
Re: Continuous and discrete uniform distributions of N
Posted:
Dec 22, 2012 2:58 PM


On 12/22/2012 11:14 AM, porky_pig_jr@mydeja.com wrote: > On Friday, December 21, 2012 11:41:44 PM UTC5, Bill Taylor wrote: >> On Dec 22, 5:23 am, FredJeffries<fredjeffr...@gmail.com> wrote:> Dirac delta, infinitesimals, irrational numbers, transfinite ordinals,> ... are legitimate not because they have been rigorously defined Yes, that is PRECISELY why they are legitimate.> No one has ever anywhere actually used the concept of a uniform> distributions on N to solve any problem. Sure they have. You can use it to calculate the probability that two randomly chosen naturals will be coprime, for example. And many others of that type.  Blunderbuss Bill ** Dogma is a bitch! (pun intended) > > So, you're saying there *exists* the uniform distribution of positive integers (or natural numbers if you wish). Well, well, well, would you please then enlighten the unwashed masses like myself and tell us that's the probability of selecting an arbitrary positive integers? > > Regards, > > PPJr.
From topological groups, I remember the concept of "amenable group".
The additive groups Z, Z^2, Z^3 and so on can be given the discrete topology, where each sigleton, say {(1, 3)} for the group Z^2, is a closed set.
Then as I recall, a theorem says some type of abelian groups are amenable.
I don't remember the type condition for abelian groups.
cf.: http://en.wikipedia.org/wiki/Amenable_group
Of course, N with addition isn't a group. It's a ? semigroup? (yes), and also a monoid if by N one means the thing that has zero in it: http://en.wikipedia.org/wiki/Monoid
Amenability means a sensible, consistent with translations, averaging procedure exists e.g. for bounded functions on the group.
For the probabilistic GCD is 1 argument for "random" elements (m, n) in N^* x N^* , I don't know if it can be recast using the amenability of Z x Z ...
gcd of (0, 0) : I know "modulo theory" is linked to ideals in rings. Anyway, since everything nonzero divides zero, it's dubious about sensibly defining gcd of (0, 0).
But then, (0, 0) is just 1 element of ZxZ, so coprimeness yes/no of (0,0) should be irrelevant to an argument based on amenability.
I don't know what that argument might look like, assuming it exists, i.e. that I'm on some sort of "right track" ...
dave
David Bernier

