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Re: CHAT: math

From:H. S. Teoh <hsteoh@...>
Date:Saturday, November 18, 2000, 23:04
On Sat, Nov 18, 2000 at 06:56:32PM -0500, John Cowan wrote:
> On Sat, 18 Nov 2000, H. S. Teoh wrote: > > > For example, between every two arbitrarily close "polynomial magnitudes", > > which correspond with real numbers, there are sub-polynomial magnitudes > > ("infinitesimals") that arise from logarithms. And between every two > > arbitrarily close logarithmic magnitudes, there are sub-logarithmic > > magnitudes ("sub-infinitesimals"?) that arise from iterated logarithms, > > and so on, ad infinitum. > > It sounds like you are just describing the (dense, continuous) real > numbers themselves, which are by no means only polynomial (= algebraic).
Oops... I knew I should've clarified what I meant :-) I'm *not* talking about the polynomial or logarithmic roots. I *do* know that the algebraic numbers are only a subset of the real numbers! :-) Rather, I'm talking about the rate of divergence or convergence to 0 (which is what I call the "magnitude" -- perhaps that's not a good term for it). OK, for the record, let me clarify what I was trying to say: Let f be a function that has polynomial magnitude p. (I'll skip the definition unless you really want me to clutter this list with it -- for now, it suffices to say that the power function x^p is one possibility for f.) Let g(x) = f(x)*log(x), where log(x) is the natural logarithm. The magnitude of g is therefore (p+lambda). (Again I omit a very lengthy definition.) Then, for *every* function g that has polynomial magnitude q > p, it holds that: p < p+lambda < q (p+lambda) is only one among a whole hierarchy of "infinitesimal" magnitudes. Let n*lambda denote the magnitude of h(x) = (log^n)(x), where log^n is the natural logarithm composed with itself n times. Then, for all functions g with polynomial magnitude q, p < p + n*lambda < q Now, let L(x) be the iterated logarithm of x. Let l denote the magnitude of L. Then: p < p + l < p + n*lambda < q for all n>=1. Hence, l behaves like an infinitesimal of infinitesimals. (I haven't actually proven this last assertion yet, but I've strong evidence that this is true.) [snip]
> For the record, his name is Paul Pedersen.
Oh, you mean I actually have to credit him? ;-) OK, I guess I'd better :-) T -- And life still goes on...