Re: CHAT: math

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

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