Re: CHAT: math

> Date:         Sat, 18 Nov 2000 18:04:35 -0500
> From:         "H. S. Teoh" <hsteoh@...>
> On Sat, Nov 18, 2000 at 06:56:32PM -0500, John Cowan wrote:
> 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.)
Are you trying to characterize the asymptotic behaviour at infinity of
any real (positive) funtion?

What magnitude do you get for  e^\sqrt{log x log log x} ? Or the
inverse Ackermann function?

Lars Mathiesen (U of Copenhagen CS Dep) <thorinn@...> (Humour NOT marked)