Re: CHAT: math

On Mon, Nov 20, 2000 at 08:16:32PM -0000, Lars Henrik Mathiesen wrote:
[snip]
> Are you trying to characterize the asymptotic behaviour at infinity of
> any real (positive) funtion?
Yes, exactly! Except that recently, I've also begun to examine asymptotic
behaviour at singularities (eg. x=0 for 1/x^r, etc.), and I'm beginning to
make a very fascinating connection between the two.

> What magnitude do you get for  e^\sqrt{log x log log x} ? Or the
> inverse Ackermann function?
mag(e^\sqrt{log x log log x}) = xi * 1/2 * (lambda + lambda^2)
        = xi * (1/2 lambda + 1/2 lambda^2)

On the surface, I *guess* it might be a magnitude between the polynomial
magnitudes 1 and (1+lambda), but I'll have to double-check that. (Not now,
I gotta hurry up and mark some assignments to be returned to my students
in 2 hours' time :-P)

Interesting that you mention the Ackermann function, though. By Ackermann
function, I assume you mean the A(x,x)? The magnitude of f(x)=A(x,x) is
asymptotically larger than any magnitude derived from an nth order
operation (addition, multiplication, tetration, pentation, ...), and
hence, it would be represented by a "very large" magnitude number which is
unreachably greater than the magnitude of any nth order operation. (By
unreachable, I mean that no amount of finite combination of the nth order
magnitudes will ever reach it.) Therefore, the inverse Ackermann function
would have an extremely small "infinitesimal" magnitude, which is "more
infinitesimal"  than the nth order inverse operation (logarithm, inverse
tetration, etc.).

The existence of such "limit" magnitudes (which don't occur isolated, BTW,
you can always find magnitudes "near" such limit magnitudes, all of which
are asymptotically larger than the "smaller" magnitudes) is what I meant
when I alluded to infinite hierarchies of magnitudes.


T

--
Caffeine underflow. Brain dumped.