19
$\begingroup$

In teaching my calculus students about limits and function domination, we ran into the class of functions

$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$

Suppose we say that $g$ weakly dominates $f$, and write $f\preceq g$, if

$$\lim_{x\to\infty}\frac{f(x)}{g(x)} \hspace{3 mm} \text{is finite}$$

We can then readily see that $(\Theta,\preceq)$ is a total order isomorphic to the lexicographic order on $\mathbb{R}^2$.

But we can get more complicated total orders with, say

$$\Theta_n=(x^{\alpha_0}(\ln{x})^{\alpha_1}(\ln\ln{x})^{\alpha_2}\cdots(\ln^{n-1} x)^{\alpha_{n-1}})_{\vec{\alpha}\in\mathbb{R}^{n}}$$

$$\Phi=\{e^{p(x)}\}_{p(x)\in\mathbb{R}[x]}$$

which are isomorphic as total orders to the lexicographic orders on $\mathbb{R}^n$ and $\operatorname{List}\mathbb{R}$

All of these complicated orders live inside what I'd call "the AP Calc linear order" $(\Omega,\preceq)$ defined as:

$$\Omega_0=\{f\in\mathscr{C}^0((\lambda,\infty))\}_{\lambda\in\mathbb{R}}$$

$$f\preceq g\Longleftrightarrow \max\left\{\left|\liminf_{x\to\infty} \frac{f(x)}{g(x)}\right|,\left|\limsup_{x\to\infty} \frac{f(x)}{g(x)}\right|\right\}<\infty$$

$$\Omega=\Omega_0/\simeq \hspace{5 mm} \text{where} \hspace{5 mm} f\simeq g \Leftrightarrow \left[f\preceq g \text{ and } g\preceq f\right]$$

where the refinement on $\preceq$ is made so as to avoid problems with things like $\sin{x}$.

This seems to be a very complicated linear order, as it includes as a suborder things like

$$\Psi=\{p_0(x)e^{p_1(x)}e^{e^{p_2(x)}}\cdots\exp^{n-1}(p_{n-1}(x))\}_{p_i(x)\in\mathbb{R}[x]\forall i}$$

My question is the following: is there any combinatorial description or universal construction, i.e. as a colimit, of the isomorphism type of $(\Omega,\preceq)$?

$\endgroup$
  • 1
    $\begingroup$ Related mathoverflow.net/questions/29624/… $\endgroup$ – David E Speyer Aug 27 '15 at 16:19
  • 1
    $\begingroup$ This is not a total order; $f$ and $g$ can oscillate between $f\gg g$ and $g\gg f$. $\endgroup$ – Eric Wofsey Aug 27 '15 at 17:49
  • 5
    $\begingroup$ A side comment: The class $\Theta$ is a very small sub-class of the logarithmico-exponential functions studied by Hardy (see also Boshernitzan's work). These have very nice properties including no oscillation: for any two distinct functions in a Hardy Field, eventually $f$ is bigger than $g$ or vice versa. $\endgroup$ – Anthony Quas Aug 27 '15 at 18:34
  • 3
    $\begingroup$ @DmitryV: That's not what I was talking about (I was also thinking like a category theorist and not caring about that). If $f/g$ oscillates between approaching $0$ and approaching $\infty$, then $f\not\leq g$ and $g\not\leq f$. Actually, if $f$ and $g$ don't have to be positive, you could just have an unbounded set where $f$ is zero and $g$ is not and an unbounded set where $g$ is zero and $f$ is not. $\endgroup$ – Eric Wofsey Aug 27 '15 at 23:39
  • 3
    $\begingroup$ With Anthony Quas, I suggest you look up "Hardy field", and also "o-minimal structure" whose germs at infinity produce Hardy fields. The question as it stands suffers from problems noted by Eric Wofsey. $\endgroup$ – Todd Trimble Aug 27 '15 at 23:47

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.