Set theory seminar -Forcing axioms and inner models V

October 12, 2008

We showed Velickovic’s result that under ${\sf MM}$ any inner model that computes $\omega_2$ correctly actually contains $H_{\omega_2}$.

The argument depends on the (weak) reflection principle (a consequence of ${\sf MM}$) and a combinatorial result due to Gitik.

It is open whether this result holds with ${\sf PFA}$ in place of ${\sf MM}$, but an attempt to settle this led to the discovery that ${\sf BPFA}$ implies the existence of a definable (in a subset of $\omega_1$) well-ordering of the reals. The well-ordering is actually $\Delta_1$ in the parameter, and the proof shows that $H_{\omega_2}$ can be decomposed as a union of small transitive structures whose height determines their reals. This “$L$-like” decomposition of $H_{\omega_2}$ is expected to continue for larger cardinals, which leads to the following:

Conjecture (Caicedo, Velickovic). Assume ${\sf MM}$ and let $M$ be an inner model that computes cardinals correctly. Then ${\sf ORD}^{\omega_1}\subset M$.

Although the conjecture is still open, there is (significant) partial evidence suggesting it. For example, we showed that if $V$ satisfies ${\sf MM}$ and is a forcing extension of an inner model that computes correctly the class of ordinals of cofinality $\omega_1$, then ${\sf ORD}^{\omega_1}\subset M$.

275- Maple worksheet on directional derivatives

October 12, 2008

Here is the Maple worksheet I tried to follow on Friday. I am including also three topographical maps to illustrate the fact that rivers follow paths of steepest descent. Unfortunately, the resolution is not ideal in any of the examples. If you know of a good free online source for decent topo maps, please let me know.

The first map (greatlakesarea) was obtained through the my-topo.com site.

The other two maps (colombia1, colombia2) were obatined through the site for the Instituto Geografico Agustin Codazzi.

Here is the Maple worksheet: directionalderivatives. Unfortunately, wordpress only allows one to upload files from a very small number of applications, so I have changed the extension from .mw to .doc, and you may want to change it back after downloading. I am somewhat rusty in my knowledge of Maple, so I don’t remember how to change the lighting in graphics directly through the code (if you know how, please let me know). To see the surfaces better, you may want to click on them so the “Plot” menu activates, and there you want to choose something like Light Scheme 1 under Lighting. You may also rotate the graphs, to see them from different angles, which helps appreciate better some of the details (like the ring of directions in the tangent plane in one of the examples).

[There is a typo, of course. At the end, where it says $5\cos\theta$, it should be $r\cos(5\theta)$.]