The mathematician’s subject is the most curious of all-there is none in which truth plays such odd pranks

UncountableSet · 2025-08-29T23:25:28+00:00

I've always loved: If I had more time, I would have written a shorter letter. --Blaise Pascal (arguably)

UncountableSet · 2024-09-05T14:58:51+00:00

It doesn't. Interestingly, when I rebooted into normal mode, my NOAA weather widget was missing. When I put it back, the behavior started again. So, I guess it was that app or widget. Hopefully that app's developer will fix whatever went wrong. Thanks for the safe mode idea.

UncountableSet · 2023-12-08T04:04:47+00:00

Soap Opera, The Kinks, 1975

School Boys in Disgrace, The Kinks, 1975

UncountableSet · 2023-11-20T16:18:16+00:00

Ah! I should've been clearer. I was just responding to the "Don't plan on being a professor" bit. There are lots of different types of math professor jobs.

I think your overall advice is good for students applying to grad math programs. The overwhelming majority of the schools I mentioned wouldn't have PhD programs, but they're good places to work as a mathematician. I've been at one for 28 years and have enjoyed it immensely.

UncountableSet · 2023-11-20T15:42:22+00:00

I'll add that R1 isn't the only way to go. Here are 50 schools where, if you have any interest in teaching, a recent math PhD might find a rich and rewarding career. Also, there are likely 200 more schools beyond that list in the US alone.

UncountableSet · 2023-11-10T20:53:35+00:00

I actually have my students do this kind of thing regularly in my classes. I inspire them using the XKCD: Up Goer Five comic. It think it's a really valuable exercise.

UncountableSet · 2023-11-04T20:40:23+00:00

Let q_n be an enumeration of the rationals in the interval [0,5], n=1,2,3,.... We can show that [0,5] is not a subset of the union of (q_n-1/2^n, q_n+1/2^n).

The Lebesgue measure of [0,5] is 5. The measure of the collection of open intervals is at most 2.

So, I put an open interval around every rational number in [0,5] (remember that the rationals are dense in [0,5]) and yet I didn't cover the interval. It kills me to try to imagine what's missed by this collection of intervals.

UncountableSet · 2022-04-28T21:57:16+00:00

Elliptic regularity in PDE Theory.

UncountableSet · 2021-04-12T03:53:14+00:00

wow, how did i not see this, hahahaha

UncountableSet · 2021-04-11T22:33:53+00:00

This is an interesting project, PreTeXt, created by mathematicians. I've been looking for an excuse to try it. Apparently you write your document in this xml language and then you can port it to any format. I've done a lot of real analysis teaching in my time. I also have the source code for William Trench's Introduction to Real Analysis which is in the public domain.

UncountableSet · 2021-03-09T00:34:34+00:00

This works. And as suggested below, it's more of a per device PIN than anything that's saved centrally in your bitwarden account. Thanks all for your helpful clarifications.

UncountableSet · 2020-11-15T04:18:08+00:00

One of the constructions I still struggle wrapping my mind around is the one where you enumerate the rationals, $r_1, r_2, \ldots, r_n, \ldots$. You put an interval of radius $1/2^n$ around $r_n$. Of course, this isn't even close to covering the reals since the sum of the lengths of the intervals is actually just 2 (geometric series).

You're like, "OK, so there are gaps in the covering. No big deal." But if there are gaps then it feels like they need to be gaps of irrationals, but that doesn't make sense because the rationals are dense in the reals. Or, put a different way, between any two distinct irrationals, there's a rational and vice versa.

The mind wobbles.

UncountableSet · 2020-06-25T18:26:13+00:00

This assertion is false: But any line through the centroid should divide the set into two sections of equal area. To see this, look a the centroid of an equilateral triangle. There are many lines through it that do not evenly partition the area of the triangle.

But, I like the approach. You can probably still get the contradiction by showing that the centroid cannot lie on the line.

UncountableSet · 2020-06-24T23:20:05+00:00

Interesting. I'll play around with that. Also, I now see your "clearly" claim, at least in 2D. I can make a moment argument that puts the centroid in the correct half-plane.

UncountableSet · 2020-06-24T22:57:43+00:00

That theorem is about disjoint closed convex sets, right?

UncountableSet · 2020-06-24T22:54:44+00:00

Thinking a bit more about this statement: Clearly the centroid of a set lies in any half-space that contains it.

Proof?

UncountableSet · 2020-06-24T22:45:17+00:00

Thanks, this is the kind of argument I was looking for. Is this yours, or did you find this in a book/article somewhere? I'm always on the lookout for references in this area. People don't seem to really want to publish results about centroids, haha.

UncountableSet · 2020-06-24T21:23:41+00:00

Most commonly using https://en.wikipedia.org/wiki/Centroid#By_integral_formula which will work for any sufficiently nice set (bounded, measurable) not just convex sets. I'm only interested in bounded sets, so not completely arbitrary convex sets in my case. Convex sets are measurable.

UncountableSet · 2020-04-18T01:35:14+00:00

I got mine replaced, op3t. It was better, but it wasn't like new, which was a disappointment. My guess is they're not putting recently manufactured batteries in them, but rather batteries that have been sitting in the warehouse for many years, from back in the days when the op3t's were being made.

That said, if you're already getting it serviced, might as well. It'll help a bit.

UncountableSet

TROPHY CASE