Pinned toot

Pinned toot

Background: A finite projective plane has point-line duality, and affine planes lack it. You can get a projective plane by adding a "line at infinity," and this is reversible.

Years ago I stumbled across this in constructing a counterexample to a graph conjecture, and I never found out if it was something people had realized or used before:

We can also recover point-line duality from an affine plane by just deleting one of the parallel classes of lines. <cont...>

Pinned toot

@jordyd

"...now lastly, set \(x=10\), ..."

Pinned toot

#proofinatoot that if a finite poset has a unique maximal \(x\), then \(x\) is maximum.

If not, there is a \(y_1||x\). \(y_1\) is not maximal, so there is \(y_2>y_1\); we cannot have \(y_2<x\), else transitivity would give \(y_1<x\), and we cannot have \(y_2>x\) because \(x\) is maximal, so \(y_2||x\). Continuing, we build a chain \(y_1<y_2<\dotsb\) (with \(y_i||x\) for all \(i\)), contradicting finiteness.

(This proof also suggests a construction of an infinite poset without the property.)

A quasi-polynomial algorithm for well-spaced hyperbolic TSP: https://arxiv.org/abs/2002.05414

This new preprint by Sándor Kisfaludi-Bak (accepted to SoCG) just came out and caught my attention. TSP is NP-hard for Euclidean points or close-together hyperbolic points. This paper shows that it's much easier when the points are widely spaced in the hyperbolic plane. The idea is to separate the input by a short line segment that the solution crosses few times and apply dynamic programming.

love it when i’m cooking, hit a rare ingredient, have to stop & poll my friends for substitutes

"help! wtf is red star anise?"

"my mom has some but she refuses to discuss it or acknowledge its presence in any way"

"it shows up in your spice drawer when you're ready. i guess you're not ready yet"

"you can use black star anise & Red #-2… yes, _minus_ two"

"you can’t buy it, it has to be wrenched from the grasp of a mortal foe"

"a metaphor, like the elusive blue rose"

"i saw it in a dream once"

Did Newton invent convex hulls? I haven't received any useful answers yet from my post to the HSM stackexchange (https://hsm.stackexchange.com/q/11401/11029) but maybe someone beyond that site knows something relevant.

Consider the algorithm "M(x): if x<0 return -x, else return M(x-M(x-1))/2". This algorithm terminates for all real x, though this is not so easy to prove. In fact, Peano Arithmetic cannot prove the statement "M(x) terminates for all natural x". Paper to come! Joint work with @jeffgerickson and @alreadydone

Ringel’s conjecture solved (for sufficiently large \(n\)): https://gilkalai.wordpress.com/2020/01/27/ringel-conjecture-solved-congratulations-to-richard-montgomery-alexey-pokrovskiy-and-benny-sudakov/

This goes back to 1963 and states that the edges of the complete graph \(K_{2n+1}\) can be partitioned into \(2n+1\) copies of your favorite \(n\)-edge tree. The new preprint claiming a proof for large-enough \(n\) is https://arxiv.org/abs/2001.02665

Nonbinary folx don't get why people think Pluto was demoted.

Pluto is happy no one's making it do planet shit anymore. You don't clear your orbit, and we're finally respecting that instead of demanding you act like your siblings.

Also it's the type specimen of its own category now, which I always thought was a really cool thing to be.

In hindsight, this should have been a sign, but 10-year-old me didn't have that context.

#NumberFields #NumberTheory Does anyone have good references other than Milne for CM fields? I'm up to my ears in them and a few basic properties in a citation-friendy format would go a long way.

It's really frustrating when I should be able to re-derive what I need, but get muddled along the way every time. This should be already done stuff.

- website
- https://bmreiniger.github.io

Combinatorist (esp. graph theorist) turned Data Scientist

Joined Jun 2017