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Anyone hear of this before, or can find something about it? It seemed kinda cool at the time, and I like this axiomatization.

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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...>

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@jordyd
"...now lastly, set $$x=10$$, ..."

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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.)

Animation of the minimum-weight matchings of increasingly many points of two colors in a unit square: twitter.com/thienan496/status/

Because the color densities fluctuate, the matching develops regions of many parallel long edges transporting excess density from one place to another. This is reflected mathematically in the fact that the expected length is $$\Theta(\sqrt{n\log n})$$ compared to $$\Theta(\sqrt{n})$$ for non-bipartite matching; see doi.org/10.1007/BF02579135

*executes arbitrary code*

How much value is there in using the $$\partial$$ symbol in opposition to $$d$$ to distinguish between "partial derivative" and, I don't know, "not-partial derivative?" I've never seen much point in it; in some cases it even seems ambiguous which one to choose.

"We will not concern ourselves with subtle foundational issues (set-theoretic issues, universes, etc.). It is true that some people should be careful about these issues. But is that really how you want to live your life?"

I FEEL YA BRO

New blog post: Relevant neighbors, 11011110.github.io/blog/2021/1

It's triggered by my new arXiv preprint (and SOSA 2022 paper) "Finding Relevant Points for Nearest-Neighbor Classification", arxiv.org/abs/2110.06163, but really the content of the post is more closely related to earlier work by @bremner @patmorin and their coauthors.

Positive thinking

I just saw a paper on arXiv by Stanley P. Y. Fung: "Is this the simplest (and most surprising) sorting algorithm ever?"

Not sure if I would answer yes, but I certainly had a bit of fun browsing through the paper.

arxiv.org/abs/2110.01111

University System of Georgia moves to gut tenure: insidehighered.com/news/2021/1

The proposed new policy includes procedures for removal of tenure under certain enumerated grounds, including failure to perform their jobs (this is pretty normal) but then adds a massive escape clause in which the board of regents can remove tenured faculty at any time as long as their reason for doing so is not one of the enumerated ones.

I recently saw a link to Chapter 1 of the MathML 3.0 spec, w3.org/TR/MathML3/chapter1.htm, using as an example the quadratic formula in both layout markup and content markup. Its totally unwieldy non-human-readable expansion obscures the fact that the MathML authors didn't even get the math right: their content markup silently replaces "±", by which the correct formula represents both solutions, with "+", giving only one of the two.

Time to re-link my old anti-MathML rant 11011110.github.io/blog/2015/0 ?

food, extremely nerdy joke

"The amount of mayonnaise people put on chopped cabbage doubles every two years."

— More slaw

Vi Hart's latest video mixes up discussions of the nature of social media, the philosophy of mathematical creativity, an exploration of symmetry, and an investigation of the spot patterns of 8-sided dice (which turn out not to all be the same) and how to visualize them.

Apparently there is one weird trick to calculating the result of rigid body collisions like this: you simply have to consider the nature of contact forces. They are inflicted at the point of contact, along the contact normal, for a brief instant. This determines the change in momentum and angular momentum up to one scalar multiple, which itself may be determined by conservation of energy (assuming the collision is elastic). This simulation here has big flaws though, because I'm a bad programmer.

Hi, my name is Ahmed, I'm pretty new to the server and I love maths. I'm going to be starting a PhD in electrical and computer engineering pretty soon (hopefully), with a focus on optimization theory. I'm currently working my way through some notes on measure-theoretic probability (Amir Dembo's lecture notes)-- if anyone likes measure theoretic probability, please talk to me!

Teaching multivariable calculus again for the first time in quite a while. Since this means teaching about the vector cross product, it brings to mind one the strangest "connection between wildly disparate-seeming things" theorems I know of:

prideout.net/blog/kauffman/

15 years later, and this still accurately portrays my work days

90s crossover shootout

A deep math dive into why some infinities are bigger than others (scientificamerican.com/article, via 3quarksdaily.com/3quarksdaily/): Martin Goldstern and Jakob Kellner carefully explore the relations between certain infinite cardinalities, all more than countable but at most the cardinality of the continuum, defined using measure-zero and nowhere-dense subsets of the plane.

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers
Article by Amnon Yekutieli
In collections: Fun maths facts, Integerology, The groups group
It is well-known that pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields, almost immediately, an...
URL: arxiv.org/abs/2101.12166v1
PDF: arxiv.org/pdf/2101.12166v1