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“Futurama theorem” slightly improved

The “Futurama theorem”, also known as Keeler’s Theorem after its creator, was a bit of maths invented for the Futurama episode The Prisoner of Benda, to solve a problem where the characters get their heads mixed up and need to swap them back without any one pair swapping heads twice. It was enthusiastically reported by the geeky press, and rightly so, because it’s a fun bit of real maths with a wonderful application. Dana Ernst has written some very good slides about the theorem, working from “what is a permutation?” up to the algorithm itself.

Anyway, some students from the University of California, San Diego have extended the result, giving a better algorithm for finding the minimum number of switches to put everyone’s head back in the right places, give optimal solutions for two particular situations, and give necessary and sufficient conditions for it being possible to represent the identity permutation as $m$ distinct transpositions in $S_n$.

Paper: http://arxiv.org/abs/1204.6086

via James Grime

Saharon Shelah has written more than 1000 papers

From David Roberts on Google+:

Saharon Shelah, the well-known Israeli set-theorist and logician, has passed 1000 papers!

http://shelah.logic.at/listb.html

The page was updated with a rush of almost twenty papers, taking him over the line. Notably, paper #1000 is not listed. +Richard Elwes and I were wondering what the topic of this (rather artificial) milestone paper would be.

Every now and then, when finding a citation for a paper, you come across one of these giants of prolificacy and their unreasonably long lists of publications. It makes me wonder why I don’t just give up and let them discover all the maths.

Shelah was the first recipient of the Erdős Prize and he is certainly following in the great man’s footsteps – though he’s still got a way to go before he can think about beating Erdős’s approximately (can’t blame him for losing count) 1525 publications.

Classic maths books reset with LaTeX on Project Gutenberg

I was going to save this for an Aperiodical Round Up but it’s such a good thing I thought I’d post it straight away. Project Gutenberg has moved on from offering just plain-text transcriptions of books: volunteers have been outstandingly generous with their time and produced LaTeX versions of many maths books, producing versions that are considerably more readable and resemble the original editions much more closely.

Not all the books in that list have been converted to LaTeX yet. Of those that have, GH Hardy’s A Course of Pure Mathematics leaps out as a good place to start. Compare it with this book still in HTML format to see the difference.

(via reddit)

Japanese researchers create a crab-based computer

This is the platonic ideal of an entry in my Interesting Esoterica collection: two scientists from Kobe University and one from UWE’s excellently-named International Center of Unconventional Computing have written a paper, Robust Soldier Crab Ball Gate, claiming that swarms of soldier crabs Mictyris guinotae can be persuaded to act as logic gates, from which a universal computer could be built. The paper first describes how they modelled swarms of crabs, then how the logic gates are implemented, and ends with data from an experiment with real soldier crabs. The AND gate worked about two thirds of the time, which isn’t bad.

It looks like this paper is a follow-up to the earlier work, Slime mould logical gates: exploring ballistic approach, which did basically the same thing on a smaller scale. I can only think that the next step must be to use humans.

I’ve given a talk about other unlikely computing machines: I can’t believe it’s a universal computer!

(via Slashdot)

Laziest torus identified

Or, in similarly simplified headlinese, “Math finds the best doughnut”. A little bit more precisely, Fernando C. Marques and André Neves claim in a preprint on the arXiv to have proved the Willmore conjecture, that the minimum achievable mean curvature of a torus is $\frac{2}{\pi^2}$.

The article I linked to is some surprisingly non-stupid coverage from the Huffington Post. It seems they have a maths professor writing a column. I will never understand that site. I don’t know if there’s a Serious Business way of framing this, but the result is nice to know.

Richard Elwes has written a very short post on Google+ with some more real-maths information about what’s going on.

Interesting Esoterica Summation, volume 3

Summing up some more interesting esoterica seems like the right thing to do at the moment, so here’s that.

A reminder: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley.

In this post the titles are links to the original sources, and I try to add some interpretation or explanation of why I think each thing is interesting below the abstract.

Continue reading “Interesting Esoterica Summation, volume 3” on cp’s mathem-o-blog