Search
Thomas Lawler
196 followers - Mathematician, Programmer, Player of Video Games
Mathematician, Programmer, Player of Video Games
196 followers
Thomas's posts
Post has shared content
When I first saw this artwork, I thought, "Huh, this reminds me of Richard Schwartz's art." Turns out that's because it is!
A fun, relatively accessible paper about a surprisingly long-standing mathematical question.
A fun, relatively accessible paper about a surprisingly long-standing mathematical question.
I think one of my earliest posts on Twitter linked to this Division by Three paper by Doyle and Conway: http://arxiv.org/abs/math/0605779
This is now a followup, Pan Galactic Division: http://arxiv.org/abs/1504.02179
It shows how for any finite set N, and any injection from the set A×N to B×N, we can construct an injection from A to B without making use of the axiom of choice, in effect allowing us to cancel by N on both sides of the map. To make things easier it views the set B as the set of numbers or pictures on cards, and N as the set of suits, and constructs the injection by induction on N using a card game. Hence the picture.
This problem has a long history going back to a "lost" proof from 1926.
This is now a followup, Pan Galactic Division: http://arxiv.org/abs/1504.02179
It shows how for any finite set N, and any injection from the set A×N to B×N, we can construct an injection from A to B without making use of the axiom of choice, in effect allowing us to cancel by N on both sides of the map. To make things easier it views the set B as the set of numbers or pictures on cards, and N as the set of suits, and constructs the injection by induction on N using a card game. Hence the picture.
This problem has a long history going back to a "lost" proof from 1926.
Post has attachment
Earlier this week I gave a talk at the New York Haskell Users Group (http://www.meetup.com/NY-Haskell/) on Category Theory. The video will become available later after it's been processed, but in the meantime, I've uploaded my slides at http://goo.gl/iSJi4j -- so if you're interested in that sort of thing, feel free check it out.
Post has attachment
So this is pretty neat: Some people decided to study the structure of the primes by treating the natural numbers as a stochastic process.
Here's the idea: You can think of the natural numbers as being a giant network, with composite numbers connected to their prime factors. You can use the ordering of the naturals to think of this as a graph that's growing over time -- each number gets added in, one by one, connecting to all its prime factors if it has any, and getting labeled as "prime" if it doesn't.
What this paper does, is it gives a very simple and intuitive randomized algorithm for generating structures that look kind of like this graph. It turns out that the structures that are generated share a lot of the large-scale structure of the primes! For example, they obey the Prime Number Theorem (http://en.wikipedia.org/wiki/Prime_number_theorem), as well as a few other theorems about the distribution of primes.
From the article it seems like this is a pretty major step forward as far as probabilistic models of the primes go; it mentions a couple of issues (notably in modeling small-scale structure; the model assigns nonzero probability to consecutive primes, oops) but overall it sounds pretty awesome.
Here's the idea: You can think of the natural numbers as being a giant network, with composite numbers connected to their prime factors. You can use the ordering of the naturals to think of this as a graph that's growing over time -- each number gets added in, one by one, connecting to all its prime factors if it has any, and getting labeled as "prime" if it doesn't.
What this paper does, is it gives a very simple and intuitive randomized algorithm for generating structures that look kind of like this graph. It turns out that the structures that are generated share a lot of the large-scale structure of the primes! For example, they obey the Prime Number Theorem (http://en.wikipedia.org/wiki/Prime_number_theorem), as well as a few other theorems about the distribution of primes.
From the article it seems like this is a pretty major step forward as far as probabilistic models of the primes go; it mentions a couple of issues (notably in modeling small-scale structure; the model assigns nonzero probability to consecutive primes, oops) but overall it sounds pretty awesome.
Post has attachment
http://whathappenedatrisd.tumblr.com/post/64797106944/that-thing-that-happened-at-risd-in-great-detail
I cannot express the degree to which I am trembling in rage at this. This sort of behavior is absolutely, entirely unacceptable. RISD should be ashamed, and the people responsible for this should be, at the very least, fired immediately.
I cannot express the degree to which I am trembling in rage at this. This sort of behavior is absolutely, entirely unacceptable. RISD should be ashamed, and the people responsible for this should be, at the very least, fired immediately.
Because I've already had two people inform me of this, I thought I'd let everyone here know before it becomes a thing: I am not on facebook. I haven't been for a long time; I deactivated my account in 2008. Apparently that wasn't enough to delete it forever, though, and now my account there has been hacked and is being used as a spambot.
If you see any posts from me on facebook, please disregard them. Better yet, unfriend my account, and then at least you won't get spam that's claiming it's from me.
If you see any posts from me on facebook, please disregard them. Better yet, unfriend my account, and then at least you won't get spam that's claiming it's from me.
Post has attachment
Today I learned that 2015 is the smallest Lucas-Carmichael number that is also a palindrome in binary. It's also one of the few palindromes in binary that has only one 0 in it: 11111011111.
A Lucas-Carmichael number is a number n that is:
1. Composite
2. Square-free (not divisible by any perfect square, e.g. 4, 9, 16, ...)
3. For every prime number p that n is divisible by, (n + 1) is divisible by (p + 1).
There aren't very many of these numbers; 2015 is the third-smallest, after 399 and 935.
Happy 11111011111 everyone :)
A Lucas-Carmichael number is a number n that is:
1. Composite
2. Square-free (not divisible by any perfect square, e.g. 4, 9, 16, ...)
3. For every prime number p that n is divisible by, (n + 1) is divisible by (p + 1).
There aren't very many of these numbers; 2015 is the third-smallest, after 399 and 935.
Happy 11111011111 everyone :)
Post has shared content
Math fun with plumbing supplies.
Post has attachment
While browsing through the references section in the HoTT book, I discovered a reference to this lovely gem. #homotopytypetheory
Post has attachment
Putting this here to increase the signal.
If you loved Myst and Riven, or love adventure games in general, or like surreal landscapes, you should consider backing Obduction on Kickstarter. It's already at over 50%, and has just over two and a half weeks left.
If you loved Myst and Riven, or love adventure games in general, or like surreal landscapes, you should consider backing Obduction on Kickstarter. It's already at over 50%, and has just over two and a half weeks left.
Wait while more posts are being loaded