Friday Random Ten, 4/13

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I’ve been swamped lately, learning to manage my new commute, and being overwhelmed by my new job. So I’ve been a bit lax about the blog; I’ve missed three weeks in a row for the friday pathological programming; and I haven’t been posting my friday random tens. I don’t have time to do a FPP post today, but I can at least inflict my strange tastes in music on you. Friday pathological programming will return next week.

  1. Navan, “Ma Labousig Ar C’hoad”: Navan is a wonderful traditional Irish
    a capella group. I caught them being interviewed on NPR the week before St. Patrick’s day, and immediately went home and tracked them down on iTunes. Beautiful stuff.
  2. Explosions in the Sky, “Catastrophe and the Cure”: post-rock, in the “Mogwai” vein. Not as good as Mogwai, but they’re definitely worth listening to.
  3. The Kells, “The Gander in the Prairie Hole”: Another traditional Irish band. This one I also discovered by accident. I recently bought myself a new tinwhistle – a beautiful low-D whistle. Everyone in the whistle community has been talking about the whistles being made by a guy named Michael Burke, so I was checking his whistles out. On his site, to show you what his whistles sound like, he has samples of recordings featuring people playing his whistles. For the Low-D whistle that I wanted, the sample was a clip from the Kells. I bought the whistle, and a Kells CD. They’re a really fantastic band, definitely highly recommended to anyone who likes Irish music. (And the whistle is an absolute delight – a good strong low-D, stable tone, just the right amount of backpressure, a really well-made tuning slide, and a brilliantly clever trick to make the lowest hole rotate, so that it’s easier to reach.)
  4. Blackfield, “1,000 People”. A track from a band that’s a spin-off of one of my favorite neo-progressive bands, Porcupine Tree. Blackfield is, in general, a bit mellower than recent PT, but the overall sound is quite similar. Great track, from a great album.
  5. Apothecary Hymn, “A Sailor Song”: Another NPR discovery. I was driving to New Jersey to visit my father in the hospital, and the New York NPR station had Apothecary Hymn in their studio performing live. They sounded like a fantastic cross between old Jethro Tull, Gentle Giant, and King Crimson. So once again, I ordered their CD as soon as I got home. Overall, I think the King Crimson resemblance doesn’t come through so much on the CD as it did in the live performance, but the Gentle Giant/Jethro Tull comparison is pretty much dead-on. Very cool stuff.
  6. Marillion, “The Other Half”: The opening track from Marillion’s new album. Rather a disappointment, I’m afraid. I’ve been a huge Marillion fan for longer than I care to admit, and I was really looking forward to this new album, because the last one was one of their best in a long time. It’s not bad, but it is rather lackluster overall.
  7. Edgar Meyer, 3rd movement (Allegro) from Bottesini’s Concerto #2 for Double Bass: Edgar Meyer is one of those dazzling musicians who can do anything,
    and make it sound good. He’s primarily a classical musician, but he’s also known
    for performing bluegrass, jazz, fold, rock… and he’s brilliant at all of them. Hearing him play a piece like the Bottesini is amazing… Seeing him do it live is even more amazing. It’s hard to believe that that kind of speed and grace, could possibly be coming from this awkward, hulking instrument. That big heavy bow is just dancing all over the place like it’s a feature, his left hand is zipping up and down the next of the base, never missing a note.
  8. Mogwai, “Stop Coming to My House”: A great track by one of my favorite post-rock bands. Mogwai is more in the rock-like side of the post-rock continuum. They’re always amazing.
  9. A Silver Mt. Zion, “Ring Them Bells (Freedom has Come and Gone)”: more post-rock, this time from a spinoff of “Godspeed You Black Emperor”. Actually, on this
    album, they don’t really go by “A Silver Mt. Zion”; they go by “Thee Silver Mt. Zion Memorial Orchestra and Tra-La-La Band”. Brilliant stuff, as you’d expect from a Godspeed spinoff. Unlike Godspeed, Mt. Zion tends to actually include lyrics, which can be interesting, but the singers voice isn’t particularly great. But still, overall, brilliant.
  10. Tony Trischka, “Run Mountain”: Classic style bluegrass, played with brilliant style by my former teacher, Tony Trischka. Tony’s amazing; he’s another one of those
    musicians who can play anything he sets his mind to. He’s also just an all-around really
    nice guy. I love to play banjo, but I’m lousy at it. I’m not anywhere close to being what I would consider good enough to take lessons from someone like him. But I used to live quite near him in New Jersey, and someone convinced me to call him about taking Banjo lessons. Much to my surprise, he was willing to take me as a student: Tony’s willing to give lessons to anyone who’s interested. I learned a lot from him. I’m still a lousy banjo player, but not nearly as lousy as before – and he taught me a lot about learning music by ear.

Normal Forms and Infinite Surreals

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When I left off yesterday, we’d reached the point of being able to write normal forms
of surreal numbers there the normal form consisted of a finite number of terms. But
typically of surreal numbers. that’s not good enough: the surreals constantly produce
infinites of all sorts, and normal forms are no different: there are plenty of surreal
numbers where we don’t see a clean termination with a zero term.

For me, this is where the surreal numbers really earn there name. There is something distinctly surreal about a number system that not has a concrete concept of infinity, but allows you to have an infinite hierarchy of infinities, resulting in numbers that have, as their simplest representation, and infinite number of terms, each of which could involve numbers which can’t be written in a finite number of symbols. It’s just totally off the wall, insane, crazy, nuts… But fun!

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The 2007 Abel Prize: Professor S. Varadhan and the Theory of Large deviations

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As an alert reader pointed out, a major mathematical prize was awarded recently. Since
2002, the government of Norway has been awarding a prize modeled on the Nobel, but in
mathematics. The prize was originally suggested by Sophus Lie, he of the Lie group, back in
1897, when he heard that Nobel was setting up his awards, and was not including
mathematics. The prize is named after Niels Abel, the Norwegian mathematician who
discovered the class of functions that are now known as Abelian functions; the same person
that Abelian groups are named after, etc.

Anyway, this year, the Abel
prize was awarded to Srinivasa Varadhan, an Indian mathematician who is currently a professor at the NYU Courant Institute.
Professor Varadhan’s specialty is probability theory – in particular, the theory of large
deviations. In honor of Professor Varadhan’s award, I thought it would be interesting to
very briefly explain what the theory of large deviations is, and why it’s so
important that it justified the award of a million dollar prize.

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Surreal Numbers and Normal Forms

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On the way to figuring out how to do sign-expanded forms of infinite and infinitesimal numbers, we need to look at yet another way of writing surreals that have infinite or infinitesimal parts. This new notation is called the normal form of a surreal
number, and what it does is create a canonical notation that separates the parts of a number that fit into different commensurate classes.

What we’re trying to capture here is the idea that a number can have multiple parts that are separated by exponents of ω. For example, think of a number like (3ω+π): it’s not equal to 3ω; but there’s no real multiplier that you can apply to 3ω that captures the difference between the two.

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Degrees and Exponents of Infinities in the Surreal Numbers

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When I first read about the sign-expanded form of the surreal numbers, my first thought was “cool, but what about infinity?” After all, one of the amazing things about the surreal numbers is the way that they make infinite and infinitessimal numbers a natural part of the number system in such an amazing way.

Fortunately, it turns out to be very easy to play with infinities in sign-expanded form: you just need to use exponents of ω. Fortunately, exponents of ω are really cool! Getting to the point where we’ve really captured the meaning of exponents of infinity, so that we can talk about general infinities in terms of sign expansion for is going to take a bit of work. So as a bit of motivation, and to give you a first taste, since 1/2 has a sign-expanded form of “+-“, (that is, integer part=0, binary fractional part or 0.1=1/2), ω/2 = +ω–ω.

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Mathematical Study of Drug Interactions in the Evolution of Antibiotic Resistance

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Orac has posted a really good description of a recent paper discussing how
interaction between different antibiotics effects the evolution of antibiotic resistance in
bacteria populations.

It’s a mathematical analysis of experimental results generated by combining drugs which normally interact poorly with one another, and analyzing the distribution of resistance in the resulting populations. It turns out that under the right conditions, you can create a situation in which the selective pressure of the combination of drugs – which are less effective when combined – can select in favor of the non-resistant variant of the bacteria!

Check out Orac’s post for details; I may also try to get a copy of the paper and post a more detailed look at the math later this week.

Sign-Expanded Surreal Numbers

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In addition to the classic {L|R} version of the surreal numbers, you can also describe surreals using something called a sign expansion, where they’re written as a sequence of “+”s and “-“s – a sort of binary representation of surreal numbers. It’s fully equivalent to the {L|R} construction, but built in a different way. This is a really cool, if somewhat difficult to grasp, construction.

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Update on Moderation and Banning

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This post is a quick moderation update, caused by recent crap going on in the comment threads involving George “First Scientific Proof of God” Shollenberger, combined with my recent change of employment.

Before changing jobs, my old employer was kind enough to allow me to write this blog, but they did not want me to ever do anything on the blog that allowed them to be identified as my employer. The idea was that since the blog was something I did on my own time, without any oversight from them, they wanted it to be clear that the blog had no connection with them. My new employer, Google, has a different attitude towards blogs. They still want me to be explicit that what I do and say here does not represent them in any way; but they don’t make me play the silly game of trying to obscure who I work for. I’m allowed to say that I work for Google – so long as I’m perfectly clear that I’m an employee, but that GM/BM has absolutely nothing to do with them. I don’t speak for them; I don’t represent them; I don’t write for the blog on company time; this blog is entirely my space, and has nothing to do with them.

Why bring all this up? Because I’ve banned George. I hate to ban people, even when they’re complete assholes. Up to now, the only person I’ve ever banned was John Davison, and he practically begged me to ban him. But George has started using our “debate” here in the comments of GM/BM to attack Google. And I’m not going to tolerate that.

Attacking me personally is entirely acceptable; it may make me angry, or frustrated, but given the kinds of things I write on this blog, I think it would be completely inappropriate, even hypocritical, for me to say that any kind of criticism or attack aimed at me personally was out of bounds.

But this blog is something I do in my free time – my own time. It has nothing to do with my employer; what I say here is said solely by me representing myself and my personal thoughts and opinions. Taking what I say here and using it to attack my employer is completely beyond the bounds of what I consider reasonable, or what I’m willing to tolerate.

George crossed that line. So he’s gone.

Basics: Binary Search

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For the basics, I wrote a bunch of stuff about sorting. It seems worth taking a moment
to talk about something related: binary search. Binary search is one of the most important
and fundamental algorithms, and it shows up in sorts of places.

It also has the amazing property that despite being simple and ubiquitous, it’s virtually
always written wrong. There’s a bit of subtlety in implementing it correctly, and virtually
everyone manages to put off-by-one indexing errors into their implementations. (Including me; last time I implemented a binary search, the first version included one of the classic errors.) The errors are so ubiquitous that even in a textbook that discusses the fact that most programmers get it wrong, they got it wrong in their example code!

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The Surreal Reals

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The Surreal Reals

I was reading Conway’s Book, book on the train this morning, and found something I’d heard people talk about, but that I’d never had time to read or consider in detail. You can use a constrained subset of the surreal numbers to define the real numbers. And the resulting formulation of the reals is arguably superior to the more traditional formulations of the reals via Dedekind cuts or Cauchy sequences.

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