Celestial Beauty

Apparently it was the longest solar eclipse in a century (people say it’s the longest solar eclipse of the century – but that isn’t saying much). Planetfunn posts some beautiful, beautiful pictures of this astronomical phenomenon:

http://planetfunn.blogspot.com/2009/07/longest-solar-eclipse-of-century-33.html

Degenerate Multivariate Gaussians

Warning: Facebook users, click “View original post” in order to actually see the notation. Also, a pretty technical post.

I need some help. So these days I’m writing some code to calculate a multivariate Gaussian probability density function. It’s a generalization of the standard Normal distribution density function, where instead of a single mean and standard deviation , is actually a -dimensional vector and you have a covariance matrix . You compute the probability of observing vector the following way:

where the is the determinant of . What do you do when the determinant is zero? You can’t calculate either the prefactor out in front or the inverse matrix in the exponent. This is not an unusual scenario, I don’t think, because sometimes in some of the dimensions of the probability space are 100% correlated with each other – but their mean value is zero (and the standard deviation is 0). In some cases, this will result in a singular , and just because of a few bad eggs, your formula breaks down.

In my implementation thus far, I’ve worked around this by ignoring the 0 rows in – but is this a valid thing to do? Are there any better solutions? Please let me know!

Sleep Solving

In Len Adleman’s Logic class I took over a year ago, he remarked that one of his favorite ways of conducting research was through sleep. Go to bed thinking really hard about a problem, he says, and in the morning you might find that you have a solution. This, of course, would be amazing if it worked! Not only are you channeling the wealth of subconscious computing power available to you (but rarely ever used), you’re doing it with no (perceivable) effort at all.

I don’t doubt that Len actually utilizes this technique! There’s simply too many times when he has come into the meeting and announced that he has a brand new – often ingenious – perspective on our latest troublesome problem – and this is only after spending the evening thinking about it. Other than his sheer intelligence, it must be some other trick – and I think that’s the sleep solving he mentioned.

All hail the new blogheads!

Let’s welcome my friend and colleague Joseph Bebel to the blogosphere! After years of coercing, he finally gave in with Presumably Correct. His last post on “Anti-Inductive Logic” is quite funny; perhaps there are other modes of logic that work equally well as induction (that is, if something has happened for the past thousand days, it’s probably going to happen tomorrow). Then again, as we all know, Nassim Taleb would raise hell with that kind of logic.

And finally, another welcome to Casey Stark (my former roommate) with his blog, The Stark Effect.

I look forward to seeing what Joe and Casey have to say, and I hope you guys do too.

Order out of chaos

Complete disorder is impossible. – T.S. Motzkin*

I first encountered Ramsey Theory in Sol Golomb’s combinatorics class, and I thought there was something pretty damn cool about it. The result we covered in class was: given an arbitrary graph of six nodes, you will be sure to find a clique of three people or an anti-clique of three people. Another way to say this is: imagine you’re in a group of six people. At least three people will mutually know each other, or at least three people will not know each other at all.

It might seem a bit obvious when put that way, but Ramsey theory generalizes this notion to any amount of people (or any size graph). If you’re in a room full of a thousand people, at least how many people are mutually acquainted (or mutually unacquainted?). It’s not so obvious. The magic of this lies in the complete arbitrariness of the situation: you can have any graph you like. For a mere 40 nodes, there are about 7 x 10²⁴⁶ different ways you can connect the nodes together to form a graph. This number is absolutely colossal: compare it to the number of elementary particles in the universe, which is approximately 10⁸⁵. The space of graphs on n nodes must be as chaotic as it gets, because you’re navigating all possible relationships between n things — and yet, Ramsey Theory claims that even in this confused and formless universe of graphs, there is a semblence of order. Order out of chaos.

The Presaging Genome

The sequencing of the human genome is surely one of the greatest achievements of mankind. This was first accomplished at the turn of the millenium, and we were told that we were on the cusp of a major revolution in medicine – perhaps the major revolution. With the capability to analyze the source code of you, you will be able to determine everything about your health: what diseases might afflict you, the probability of cancer, Alzheimers, possibly how long you will live (through natural means). Imagine, then, the explosive rise of the preventive medicine industry. You might get Parkinson’s 40 years from now? Not anymore – just start taking these personalized pills, designed just for your body and genes. It seems like we’re on the brink of a golden age of health and medicine.

Not so fast.

A Midsummer Midday’s Muse

I’m not travelling around beautiful Europe this summer, but I’m part of this amazing adventure, my work. I may be trapped in the grime and grit of central LA, butI’m off wandering the mystical, pristine sierras of mathematics; I’m navigating uncharted oceans of science, sighting new lands of thought. I feel obliged to quote Mr. Mark Twain: “To do something, say something, see something, before any body else — these are the things that confer a pleasure compared with which other pleasures are tame and commonplace, other ecstasies cheap and trivial.” What a joy it is to wake every morning! Who knows what one may find today?

Primes on Mars

Let’s say you were part of the first manned mission to Mars. You eagerly land your spacecraft, step out and survey the rust-red landscape sprawled before you. After a few hours of hiking around you encounter a very peculiar sight: two rows of rocks, each a kilometer long, perfectly straight. However, it is not just uniform lines of rocks, there are intervals of spacings interspersed about – but the rocks are collinear nonetheless. Furthermore, after some moments of contemplation, you realize that if interpreted as a binary code, the first two rows represents two primes p and q. That’s not all though – the product pq turns out to be a Mersenne number of a huge number of digits! What a find!

All intuition seems to scream: “This is no accident!” But why is it more plausible that an advanced alien civilization had purposefully organized the rocks in that fashion, than the idea that natural processes had somehow tossed the pebbles into straight, parallel lines?

Towards Hilbert’s 10th problem

Gregory Chaitin

I’m currently reading through Gregory Chaitin’s wonderful book Meta Math! The Quest for Omega (it totally sounds like a sci-fi novel). Perhaps I will post a review of it someday – but that requires finishing the book first, and I am barely through the second chapter.

The second chapter talks about the relation between Kurt Gödel’s incompleteness, Hilbert’s 10 problem, and Alan Turing’s halting problem. Very briefly, Gödel incompleteness is a statement about the, well, incompleteness of “sufficiently powerful” axiomatic systems in mathematics. In a similar vein, Turing’s halting problem is whether one can algorithmically decide if a computer program will halt or not – it turns out one cannot. Matiyasevich and others showed that the undecidability of the halting problem implies the answer to Hilbert’s 10 problem, whether one can algorithmically decide if a system of Diophantine equations are solvable or not – it turns out one cannot.

Spring Time for Hit- I mean, Henry.

High: 81° F, slight breeze, nary a wisp in the sky; it was simply too nice to stay inside today. Totin’ me notebook and violin, I sought the beautiful serenity that spring in Southern California would bring me. It would be just me and my equations, Brazilian jazz too. It helps that it’s a Sunday, so it feels like I have the entire campus to myself. It’s quiet.