Math

I added two technical articles to my personal web site this evening.

Step size for numerical differential equations is a one-page set of notes on how to select the optimal step size when numerically solving ODEs (ordinary differential equations).

Separation of convex sets in linear topological spaces is a highly technical article I wrote a long time ago. I decided to put it on the web in case someone finds it useful. It’s a fairly obscure topic, but this paper covers it thoroughly.

I maintain two pages for articles, one for informal notes and one for academic publications. I went back and added my old PDE papers to my academic publications page this evening.

In 1986, Lawrence Leemis published a paper containing a diagram of 43 probability distribution families. The diagram summaries connections between the distributions with arrows: chi-squared is a special case of gamma, Poisson is a limiting case of binomials, the ratio of two standard normals is a Cauchy, etc. It’s a very handy reference, a sort of periodic table for statisticians. His diagram and variations have appeared in several text books over the last 20 years, such as Casella and Berger.

Now Leemis has published an expanded version containing 76 probability distributions. The paper is in the February 2008 issue of American Statistician and is also available online. The heart of the article is the diagram on page 3.

This morning I posted some notes on orthogonal polynomials and Gaussian quadrature.

“Orthogonal” just means perpendicular. So how can two polynomials be perpendicular to each other? In geometry, two vectors are perpendicular if and only if their dot product of their coordinates is zero. In more general settings, two things are said to be orthogonal if their inner product (generalization of dot product) is zero. So what was a theorem in basic geometry is taken as a definition in other settings. Typically mathematicians say “orthogonal” rather than “perpendicular.” The basic idea of lines meeting at right angles acts as a reliable guide to intuition in more general settings.

Two polynomials are orthogonal if their inner product is zero. You can define an inner product for two functions by integrating their product, sometimes with a weighting function.

Orthogonal polynomials have remarkable properties that are easy to prove. Last week I posted some notes on Chebyshev polynomials. The notes posted today include Chebyshev polynomials as a special case and focus on the application of orthogonal polynomials to quadrature. (“Quadrature” is just an old-fashioned word for integration, usually applied to numerical integration in one dimension.) It turns out that every class of orthogonal polynomials corresponds to an integration rule.

Male honeybees are born from unfertilized eggs. Female honeybees are born from fertilized eggs. Therefore males have only a mother, but females have both a mother and a father.

Take a male honeybee and graph his ancestors. Let B(n) be the number of bees at the n^th level of the family tree. At the first level of the tree is our male honeybee all by himself, so B(1) = 1. At the next level of our tree is his mother, all by herself, so B(2) = 1.

Pick one of the bees at level n of the tree. If this bee is male, he has a mother at level n+1, and a grandmother and grandfather at level n+2. If this bee is female, she has a mother and father at level n+1, and one grandfather and two grandmothers at level n+2. In either case, the number of grandparents is one more than the number of parents. Therefore B(n) + B(n+1) = B(n+2).

To summarize, B(1) = B(2) = 1, and B(n) + B(n+1) = B(n+2). These are the initial conditions and recurrence relation that define the Fibonacci numbers. Therefore the number of bees at level n of the tree equals F(n), the n^th Fibonacci number.

This is a more realistic demonstration of Fibonacci numbers in nature than the oft-repeated rabbit problem.

I posted a four-page set of notes on Chebyshev polynomials on my web site this morning. These polynomials have many elegant properties that are simple to prove. They’re also useful in applications.

Mr. Chebyshev may have the honor of the most variant spellings for a mathematician’s name. I believe “Chebyshev” is now standard, but his name has been transliterated from the Russian as Chebychev, Chebyshov, Tchebycheff, Tschebyscheff, etc. His polynomials are denoted T_n(x) based on his initial in one of the older transliterations.

Mark Dominus brought up an interesting question last month: have there been major screw-ups in mathematics? He defines a “major screw-up” to be a flawed proof of an incorrect statement that was accepted for a significant period of time. He excludes the case of incorrect proofs of statements that were nevertheless true.

It’s remarkable that he can even ask the question. Can you imagine someone asking with a straight face whether there have ever been major screw-ups in, say, software development? And yet it takes some hard thought to come up with examples of really big blunders in math.

No doubt there are plenty of flawed proofs of false statements in areas too obscure for anyone to care about. But in mainstream areas of math, blunders are usually uncovered very quickly. And there are examples of theorems that were essentially correct but neglected some edge case. Proofs of statements that are just plain wrong are hard to think of. But Mark Dominus came up with a few.

Yesterday he gave an example of a statement by Kurt Gödel that was flat-out wrong but accepted for over 30 years. Warning: reader discretion advised. His post is not suitable for those who get queasy at the sight of symbolic logic.

Bart Kosko in his book Noise argues is that thick-tailed probability distributions such as the Cauchy distribution are common in nature. This is the opposite of what I was taught in college. I remember being told that the Cauchy distribution, a distribution with no mean or variance, is a mathematical curiosity more useful for constructing academic counterexamples than for modeling the real world. Kosko disagrees. He writes

… all too many scientists simply do not know that there are infinitely many different types of bell curves. So they do not look for these bell curves and thus they do not statistically test for them. The deeper problem stems from the pedagogical fact that thick-tailed bell curves get little or no attention in the basic probability texts that we still use to train scientists and engineers. Statistics books for medicine and the social sciences tend to be even worse.

We see thin-tailed distributions everywhere because we don’t think to look for anything else. If we see samples drawn from a thick-tailed distribution, we may throw out the “outliers” before we analyze the data, and then a thin-tailed model fits just fine.

How do you decide what’s an outlier? Two options. You could use your intuition and discard samples that “obviously” don’t belong, or you could use a formal test. But your intuition may implicitly be informed by experience with thin-tailed distributions, and your formal test may also circularly depend on the assumption of a thin-tailed model.

Here’s a web site where you can type in some TeX code, click a button, and get back a GIF with a transparent background. Handy for pasting equations into HTML.

http://www.artofproblemsolving.com/LaTeX/AoPS_L_TeXer.php

For example:

Independence in probability can be both intuitive and mysterious. Intuitively, two events are independent if they have nothing to do with each other. Suppose I ask you to guess whether the next person walking down the street is left handed. I stop this person, and before I ask which hand he writes with, I ask him for the last digit of his phone number. He says it’s 3. Does knowing the last digit of his phone number make you change your estimate of the chances this stranger is a south paw? No, phone numbers and handedness are independent. Presumably about 10% of right-handers and the same percentage of left-handers have this distinction. Even if the phone company is more likely or less to assign numbers ending in 3, there’s no reason to believe they take customer handedness into account when handing out numbers. On the other hand, if I tell you the stranger is an artist, that should change your estimate: a disproportionate number of artists are lefties.

Formally, two events A and B are independent if P(A and B) = P(A) P(B). This implies that P(A | B), the probability of A happening given that B happened, is just P(A). Similarly P(B | A) = P(B).  Knowing whether or not one of the events happened tells you nothing about the likelihood of the other. Knowing someone’s phone number doesn’t help you guess which hand they write with, unless you use the phone number to call them and ask about their writing habits.

Now lets extend the definition to more events. A set of events is mutually independent if the probability of any subset of two or more events is the product of the probabilities of each event separately.

So, let’s look at three events: A, B, and C. If we know P(A and B and C) = P(A) P(B) P(C), are the three events mutually independent? Not necessarily. It is possible for the above equation to hold and yet P(A and B) is not equal to P(A) P(B). The definition of mutual independence requires something of every subset of {A, B, C} with two or more elements, not just the subset consisting of all elements. So we have to look at the subsets {A, B}, {B, C}, and {A, C} as well.

What if A and B are independent, B and C are independent, and A and C are independent? In other words, every pair of events is independent. Is that enough for mutual independence? Surprisingly, the answer is no. It is possible to construct a simple example where

• P(A and B) = P(A) P(B)
• P(B and C) = P(B) P(C)
• P(A and C) = P(A) P(C)

and yet P(A and B and C) does not equal P(A) P(B) P(C).

There are no short cuts to the definition of mutual independence.

Edsgar Dijkstra quipped that software testing can only prove the existence of bugs, not the absense of bugs. His research focused on formal techniques for proving the correctness of software, with the implicit assumption that proofs are infallible. But proofs are written by humans, just as software is, and are also subject to error. Donald Knuth had this in mind when he said “Beware of bugs in the above code; I have only proved it correct, not tried it.” The way to make progress is to shift from thinking about the possibility of error to thinking about the probability of error.

Testing software cannot prove the impossibility of bugs, but it can increase your confidence that there are no bugs, or at least lower your estimate of the probability of running into a bug. And while proofs can contain errors, they’re generally less error-prone than source code. (See a recent discussion by Mark Dominus about how reliable proofs have been.) At any rate, people tend to make different kinds of errors when proving theorems than when writing software. If software passes tests and has a formal proof of correctness, it’s more likely to be correct. And if theoretical results are accompanied by numerical demonstrations, they’re more believable.

Leslie Lamport wrote an article entitled How to Write a Proof where he addresses the problem of errors in proofs and recommends a pattern of writing proofs which increases the probability of the proof being valid. Interestingly, his proofs resemble programs. And while Lamport is urging people to make proofs more like programs, the literate programming folks are urging us to write programs that are more like prose. Both are advocating complementary modes of validation, adding machine-like validation to prosaic proofs and adding prosaic explanations to machine instructions.

When I first saw integration techniques in calculus, I thought they were a waste of time because software packages could do any integral I could do by hand. Besides, you can always just use Simpson’s rule to compute integrals numerically.

In short, I thought symbolic integration was useless and numerical integration was trivial. Of course I was wrong on both accounts. I’ve solved numerous problems at work by being able compute an integral in closed form, and I’ve had a lot of fun cracking challenging numerical integration problems.

Many of the things I thought were impractical when I was in college have turned out to be very practical. And many things I thought would be supremely useful I have yet to apply. Paying too much attention to what is “practical” can be self-defeating. Pragmatism is impractical.