On publishing your trash can

I'm rereading de Bruijn's book Asymptotic Methods in Analysis (which, sadly, appears to be out of print again!) -- one of the great mathematical expositions, of asymptotic methods in analysis as they stood at midcentury. It's one of the most readable math texts I know.

de Bruijn writes in the preface:

Many things in this book are not presented in the shortest possible form, as an attempt has been made to reveal, to a certain extent, the motives that lead to certain methods. Naturally one cannot go too far in this respect; a mathematician cannot possibly publish his waste-paper basket.

This seems worth remembering; terseness is not always a virtue.

The Archimedean property of reality

The Archimedean property of reality: if you put one foot in front of the other for long enough, you can travel an arbitrarily long distance.

This is, of course, the Archimedean property of the real numbers -- that is, for any two positive real numbers x and y, there exists an integer n such that nx > y -- if we can make the assumption that step length is constant.

It's also what got me through some hard times in the first year of grad school, when I doubted my mathematical abilities; at least I could successfully walk to and from school each day. The proverb "a journey of a thousand miles begins with a single step" comes to mind, but my walk is about a mile and a half.

The uniform distribution as a sum?

Yesterday, I was asked the following question: the sum of two uniformly distributed random variables with the same support has a triangular distribution. Is there a random variable X such that X + Y has a distribution which is uniform, where X and Y are independent and identically distributed?

I don't know the answer, but I started thinking as follows. First, it's enough to show that there aren't independent identically distributed X, Y, such that X+Y has a distribution uniform on [-1, 1]; linearly scaling gets the general result. Now, the characteristic function of a uniform distribution on [-1, 1] is φ(t) = (sin t)/t. The characteristic function of X+Y is the product of the characteristic functions of X and Y. (If you're more familiar with analysis than probability, note that characteristic functions are basically Fourier transforms, and the probability density function of X+Y is the convolution of the probability density functions of X and Y.) Thus, if X exists it has characteristic function ψ(t) = [(sin t)/t]^1/2 -- this is already a bit problematic, because we want ψ to be continuous, but even with that restriction we still have to specify which square root is being taken on each of the intervals ... [-3π, -2π], [-2π, -π], [-π, π], [π, 2π], [2π, 3π] ... (Informally, we have to make a new choice every time (sin t)/t goes through 0.

At this point I think one wants to use Bochner's theorem, which says that the functions which are characteristic functions of measures on the real line are exactly the positive definite functions -- but how does one show that this function is positive definite?

The other thing to do is to look at the discrete analogue; consider the probability generating function of a random variable which is uniformly distributed on the set {0, 1, ..., n-1}. This is χ(x) = (1+x+x²+...+x^n-1)/n. Now, if this random variable were the sum of two independent identically distributed random variables, its p.g.f. would be the square of a polynomial with positive real coefficients. It's not.

But what about the continuous case?

Mathematica est omnis divisa in partes tres

That is, "all mathematics is divided into three parts". I don't actually know Latin. I fear I may have said that Mathematica, the software, is divided into three parts.

This occurred to me this morning, when I saw the following three-fold characterization of the conic sections, which may amuse. (I'm basically copying this from Leila Schneps' lecture notes.)

Conic sections can be defined geometrically, analytically, or algebraically.

• Geometrically: a conic section is the curve obtained when a plane intersects with a cone. The conic section is an ellipse, parabola, or hyperbola according to whether the plane is less steeply slanted, exactly as steeply slanted, or more steeply slanted than the cone's generating line.


• Analytically: given a point F the focus and a line D the directrix, a conic section is the locus of points P such that d(P,F)/d(P,D) is a constant e, the eccentricity. The conic section is an ellipse, parabola, or hyperbola according to whether the eccentricity is less than, equal to, or greater than 1.


• Algebraically: a conic section is the solution set of Ax² + Bxy + Cy² + Dx + Ey + F = 0. The conic section is an ellipse, parabola, or hyperbola according to whether B² - 4AC, the discriminant, is negative, zero, or positive.

The three definitions are equivalent.

It's fitting that the quote is in Latin, because I don't know Latin, and I also don't believe that mathematics is divided into only these three parts. In particular, you all know that I like probability and combinatorics, which don't naturally fit into this tripartite scheme. But it's a nice division of what one might call "classical" mathematics (it's roughly the content of the semi-standard first-year graduate mathematics curriculum, for example).

Calling the second definition "analytic" is a bit of a stretch, though.