A conformal map between two regions in the plane preserves angles [1]. If two curves meet at a given angle in the domain, their images will meet at the same angle in the range. Two subsets of the plane are conformally equivalent if there is a conformal map between them. The Riemann mapping theorem says […]

# Category: Math

## Stone-Weierstrass on a disk

A couple weeks ago I wrote about a sort of paradox, that Weierstrass’ approximation theorem could seem to contradict Morera’s theorem. Weierstrass says that the uniform limit of polynomials can be an arbitrary continuous function, and so may have sharp creases. But Morera’s theorem says that the uniform limit of polynomials is analytic and thus […]

## Distribution of zip code population

There are three schools of thought regarding power laws: the naive, the enthusiasts, and the skeptics. Of course there are more than three schools of thought, but there are three I want to talk about. The naive haven’t heard of power laws or don’t know much about them. They probably tend to expect things to […]

## Landau kernel

The previous post was about the trick Lebesgue used to construct a sequence of polynomials converging to |x| on the interval [-1, 1]. This was the main step in his proof of the Weierstrass approximation theorem. Before that, I wrote a post on Bernstein’s proof that used his eponymous polynomials to prove Weierstrass’ theorem. This […]

## Lebesgue’s proof of Weierstrass’ theorem

A couple weeks ago I wrote about the Weierstrass approximation theorem, the theorem that says every continuous function on a closed finite interval can be approximated as closely as you like by a polynomial. The post mentioned above uses a proof by Bernstein. And in that post I used the absolute value function as an […]

## Proving that a choice was made in good faith

How can you prove that a choice was made in good faith? For example, if your company selects a cohort of people for random drug testing, how can you convince those who were chosen that they weren’t chosen deliberately? This is something I’ve helped companies with. It may be impossible to prove that a choice […]

## Detecting a short period in an RNG

The last couple posts have been looking at the Cliff random number generator. Yesterday I posted about testing the generator with the DIEHARDER test suite, the successor to George Marsaglia’s DIEHARD battery of RNG tests. This morning I discovered something about the Cliff RNG which led to discovering something about DIEHARDER.The latter is more important: […]

## Testing Cliff RNG with DIEHARDER

My previous post introduced the Cliff random number generator. The post showed how to find starting seeds where the generator will start out by producing approximately equal numbers. Despite this flaw, the generator works well by some criteria. I produced a file of 100 million 32-bit integers by multiplying the output values [1], which were […]

## Fixed points of the Cliff random number generator

I ran across the Cliff random number generator yesterday. Given a starting value x0 in the open interval (0, 1), the generator proceeds by xn+1 = | 100 log(xn) mod 1 | for n > 0. The article linked to above says that this generator passes a test of randomness based on generating points on […]

## Uniform approximation paradox

What I’m going to present here is not exactly a paradox, but I couldn’t think of a better way to describe it in the space of a title. I’ll discuss two theorems about uniform convergence that seem to contradict each other, then show by an example why there’s no contradiction. Weierstrass approximation theorem One of […]

## Nearly parallel is nearly transitive

Let X, Y, and Z be three unit vectors. If X is nearly parallel to Y, and Y is nearly parallel to Z, then X is nearly parallel to Z. Here’s a proof. Think of X, Y, and Z as points on a unit sphere. Then saying that X and Y are nearly parallel means […]

## Angles in the spiral of Theodorus

The previous post looked at how to plot the spiral of Theodorus shown below. We stopped the construction where we did because the next triangle to be added would overlap the first triangle, which would clutter the image. But we could certainly have kept going. If we do keep going, then the set of hypotenuse […]

## How to plot the spiral of Theodorus

You may have seen the spiral of Theodorus. It sticks a sequence of right triangles together to make a sort of spiral. Each triangle has a short side of length 1, and the hypotenuse of each triangle becomes the long leg of the next triangle as shown below. How would you plot this spiral? At […]

## Encryption as secure as factoring

RSA encryption is based on the assumption that factoring large integers is hard. However, it’s possible that breaking RSA is easier than factoring. That is, the ability to factor large integers is sufficient for breaking RSA, but it might not be necessary. Two years after the publication of RSA, Michael Rabin created an alternative that […]

## Accelerating convergence with Aitken’s method

The previous post looked at Euler’s method for accelerating the convergence of a slowly converging alternating series. Both hypotheses are necessary. The signs must alternate between terms, and applying the method to a series that is already converging quickly can slow down convergence. This post looks at Aitken’s method for speeding up the convergence of […]

## Accelerating an alternating series

The most direct way of computing the sum of an alternating series, simply computing the partial sums in the terms get small enough, may not be the most efficient. Euler figured this out in the 18th century. For our demo we’ll evaluate the Struve function defined by the series Note that the the terms in […]

## Beating the odds on the Diffie-Hellman decision problem

There are a couple variations on the Diffie-Hellman problem in cryptography: the computation problem (CDH) and the decision problem (DDH). This post will explain both and give an example of where the former is hard and the latter easy. The Diffie-Hellman problems The Diffie-Hellman problems are formulated for an Abelian group. The main group we […]

## Magic square links and errata

Someone pointed out that what I called a knight’s tour magic square is technically a semi-magic square: the rows and columns add up to the same constant, but the diagonals do not. It turns out there are no strict magic squares formed by knight’s tours. This was proved in 2003. See a news article here. […]

## Quaternion reference in the Vulgate

To contemporary ears “quaternion” refers to a number system discovered in the 19h century, but there were a couple precedents. Both refer to something related to a group of four things, but there is no relation to mathematical quaternions other than that they have four dimensions. I’ve written before about Milton’s use of the word […]

## Fame, difficulty, and usefulness

Pierre Fermat is best known for two theorems, dubbed his “last” theorem and his “little” theorem. His last theorem is famous, difficult to prove, and useless. His little theorem is relatively arcane, easy to prove, and extremely useful. There’s little relation between technical difficulty and usefulness. Fermat’s last theorem Fermat’s last theorem says there are […]