Ten years ago I wrote a blog post that concludes with this observation: The ideas of being relatively prime, independent, and perpendicular are all related, and so it makes sense to use a common symbol to denote each. History Graham, Knuth, and Patashnik proposed using ⊥ for relatively prime numbers …

March 27, 2020, 3:28 p.m.

It’s very satisfying to know that you have a good solution even under the worst circumstances. Worst-case thinking doesn’t have to be concerned with probabilities, with what is likely to happen, only with what could happen. But whenever you speak of what could happen, you have to limit your universe …

March 24, 2020, 5:48 p.m.

A few days ago I wrote a post about copulas and operations on them that have a group structure. Here’s another example of group structure for copulas. As in the previous post I’m just looking at two-dimensional copulas to keep things simple. Given two copulas C1 and C2, you can …

March 23, 2020, 12:12 a.m.

One of the new features in Mathematica 12.1 is the function Asymptotic. Here’s a quick example of using it. Here’s an asymptotic series for the log of the gamma function I wrote about here. If we ask Mathematica Asymptotic[LogGamma[z], z -> Infinity] we get simply the first term: But we …

March 21, 2020, 1:59 a.m.

The GNU MPFR library is a C library for extended precision floating point calculations. The name stands for Multiple Precision Floating-point Reliable. The library has an R wrapper Rmpfr that is more convenient for interactive use. There are also wrappers for other languages. It takes a long time to install …

March 18, 2020, 2:54 p.m.

Suppose you store a floating point number in memory, print it out in human-readable base 10, and read it back in. When can the original number be recovered exactly? D. W. Matula answered this question more generally in 1968 [1]. Suppose we start with base β with p places of …

March 16, 2020, 11:20 p.m.

You don’t often see references to group theory in a statistics book. Not that there aren’t symmetries in statistics that could be described in terms of groups, but this isn’t often pointed out. Here’s an example from An Introduction to Copulas by Roger Nelsen. Show that under composition the set …

March 16, 2020, 7:05 p.m.

Chebyshev polynomials satisfy a lot of identities, much like trig functions do. This point will look briefly at just one such identity. Chebyshev polynomials Tn are defined for n = 0 and 1 by T0(x) = 1 T1(x) = x and for larger n using the recurrence relation Tn+1(x) = …

March 15, 2020, 7:12 p.m.

The Markov brother you’re more likely to have heard of was Andrey Markov. He was the Markov of Markov chains, the Gauss-Markov theorem, and Markov’s inequality. Andrey had a lesser known younger brother Vladimir who was also a mathematician. Together the two of them proved what is known as the …

March 14, 2020, 2:46 p.m.

It is widely believed that π is a “normal number,” which would mean that you can find any integer sequence you want inside the digits of π, in any base, if you look long enough. So for Pi Day, I wanted to find c0ffee inside the hexadecimal representation of π. …

March 14, 2020, 6:05 a.m.

In the previous post I mentioned that Remez algorithm computes the best polynomial approximation to a given function f as measured by the maximum norm. That is, for a given n, it finds the polynomial p of order n that minimizes the absolute error || f – p ||∞. The …

March 11, 2020, 12:25 p.m.

The best polynomial approximation, in the sense of minimizing the maximum error, can be found by the Remez algorithm. I expected Mathematica to have a function implementing this algorithm, but apparently it does not have one. It has a function named MiniMaxApproximation which sounds like Remez algorithm, and it’s close, …

March 10, 2020, 10:23 p.m.

3Blue1Brown explains exponential growth and epidemics and answers the question of when the…Tags: 3Blue1Brown, math, video

March 10, 2020, 2:57 p.m.

A maximum distance separable code, or MDS code, is a way of encoding data so that the distance between code words is as large as possible for a given data capacity. This post will explain what that means and give examples of MDS codes. Notation A linear block code takes …

I recently stumbled on a formula for the largest gap between consecutive items in a row of Pascal’s triangle. For n ≥ 2, where For example, consider the 6th row of Pascal’s triangle, the coefficients of (x + y)6. 1, 6, 15, 20, 15, 6, 1 The largest gap is …

March 2, 2020, 12:19 p.m.

Take a positive integer x, square each of its digits, and sum. Now do the same to the result, over and over. What happens? To find out, let’s write a little Python code that sums the squares of the digits. def G(x): return sum(int(d)**2 for d in str(x)) This function …

Heron’s formula computes the area of a triangle given the length of each side. where If you have a very thin triangle, one where two of the sides approximately equal s and the third side is much shorter, a direct implementation Heron’s formula may not be accurate. The cardinal rule …

I recently stumbled on a paper [1] that looks at a cubic equation that comes out of a problem in orbital mechanics: σx³ = (1 + x)² Much of the paper is about the derivation of the equation, but here I’d like to focus on a small part of the …

Feb. 27, 2020, 12:59 a.m.