John D. Cook

Suppose you have software to compute one hypergeometric function. How many more hypergeometric functions can you compute from it? Hypergeometric functions satisfy a lot of identities, so you can bootstrap one such function into many more. That’s one reason they’re so useful in applications. For this post, I want to focus on just three formulas, the so-called […]

Mark Kac asked in 1966 whether you can hear the shape of a drum. In other words, can two drums, made of the same material, produce the exact same sound but have different shapes? More formally, Kac asked whether the eigenvalues of the Laplace’s equation with zero boundary conditions uniquely determine the shape of a […]

The relationship between the discrete Laplace transform and discrete Fourier transform is not quite the same as that between their continuous counterparts. Continuous Fourier and Laplace transforms The continuous versions of the Fourier and Laplace transforms are given as follows. Fourier transform: Laplace transform: The Fourier transform is defined several ways, and I actually prefer […]

“Reproducible” and “randomized” don’t seem to go together. If something was unpredictable the first time, shouldn’t it be unpredictable if you start over and run it again? As is often the case, we want incompatible things. But the combination of reproducible and random can be reconciled. Why would we want a randomized controlled trial (RCT) to […]

The Joukowsky transformation, or Joukowsky map, is a simple function that comes up in aerospace and electrical engineering calculations. (Here z is a complex variable.) Despite its simplicity, it’s interesting to look at in detail. Mapping lines and circles Let z = r exp(iθ) and let w = u + iv be its image. Writing the Joukowsky transformation in terms of its real and complex parts makes it […]

The birthday problem, sometimes called the birthday paradox, says that it’s more likely than you’d expect that two people in a group have the same birthday. Specifically, in a random sample of 23 people, there’s about a 50-50 chance that two people share the same birthday. The birthday problem makes a nice party trick, but […]

Long run or broken software? I got a call one time to take a look at randomization software that wasn’t randomizing. My first thought was that the software was working as designed, and that the users were just seeing a long run. Long sequences of the same assignment are more likely than you think. You […]

You can get upper and lower bounds on the diameter of a connected graph G from its spectrum. If G has r distinct eigenvalues—whether of the adjacency matrix A, Laplacian L, or signless Laplacian Q—then the its diameter d is at most r-1. (Definitions of these matrices here.) If G has n vertices then we […]

You can tell whether a graph is regular, or nearly regular, by looking at its eigenvalues. Let G be a graph with n vertices and m edges and let L be the Laplacian matrix of G. Then the sum of the eigenvalues of L is 2m. (The diagonal of L contains the degrees of each node, so it’s trace is twice the […]

Not that long ago number theory was considered strictly pure math. Then came applications to cryptography. Now number theory is at the foundation of the online economy. What are the next areas of pure math to find widespread application? Some people are saying algebraic topology and category theory. [I saw a cartoon to this effect the other day […]

The other day I ran across a line from Peter Thiel saying that if you have a plan for where you’d like to be in ten years, ask yourself if you could get there in six months. I don’t think he’s simply saying see if you can do everything 20 times faster. If you estimate something […]

Here are a couple elegant series involving the golden ratio φ = (1 + √5)/2. First, reciprocals of integer powers: Then reciprocals of odd powers: Both are easy to prove since they’re geometric series. More posts related to the golden ratio: Imaginary gold Oscillating Fibonacci ratios Golden strings and the rabbit constant Power method and […]

One way to get some understanding of a graph is to peel away nodes, starting with the least connected. The k-core of a graph is the maximal subgraph such that every vertex has degree at least k. The k-shell is the set of vertices that are part of the k-core but not part of the […]

Last year I wrote a couple posts about what was then the largest known prime, 257885161 – 1. Now there’s a new record, P = 274207281 – 1. For most of the last 500 years, the largest known prime has been a Mersenne prime, a number of the form 2p – 1 where p is itself prime. Such numbers […]

The vertices of a bipartite graph can be divided into two sets where edges only go from one set to the other; there are no edges within each set. For example, in the graph below edges connect blue to green nodes, but not blue to blue or green to green. In this example there are two […]

Create a graph by first making a set of n vertices and then connecting nodes i and j, i > j, with probability p. This is known as the Erdős-Rényi graph Gn,p. Here’s such a graph with n = 100 and p = 0.1. Then the spectrum of the adjacency matrix A of the graph bounces around the spectrum of the expected […]

A Pythagorean triple is a list of positive integers (a, b, c) such that a2 + b2 = c2. Euclid knew how to find all Pythagorean triples: pick two positive integers m and n with m > n and let a = m2 – n2, b = 2mn, c = m2 + n2. Now what if we look at matrices with […]

The other day someone asked me to look at the search data for Debug Pest Control, a pest management company based in Rhode Island. One of the things I wanted to visualize was how the search terms overlapped with each other. To do this, I created a graph where the nodes are the keywords and edges join nodes that […]

A graph doesn’t have any geometric structure unless we add it. The vertices don’t come with any position in space. The same graph can look very different when arranged different ways. Spectral coordinates are a natural way to draw a graph because they are determined by the properties of the graph, not arbitrary aesthetic choices. Construct the Laplacian […]

You can arrange a standard set of dominoes into a magic square of sorts. There are 28 dominoes, each with two ends, so the number of ends isn’t a perfect square. But if you ignore the row of blanks at the bottom, you have a 7 by 7 square where every row, column, and diagonal […]

A statistician could help a lawyer answer the following questions. Was this data collected in a proper way? Does common sense apply here, or is there something subtle going on? What conclusions can we draw from the data? Is this analysis routine or is there something unusual about it? How much confidence can we place […]

The discrete Fourier transform (DFT) of length N multiplies a vector by a matrix whose (j, k) entry is ωjk where ω = exp(-2πi/N), with j and k running from 0 to N – 1. Each element of the matrix is a rotation, so if N = 12, we can represent each element by an hour on a clock. The angle […]

A few examples help build intuition for what the eigenvalues of the graph Laplacian tell us about a graph. The smallest eigenvalue is always zero (see explanation in footnote here). For a complete graph on n vertices, all the eigenvalues except the first equal n. The eigenvalues of the Laplacian of a graph with n […]

When you add an edge to a graph, each of the eigenvalues of the graph Laplacian either increases or stays the same. As I wrote in my previous post, the smallest eigenvalue is always 0. The second smallest eigenvalue is 0 if and only if the graph is disconnected. So no edge will increase the […]

If a graph can be split into two components with no links between them, that’s probably easy to see. It’s also unlikely, unless there’s a good reason for it. The less obvious and more common case is a graph that can almost be split into two components. Spectral graph theory, looking at the eigenvalues of the graph Laplacian, […]

Statisticians use n to denote the number of subjects in a data set and p to denote nearly everything else. You’re supposed to know from context what each p means. In the phrase “big n, little p” the symbol p means the number of measurements per subject. Traditional data sets are “big n, little p” […]

I only connect to people on LinkedIn that I know. This almost always means people I have met face-to-face or at least talked to over the phone. If you’d like to connect on LinkedIn and we haven’t met, please contact me to set up a phone call. I look forward to talking to you.

Fourier series and Fourier transforms may seem more different than they are because of the way they’re typically taught. Fourier series are presented more as a representation of a function, not a transformation. Here’s a function on an interval. We can write it as a sum of sines and cosines, just as we can write […]

Here’s an interesting line from Brad Osgood: Isn’t it a little embarrassing that multibillion dollar industries seem to depend on integrals that don’t converge? In context, he’s not saying that huge companies are blithely using bad math. Some are, but that’s not what he’s getting at here. His discussion is an example of coming full […]

2016 appears in π starting at the 7173rd decimal place: You can confirm this with Mathematica or Wolfram Alpha: Mod[ Floor[10^7177 pi] , 10000] I found it using the following Python code: >>> from sympy import pi >>> digits = str(pi.evalf(10000))[2:] >>> digits.find('2016') 7173 By the way, it’s also true that 2016 = 1 + 2 […]

I heard of someone who had a commitment to get rid of something every Thursday. I don’t know anything about how they carried that out. It could mean throwing out or donating to charity a physical object each Thursday. Or maybe it could be handing over a responsibility or letting go of an ambition. It could be a […]

Here are this blog’s most popular technology posts of 2015: The most important skill in software development Automate to save mental energy, not time The success of object oriented programming Learning (needlessly) hard technology And here are the most popular math posts of 2015: Defining zero factorial Life lessons from differential equations Distance to Mars

Automation doesn’t always save as much time or effort as we expect. The xkcd cartoon above is looking at automation as an investment. Does the work I put in now eventually save more work than I put into it? Automation may be well worth it even if the answer is “no.” Automation can be like […]

The sha function, also known as the Dirac comb, is denoted with the Cyrillic letter sha (Ш, U+0428). This letter was chosen because it looks like how people visualize the function, a long series of vertical spikes. The function is called the Dirac comb for the same reason. This function is very important in Fourier […]

Suppose project completion time follows a Pareto (power law) distribution with parameter α. That is, for t > 1, the probability that completion time is bigger than t is t-α. (We start out time at t = 1 because that makes the calculations a little simpler.) Now suppose we know that a project has lasted […]

There are a couple common uses of the term distribution in math. The most familiar is probability distribution, such as a beta distribution, a Poisson distribution, etc. Less familiar but still common is distributions in the sense of generalized functions, like the Dirac delta distribution. Anybody with much exposure to math will have heard of a […]

I’m making a couple changes to my Twitter accounts. First, I’m winding down @PerlRegex. I’ll stop tweeting there when my scheduled tweets run out. I suggest that everyone who has been following @PerlRegex start following @RegexTip instead. The latter is more general, but is mostly compatible with Perl. Second, I’m reviving my @DSP_Fact. I stopped […]

I was listening to a classic music station yesterday, and I heard the story of a professional pianist whose hand was injured in an accident. He then started learning trumpet and two years later he was a professional trumpeter. I didn’t catch the musician’s name. I was not surprised that a professional in one instrument could become […]

In the previous post, we looked at an elegant equation involving integrals of the sinc function and computed the corresponding integrals for the jinc function. It turns out the analogous equation holds for sums as well: As before, we’d like to compute these two sums and see whether we can compute the corresponding sums for the […]

The sinc function is defined by sinc(x) = sin(x)/x. Philip Woodward introduced the name of the function in 1952, saying it “occurs so often in Fourier analysis and its applications that it does seem to merit some notation of its own.” Here’s an elegant equation involving the integrals of the sinc function: When I ran […]

Almost three years ago I put up a web page to let you go back and forth between Unicode code points and LaTeX commands. Here’s the page and here’s a blog post explaining it. I’ve expanded the data the page uses by merging in data from the STIX Project. More queries should return successfully now. […]

Pi The other day I ran across the following continued fraction for π. Source: L. J. Lange, An Elegant Continued Fraction for π, The American Mathematical Monthly, Vol. 106, No. 5 (May, 1999), pp. 456-458. While the continued fraction itself is interesting, I thought I’d use this an example of how to typeset and compute […]

There are six or eight ways to define a Fourier transform. The differences in the various conventions are minor, but they lead to differences in the basic results. So whenever you look up a result, you have to make sure the reference’s definition matches the one you’re expecting. Or maybe you re-derive the result. This is good […]

This is what the book Social Media Mining calls the Big Data Paradox: Social media data is undoubtedly big. However, when we zoom into individuals for whom, for example, we would like to make relevant recommendations, we often have little data for each specific individual. We have to exploit the characteristics of social media and […]

Richard Guy’s Strong Law of Small Numbers says There aren’t enough small numbers to meet the many demands made of them. In his article by the same name [1] Guy illustrates his law with several examples of patterns that hold for small numbers but eventually fail. One of these examples is 3! – 2! + 1! = […]

Here are some of the non-technical books I’ve mentioned in blog posts this year. I posted the technical list a couple days ago. Maybe I should say “less technical” rather than “non-technical.” For example, Surely You’re Joking, Mr. Feynman is a book about a physicist, but it’s at least as much a human interest book as […]

“Nature does not consist entirely, or even largely, of problems designed by a Grand Examiner to come out neatly in finite terms, and whatever subject we tackle the first need is to overcome timidity about approximating.” H. and B. S. Jeffreys, Methods of Mathematical Physics, 2nd ed., Cambridge University Press, 1950, p. 8. Related post: […]

RSS feeds RSS lets you subscribe to blogs. It also lets you read posts in peace, free from distracting peripheral ads. This explains why Google would kill off the world’s most popular RSS reader. Blogs used to display an icon linking to the site’s RSS feed, and any still do. Blogging software still creates RSS feeds, […]

Here are some of the books I’ve mentioned in blog posts this year. One of these books may be just the present for a geek in your life. This post looks at technical books: math, science, engineering, programming. I’ll have a follow-up post with non-technical books I’ve written about. (Update: here’s the non-technical list.) Programming Book […]

I saw an impressive chart once of how numerical linear algebra algorithm efficiency have improved over time. I haven’t been able to find that chart, but here is something similar. Thanks to Naveen Palli for pointing this out. Even more remarkable [than Moore’s law] — and even less widely understood — is that in many areas, performance gains due […]