John D. Cook

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. Suppose A is a real symmetric matrix. The rank-trace inequality says where tr is the trace operator, the sum of the elements along the diagonal of the matrix. Terse proof Here’s the proof in a nutshell: diagonalize A and use the Cauchy-Schwarz inequality. Detailed proof Now let’s unpack that. Any real symmetric matrix A is similar to a matrix D with the eigenvalues of A along…

Suppose you want to know the rank of an n × n matrix A, the number of linearly independent rows of A, or equivalently the number of linearly independent columns. There are at least three difficulties. Difficulties in computing rank First of all, rank is not a continuous function of a matrix. Since rank is an integer, an arbitrarily small change in the matrix could cause a discrete change in the rank [1]. A small error in computing A could produce a matrix with a different rank. Second, finding…

NVIDIA has offered by buy Hugging Face for $12,930,300,000. 129303 is the Unicode code point for the Hugging Face emoj (U+1F917), which you can verify with the following Python code. >>> import unicodedata >>> 129303 == 0x1F917 True >>> unicodedata.name(chr(0x1F917)) 'HUGGING FACE' Related posts Prevent characters from displaying as emoji Unicode, Tolkien, and Privacy Unicode surrogates Making flags in Unicode The post Hugging Face Easter Egg first appeared on John D. Cook.

Eric Lu announced on X today that he has factored RSA-260, a number N with 260 digits (862 bits) that is the product of two large primes [1]. RSA numbers are challenge problems posed to gauge the security of RSA encryption, which rests on the difficulty of factoring large numbers [2]. The naming scheme is confusing because RSA-n might have n digits or n bits. For example, RSA-768 is smaller than RSA-260 because the former has 768 bits and the latter has 260 digits. RSA-260 is the largest RSA…

I just found out Brian Beckman and I got a patent on work we did for GSI Technology [1]. Nearly all the work I do is under an NDA, so I don’t often get a chance to talk about my projects. This work is public now that it’s in a patent; I suppose it has been public since the application was published. Brian did most of the work on the project. My contribution was to mathematically formalize low-level operations on sheets of bits using linear algebra over a binary field. Lots of Hadamard products…

I watched a few videos this morning, looking for ideas of what I could use AI to do. In one video, someone had an agent monitor tech news sites every 30 minutes to notify him of a variety of developments. No doubt that’s less effort than visiting a bunch of sites every half hour. But not monitoring tech news in real time takes even less effort. Another video mentioned having Grok Bot order a sandwich through DoorDash. If I want a sandwich, I make a sandwich. One video showed how to manage…

This post provides a couple examples to go along with two earlier posts. The pattern we’re illustrating is families of polynomials pn(x) that each satisfy a differential equation and a three-term recurrence. The differential equations have a second solution qn(x) that is the larger solution with respect to x but the smaller solution with respect to n. In both the examples below pn(x) is a polynomial, and so bounded on the interval [−1, 1], and qn(x) is not a polynomial, with singularities at…

AI-powered coding agents increase productivity for many developers. But do these agents produce good-quality code? Some say this doesn’t matter, we are heading toward dark software factories with source code never inspected, and maybe we should eliminate source code altogether—“source code is the new assembly code”. Others have a different view. Humans sometimes need to debug source code. Source code may need to be audited by humans for compliance. Code should be clear enough for a human to…

When you’re interested in studying a family of functions, it can be useful to look at a differential equation that the functions solve. This is a theme I’ve written about several times, most recently here and here, but also three years ago here. Orthogonal polynomials are mathematically elegant as well as very useful in applications [1]. Various families of orthogonal polynomials satisfy various differential equations. These equations have a polynomial and non-polynomial solutions. What use are…

When I hear the term ultraspherical I think of something extremely spherical. For example, a baseball is spherical, but a billiard ball is more spherical. Maybe a highly polished billiard ball is ultraspherical. Using this line of thought, the term ultraspherical polynomial is inexplicable. This is an example of the arcane terminology I wrote about recently. In this post I’ll explain what it conveys. A spherical polynomial is a polynomial that naturally falls out of solving Laplace’s equation…

The previous post gave several examples of three-term recurrence relations for special functions. These relations can be computationally useful, but they have to be applied carefully. Several years ago I wrote a post on stable and unstable recurrences. In that post I show that the stability of the recurrence relation for Bessel functions produces depends on which kind of Bessel function and which direction the recurrence is applied. In the forward direction, computing higher order values from…

There many examples of families of functions where each function can be computed as a linear combination of the two previous terms where a and b are functions of x but not on n. This is called a three-term recurrence formula. It’s amazing how often you can run into three-term recurrence formulas. There are theorems that give conditions for such recurrences to hold, but I haven’t reached the bottom of that rabbit hole [1]. For this post I just want to give examples. Bessel functions of the first…

Probability density function must integrate to 1, and so if you know a density function up to a constant, the constant is determined. When you’re looking at a probability density f(x) for the first time, it helps to ignore the normalizing constant. Concentrate on the part of the function involving x and know that the normalizing constant is whatever it has to be. For example, about half of the ink that it takes to write down a beta or chi-squared density is devoted to the normalization…

Special functions often have arcane names that not very helpful without some context. The previous post goes into some reasons for this. This post will expand on a point at the end of the post about “modified” functions. Things are given their names for a reason. Discovering that reason helps you understand their motivation and use. Pure math perspective For each integer n, the modified Bessel function In is essentially the Bessel function Jn evaluated along the imaginary axis. Specifically,…

Special functions are special because they’re useful. They can also be shrouded in arcane terminology. These two facts are related. The more widely useful a function is, the more likely it is that the function will be discovered independently multiple times. Independent discoveries lead to varying definitions and notations. For example, there are two widely used definitions of Hermite polynomials, one used in probability and another used in physics, that only differ by a scaling factor. This…

Suppose you wanted to find the distance between Earth and Mars over time. To first approximation, both planets orbit the sun in elliptic orbits in the same plane. If you wanted to be more accurate, you’d need to take into account the fact that the orbit of Mars is tilted about 1.85° relative to the Earth’s orbit. How much difference does that make? To simplify things, let’s assume the Earth orbits the sun in a circle of radius 1 and Mars orbits the sun in a circle of radius 1.5. The distance…

There’s a pizza shop near my home with a sign out front that says “Coming Soon.” When I drove by it this morning I thought about how you would model the time until an event happens that is “coming soon.” Suppose I look at the sign one day and guess how many days until the pizza shop will open. When I drive by a week later and guess again, should my guess be smaller? You might argue that the shop will open some day, fixed in time but unknown to me, and so every day I’m one day closer to the…

A few weeks ago I wrote about the number system used in labeling spreadsheet columns. Labels run from A through Z, then AA through AZ, etc. This looks a lot like base 26, but it’s not quite the same. It has no analog of zero. If Z were like zero, Y would be followed by AZ. The Excel labeling system is not base 26, but what’s called bijective base 26. If you found fragments of writing from an ancient culture and inferred that five symbols were used as digits, how could you distinguish base 5…

I like AI-generated ASCII diagrams. Because nobody would ask AI to generate ASCII diagrams, and so, it’s congruous. I like incongruity [1]. Aside from the incongruity of using a gazillion-parameter neural network to make 1970’s style ASCII art, ASCII diagrams have some uses. They’re absolutely tiny compared to image files. But more importantly they can be inserted into plain text files, such as source code or markdown. A diagram embedded directly into a source file cannot become separated from…

A new paper just came out, The Maximum-Area Small Polygon Problem. The paper solves the problem of finding, for each n, the n-gon with diameter 1 and maximum area. For odd n, the solution is what you might expect: a regular n-gon. I would expect this to be the solution for even n as well, but it’s not. In 1974 [1] Ron Graham found a solution for n = 6, a hexagon with unit diameter and area larger than a regular hexagon with unit diameter. Polygons with diameter ≤ 1 are called “small”, and he…

The imbalance conjecture is now a theorem. James Alexander Schreib and Yousof Yavari posted a proof last week. What does the conjecture theorem say? Start with a graph G and for every edge, calculate the absolute value of the difference of the degree of each end. Then the theorem says there exists another graph H whose vertices have degrees corresponding to the differences of degrees in G. For example, let G be the graph below. The edges from the top red vertex A to each of the blue vertices…

Suppose you have a planet in an elliptical orbit around a star. The math is identical for any light object orbiting a heavy object, such as a moon or satellite orbiting a planet, but we’ll call the heavy object a star and the light object a planet. The center of the star is not quite the center of the orbit. The planet moves along an ellipse with the star at one focus of that ellipse. Let a be the semi-major axis of planet’s orbit, the maximum distance from the center of the ellipse to a point…

The first post in the recent series of posts on Hadamard matrices describes a way of constructing new Hadamard matrices from two other Hadamard matrices by taking their Kronecker product. Starting with a Hadamard matrix H0 and a Hadamard matrix G, you can construct a sequence of Hadamard matrices by Hn+1 = G ⊗ Hn for positive integers n. This is known as the generalized Sylvester method. Let pn be the proportion of 1s in Hn and let q be the proportion of 1s in G. Then you can show that the…

Error correcting codes are most simply described in terms of the errors they can certainly correct. For example, the Hadamard code used for the Mariner 9 probe to Mars encoded each 6-bit pixel to a 32-bit codeword in such a way that the original pixel could be recovered if no more than 7 bits were corrupted in transit. What is the probability that a pixel could be repaired if corrupted? That depends on your probability model. We will assume that the probability of each bit being flipped is p…

Hadamard matrices are in the news following the recent announcement of a newly discovered Hadamard matrix. I’ve written three posts on Hadamard matrices recently, one as a sort of introduction and two on applications: the error correcting code used in the Mariner 9 probe and constructing sphere packings. A Hadamard matrix is an orthogonal matrix with all entries equal to ±1. Jacques Hadamard conjectured that there exist Hadamard matrices of order 4n for all positive integers n. It’s necessary…

Yesterday Levent Alpöge announced that he and his colleagues had discovered a new Hadamard matrix using Claude AI. That motivated a post I wrote this morning on how to construct Hadamard matrices. I mentioned in that post that these matrices arise in applications. This evening I gave an example, describing how NASA used a Hadamard matrix of order 32 to transmit photos from the Mariner 9 spacecraft in 1971. This post will give another application: sphere packing. Conway and Sloane [1] give a…

NASA set Mariner 9 to photograph Mars in 1971. The images had to be encoded for transmission using an error-correcting code, otherwise they would be significantly corrupted when they were received on Earth. The images were encoded for transmission using a code based on Hadamard matrices, specifically a (32, 6, 16) Hadamard code. This means that each 6-bit pixel value was encoded as a 32-bit code word, with all code words differing in at least 16 positions. The previous post explained a way to…

A Hadamard matrix is an orthogonal matrix whose entires are all either 1 or − 1. For example is a Hadamard matrix of order 2. True to Stigler’s law of eponymy, James Joseph Sylvester investigated Hadamard matrices before Jacques Hadamard. Sylvester saw how to bootstrap the example above into more examples. If H is a Hadamard matrix, then the partitioned matrix Sylvester’s construction can be generalized as follows. If Hm is a Hadamard matrix of order m and Hn is a Hadamard matrix of order n,…

Suppose you’ve never worked at the command line and you’re reading a book about the bash shell. You read that !$ is a shortcut to refer to the last word of the previous command. That little fact will almost certainly not stick in your head for a couple reasons. First, you probably see no need for such a shortcut. Second, the syntax seems completely arbitrary. But then after you use the command line a while, you might begin to notice a pattern. You often have to run a pair of commands with the…

I was reading a blog post on boat names because it was on Hacker News this morning. The post contained a link to a data set on dog names in NYC and I poked around the data a little. The top names were not at all what I expected, but then again this is limited to NYC; it’s not a sample across the US. These were the top 10 names: Bella Luna Max Charlie Coco Lola Rocky Milo Teddy Lucy I wondered if the name frequencies might fit a power-law distribution. They do not, but they follow a log-normal…