John D. Cook

When I wrote about the harmonic triangle yesterday, I displayed the triangle left-aligned using fixed-width text because that was easier than producing the format you usually see for Pascal's triangle and related triangles. 1 1/2 1/2 1/3 1/6 1/3 1/4 1/12 1/12 1/4 1/5 1/20 1/30 1/20 1/5 ... I asked Claude to write a [...]The post LaTeX package for Pascal-like triangles first appeared on John D. Cook.

G. D. Chakerian [1] discovered the following generalization of the Pythagorean theorem. Take any triangle and attach squares to each side. From each triangle vertex, extend a line perpendicular to the opposite side and extend the line through the square, as shown in the figure below. Then the areas of the rectangles with the same [...]The post A generalization of the Pythagorean theorem first appeared on John D. Cook.

What's a connection between Edgar Allan Poe and Friedrich Wilhelm Bessel? Poe's story The Pit and the Pendulum describes a pendulum with a semicircular blade being lowered on a cable. This is an interesting variation on the usual harmonic oscillator because the length L of the pendulum is not constant but rather an increasing function [...]The post Poe, Bessel, and Drones first appeared on John D. Cook.

Pascal's triangle is the most famous triangular arrangement of numbers. I've written about other numerical triangles, such as Bell's triangle and the trinomial triangle. There's another well-known triangle that somehow I never heard of until recently: the harmonic triangle, first explored by Leibniz around 1700. To create the harmonic triangle, start with the harmonic series [...]The post Harmonic triangle first appeared on John D. Cook.

A few days ago I wrote about topological models of modal logic. If you find general topology more familiar than modal logic, this correspondence could be helpful. If you think the motivation for general topology is evident but the motivation for modal logic is suspect, the correspondence between the two fields might relieve some of [...]The post Categorical models of logics first appeared on John D. Cook.

If you've followed this blog for a while, you may know that I do a lot of work with data privacy and that I have an interest in modal logic. Recently these two worlds collided: I became aware of work that uses modal logic to reason about data privacy. The story begins with Helen Nissenbaum's [...]The post Privacy policies and modal logic first appeared on John D. Cook.

The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function (s) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to . The Quasi Riemann Hypothesis (QRH) says that there exists a constant < 1 [...]The post Consequences of progress toward the Riemann Hypothesis first appeared on John D. Cook.

The Fast Fourier Transform (FFT) algorithm can compute the discrete Fourier transform of a sequence of lengthn in time O(n logn). OpenAI recently posted a paper saying there is an algorithm that could compute the discrete Fourier transform in O(n (logn)1 - ) time for = 10-13. This result is amazing. It seemed that [...]The post Faster Fourier Transform first appeared on John D. Cook.

For a real numberx, the irrationality index (x) is a way of measuring how wellx can be approximated by rational numbers. Ifx is rational, (x) = 1. Ifx is irrational, (x) >= 2. OpenAI recently published a proof that () = 2. Almost all real numbers have irrationality exponent 2, so the new result says [...]The post Irrationality exponent of first appeared on John D. Cook.

I was skimming through David Well's book The Penguin Dictionary of Curious and Interesting Geometry and was surprised by this entry: Lissajous figures orBowditch curves First discussed by Nathaniel Bowditch in 1815, and later by Jules Antoine Lissajous in 1857 ... I wrote a couple posts about Lissajous curves years ago, and recently I wrote [...]The post Lissajous and Bowditch first appeared on John D. Cook.

Godel's incompleteness theorem illustrated the need to distinguish between what is true and what is provable. There are true statements that cannot be proven. Let p denote the assertion thatp is provable in Peano arithmetic. The logic with this interpretation for the operator is the Godel-Lob logic, also called provability logic. This is a [...]The post A topological model for provability logic first appeared on John D. Cook.

An earlier post presented an elegant plane geometry theorem discovered by the 19th century school teacher Auguste Miquel. This post presents his pentagon theorem. Start with a pentagon. It may be irregular, but it needs to be convex. Extend each of the sides of the pentagon to form a star, then draw give circles, one [...]The post Miquel's pentagon theorem first appeared on John D. Cook.

The previous post discussed a superficial connection between modal logic and topology, that both use the terms regular andnormal to indicate added sets of axioms. McKinsey and Tarski developed a deeper connection between modal logic and topology that we'll discuss here. Starting with a topological spaceX and a proposition p, define [[p]] as the set [...]The post Topological models of modal logic first appeared on John D. Cook.

You can't say much about modal logic in general. You have to be more specific to get anywhere. You have to choose some axioms. Ideally the axioms you need for your application correspond to a named set of axioms that has been studied before. The situation is similar in point-set topology. You can't say very [...]The post Modal logic and topology first appeared on John D. Cook.

Euclidean geometry dates back at least to Euclid (circa 300 BC), and so you might think it's been pretty well picked over by now. And yet people still occasionally discover new plane geometry theorems. Some of these new theorems are complicated, asking question that the ancients would not have asked. But once in a while [...]The post Miquel's pivot theorem first appeared on John D. Cook.

Despite the predictions that no one would ever put build data centers in space, Google is starting on Thursday. Google's prototype satellite will be one of 130 payloads on SpaceX's Transporter 18 mission on October 1. The server will follow a dawn-dusk orbit, a special case of a sun-synchronous orbit (SSO), following the terminator line [...]The post Servers in dawn-dusk orbit first appeared on John D. Cook.

In the novel Carry On, Mr. Bowditch, a sailor asked Nathaniel Bowditch to teach him how to do navigational calculations, but the man only knows how to add and subtract by counting on his fingers. He had not heard of multiplication. Bowditch is surprised, but realizes if he made tables of logs of trig functions, [...]The post Navigation with only addition, subtraction, and tables first appeared on John D. Cook.

A couple days ago a friend told me about the book Carry On, Mr. Bowditch, a fictional account of the life of Nathaniel Bowditch (1773-1838). I've been listening to the book on Audible, and apparently it's only lightly fictionalized. Bowditch was a self-educated mathematician and astronomer, best known for his book The American Practical Navigator, [...]The post Nathaniel Bowditch first appeared on John D. Cook.

Suppose you want to solve a triangle. You know two sides and the angle between them. Then you can solve for the third side using the law of cosines. Now suppose you want to solve abig triangle, a triangle on the surface of the earth so large that the curvature of the earth matters. You [...]The post Haversine law first appeared on John D. Cook.

Nothing grows exponentially forever. What appears to be an exponential curve often turns out to be some sort of S curve, such as a logistic curve. Suppose you're collecting data on the left side of the curve. If there's even a small amount of error in your data, you won't be able to predict the [...]The post Why fitting a logistic is nearly impossible from early data first appeared on John D. Cook.

There's a common saying in discussion of fractals that the length of a coastline depends on how small a device you use to measure it. I thought this was a hypothetical, say as applied to the steps in the construction of the Koch snowflake. But the saying has its roots in actually surveying. Lewis Fry [...]The post Empirical fractal first appeared on John D. Cook.

I recently bought a copy of Los Alamos Rolodex, a book displaying business cards from Los Alamos Nation Labs from 1967 to 1978. You can find some examples of the cards here. One of the cards in the book is for Eugene Frank, President of B & F Instruments. His card lists his phone number [...]The post Phone words first appeared on John D. Cook.

I've written three posts on cosine similarity lately. The first looked at interpreting cosine similarity. The second looked at an approximation related to the first. The third looked at how ranking according to cosine similarity works better than cosine similarity itself. Normalized word vectors are points on a high dimensional sphere, and geometry in high [...]The post Converting between cosine similarity and concentration ratio first appeared on John D. Cook.

Yesterday I wrote about the canonical example of how vector embeddings of words add: king" - man" + woman" queen" This should be interpreted as saying that the word vector forking, minus the word vector forman, plus the word vector forwoman, is in some sense close to the word vector forqueen. This post will [...]The post Coffee + milk latte first appeared on John D. Cook.

The product of four consecutive Fibonacci numbers equals the product of two consecutive integers. For example, 3 * 5 * 8 * 13 = 39 * 40. I ran across this theorem in a note [1] that says The product of any four consecutive Fibonacci numbers is twice a triangular number." Since triangular numbers have [...]The post Fibonacci product first appeared on John D. Cook.

The previous post looked at how to interpret cosine similarity, or equivalently angles between word vectors. In a high-dimensional space, randomly chosen vectors are likely nearly perpendicular, and so relatively large angles, such as 50, indicate very closely related words. Another way to look at this, as explained in the previous post, is that in [...]The post Simple approximation for spherical cap area first appeared on John D. Cook.

Machine learning represents words as vectors and measures the similarity of words by the angles between the vectors. For vectorsxandy, where is the angle between the vectors, and so This is the cosine similarity between the words represented by x and y. Small angles have large cosines, and so words with larger cosine similarities [...]The post What counts as a large cosine similarity? first appeared on John D. Cook.

Suppose I give you ann-digit number and ask you what it represents. This seems impossible, and in theory itis impossible. But in practice it's often possible. Apps on a phone may automatically interpret a 10-digit number as a phone number or a 16-digit number as a package tracking number. And very often these interpretations are [...]The post Guessing the meaning of a number first appeared on John D. Cook.

The Greek letter (beta) and the German letter (eszett) look similar, especially in some fonts. Now suppose an OCR program sees some character that could be a beta or could be an eszett. It could calculate some kind of distance between between the pixel pattern of the character and the pixel patterns of [...]The post Bayesian OCR first appeared on John D. Cook.

The idea of using computers to assist with proofs is not new. The first major computer-assisted proof was published in 1976, the proof of the four color theorem by Kenneth Appel and Wolfgang Haken. The authors reduced the proof of the four color theorem to verifying calculations on 1,834 configurations, each checked by a computer [...]The post A 50-year-old computer-assisted proof first appeared on John D. Cook.

A rising tide may lift all boats, but the AI tide lifts some boats much more than others. By all accounts, the best programmers have had the biggest productivity boost from AI. And top tier mathematicians are using AI to settle long-standing mathematical conjectures. AI is a powerful tool, but tools don't come to life [...]The post AI is an intelligence multiplier first appeared on John D. Cook.

Yesterday OpenAI announced a proof that settled a long-standing question about the Navier-Stokes equations from fluid dynamics. The announcement has created a lot of buzz, as one would expect. But there's an aspect of OpenAI's work that I haven't seen anyone talk about: they posted a Lean 4 formal proof at the same time as [...]The post The part of Navier-Stokes no one is talking about first appeared on John D. Cook.

There are rumors that a long-standing math problem, one of the Millennium Prize problems, has been solved. The problem concerns technical properties of solutions to the Navier-Stokes equations [1], a set of equations that describe the dynamics of fluid flow. Popular accounts of the problem are often oversimplified and misleading. Some reports will speak of [...]The post Navier-Stokes in the news first appeared on John D. Cook.

The Online Etymological Dictionary gives the following etymology forgrok: grok (v.) understand empathically," 1961, an arbitrary formation by U.S. science fiction writer Robert A. Heinlein (1907-1988) in his book Stranger in a Strange Land." In the book it is a transliteration of a Martian word and is said to mean etymologically to drink." It attained [...]The post Ngram error rate first appeared on John D. Cook.

The previous post discussed the motivation for and application of the rank-trace theorem. This post will give a proof. SupposeA 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: diagonalizeA [...]The post Proof of the rank-trace theorem first appeared on John D. Cook.

Suppose you want to know the rank of an n * n matrix A, the number of linearly independent rows ofA, 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 [...]The post Computing a lower bound on matrix rank first appeared on John D. Cook.

NVIDIA has offered to 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 [...]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 [...]The post New RSA number factored first appeared on John D. Cook.

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 [...]The post Patented application of linear algebra first appeared on John D. Cook.

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 [...]The post Making the unnecessary easier first appeared on John D. Cook.

This post provides a couple examples to go along with two earlier posts. The pattern we're illustrating is families of polynomialspn(x) that each satisfy a differential equation and a three-term recurrence. The differential equations have a second solutionqn(x) that is the larger solution with respect tox but the smaller solution with respect ton. In both [...]The post Second solutions first appeared on John D. Cook.

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 [...]The post What is the quality of software that AI writes? first appeared on John D. Cook.

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 hereand here, but also three years ago here. Orthogonal polynomials are mathematically elegant as well as very useful in applications [1]. [...]The post Junk solutions first appeared on John D. Cook.

When I hear the term ultrasphericalI 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 polynomialis inexplicable. This is an example of the arcane terminology I wrote about recently. In [...]The post Ultraspherical first appeared on John D. Cook.

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 [...]The post Numerical (in)stability of recurrence relations first appeared on John D. Cook.

There many examples of families of functions where each function can be computed as a linear combination of the two previous terms wherea andb are functions ofx and possibly 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 [...]The post Three-term recurrences first appeared on John D. Cook.

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 densityf(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 [...]The post The von Mises-Fisher distribution first appeared on John D. Cook.

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 reasons. Discovering those reasons may help you understand their motivation and [...]The post What exactly is modified about a modified Bessel function? first appeared on John D. Cook.

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 [...]The post Why special function terminology is arcane first appeared on John D. Cook.

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 [...]The post The difference orbit inclination makes first appeared on John D. Cook.