John D. Cook

The time it takes for the earth to orbit the sun is not an integer multiple of the time it takes for the earth to rotate on its axis, nor is it a rational number with a small denominator. Why should it be? Much of the complexity of our calendar can be explained by rational [...]The post Gregorian Calendar and Number Theory first appeared on John D. Cook.

Here's something I posted on X a couple days ago: There's no direct connection between AI and cryptocurrency, but they have a similar vibe. They both leave you wondering whether the emperor is sumptuously clothed, naked, or a mix of both. Maybe he's wearing a hospital gown with gold threads. In case you're unfamiliar with [...]The post Golden hospital gowns first appeared on John D. Cook.

Yesterday I needed to write a regular expression as part of a client report. Later I was curious whether an LLM could have generated an equivalent expression. When I started writing the prompt, I realized it wasn't trivial to tell the LLM what I wanted. I needed some way to describe the pattern that the [...]The post LLMs and regular expressions first appeared on John D. Cook.

A logarithmic scale is very useful when you need to plot data over an extremely wide range. However, sometimes even a logarithmic scale may not reduce the visual range enough. I recently saw a timeline-like graph that was coiled into a spiral, packing more information into a limited visual window [1]. I got to thinking [...]The post Coiled logarithmic graph first appeared on John D. Cook.

How many ways can you select six points in the plane so that every subset of three points forms the vertices of an isosceles triangle? This is a question asked by Erds and recently resolved. One solution is to choose the five vertices of a regular pentagon and the center. It's easy to verify that [...]The post Solution to a problem of Erds first appeared on John D. Cook.

This post was motivated by an exercise in [1] that says Prove that for the hyperbolic functions ... formulas hold similar to those in Section 2.3 with all the minuses replaced by pluses. My first thought was that this sounds like Osborn's rule, a heuristic for translating between (circular) trig identities and hyperbolic trig identities. [...]The post Multiple angles and Osborn's rule first appeared on John D. Cook.

The vis-viva equation greatly simplifies some calculations in orbital mechanics. It is reminiscent of how conservation of energy can sometimes trivialize what appears to be a complicated problem. In fact, the vis-viva equation is derived from conservation of energy, but the derivation is not trivial. Which is good: the effort required in the derivation implies [...]The post The vis-viva equation first appeared on John D. Cook.

The most efficient maneuver for transferring from one circular orbit to another circular orbit of roughly the same size is the Hohmann transfer orbit. It requires two burns: one to leave the initial circular orbit into an elliptical orbit, and another to leave the elliptic orbit for the new circular orbit. If the new orbit [...]The post Efficiently transferring to a much higher orbit first appeared on John D. Cook.

Shortly after I started using a MacBook I remapped the keys so that they function the same way on Mac OS, Windows, and Linux. The key in the lower left corner, for example, behaves the same way across operating systems, as does the key to the left of the space bar. Note that I'm going [...]The post Rotating MacBook keys first appeared on John D. Cook.

We tend to think that the effort required to generate a solution and verify a solution are roughly equal, assuming that you need to retrace the generation steps to verify that they are correct. But sometimes verification can be far easier than generation [1]. Factoring For example, suppose I generate two 1000-digit prime numbers, multiply [...]The post Asymmetric generation / verification costs first appeared on John D. Cook.

Someone sent me an AI-generated Dogecoin anthem: To Da Moon. Here's the audio. And here are the lyrics: Yo, it started as a joke, now we in the game, Dogecoin rocket, yeah, remember the name. Crypto vibes, makin' history soon, Strapped to the rocket, we're goin' to the moon. Elon on the tweets, got [...]The post Dogecoin anthem first appeared on John D. Cook.

When I started this blog, almost 17 years ago, I posted nearly every day. The first time I went a couple days without posting I got a message from someone asking if everything was OK. I've slowed down since then, and even more lately. Last week I was busy with professional work, and this week [...]The post Blogging pace first appeared on John D. Cook.

A client emailed me a screenshot of a table rather than pasting the table as text into an email. I thought about using an LLM to convert it to text, but the table is confidential client information and so I shouldn't upload it anywhere. I searched for a command line utility to do OCR and [...]The post Confidential OCR first appeared on John D. Cook.

A perfect number is a positive integer equal to the sum of its proper divisors, all divisors less than itself. The first three examples are as follows. 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14 496 = 1 + 2 + 4 + 8 + [...]The post Perfect numbers first appeared on John D. Cook.

I stumbled on a post on X this morning, a commentary on the photo of RFK eating food from McDonalds that has been making the rounds. This photo divides Puritans from Southerners. Puritans think because RFK Jr is on the side of health food he can never commit such a sin." Southerners think a rare [...]The post Food and Grace first appeared on John D. Cook.

I've had a Bluesky account for over a year, but never posted much on it. Recently I noticed I'd gotten more followers on Bluesky and thought I might try posting there more often. I am not moving to Bluesky. I have orders of magnitude more followers on X than on Bluesky and so I will [...]The post Bluesky account first appeared on John D. Cook.

The basic idea of GPS is that if you know the distance to several satellites, you can figure out your position. But you don't actually know, or need to know, the distance to the satellites: you know the time (according to each satellite's clock) when the signals were sent, and you know the time (according [...]The post The mathematics of GPS first appeared on John D. Cook.

GPS satellites all orbit at the same altitude. According to the FAA, GPS satellites fly in circular orbits at an altitude of 10,900 nautical miles (20,200 km) and with a period of 12 hours. Why were these orbits chosen? You can determine your position using satellites that are not in circular orbits, but with circular [...]The post GPS satellite orbits first appeared on John D. Cook.

A few weeks ago I wrote about the Mellin transform. Mitchell Wheat left comment saying the transform seems reminiscent of Ramanujan's master theorem, which motivated this post. Suppose you have a function f that is nice enough to have a power series. Now focus on the coefficients ak as a function of k. We'll introduce [...]The post Ramanujan's master theorem first appeared on John D. Cook.

Here's a simple calculation that I've done often enough that I'd like to save the result for my future reference and for the benefit of anyone searching on this. A linear combination of sines and cosines a sin(x) + b cos(x) can be written as a sine with a phase shift A sin(x + ). [...]The post Linear combination of sine and cosine as phase shift first appeared on John D. Cook.

Suppose you want to write a shell script searches the current directory for files that have a keyword in the name of the file or in its contents. Here's a first attempt. find . -name '*.py' -type f -print0 | grep -i "$1" find . -name '*.py' -type f -print0 | xargs -0 grep -il [...]The post Resolving a mysterious problem with find first appeared on John D. Cook.

I recently stumbled upon the Postage Stamp Problem. Given two relatively prime positive numbers a and b, show that any sufficiently large number N, there exists nonnegative integers x and y such that ax + by = N. I initially missed the constraint that x and y must be positive, in which result is well [...]The post The Postage Stamp Problem first appeared on John D. Cook.

I've read some math publications from around a century or so ago, and I wondered if I could pull off being a math professor if a time machine dropped me into a math department from the time. I think I'd come across as something of an autistic savant, ignorant of what contemporaries would think of [...]The post Impersonating an Edwardian math professor first appeared on John D. Cook.

Before Copernicus promoted the heliocentric model of the solar system, astronomers added epicycle on top of epicycle, creating ever more complex models of the solar system. The term epicycle is often used derisively to mean something ad hoc and unnecessarily complex. Copernicus' model was simpler, but it was less accurate. The increasingly complex models before [...]The post Maybe Copernicus isn't coming first appeared on John D. Cook.

Suppose you want to interpolate a set of data points with a combination of sines and cosines. One way to approach this problem would be to set up a system of equations for the coefficients of the sines and cosines. If you have N data points, you will get a system of N equations in [...]The post Trigonometric interpolation first appeared on John D. Cook.

This is a quick note to mention a connection between two recent posts, namely today's post about moments and post from a few days ago about the Laplace transform. Letf(t) be a function on [0,) and F(s) be the Laplace transform of f(t). Then the nth moment of f, is equal to then nth derivative [...]The post Moments with Laplace first appeared on John D. Cook.

It's fascinating that there's such a thing as the World Jigsaw Puzzle Championship. The winning team of the two-person thousand-piece puzzle round can assemble a Ravensburger puzzle in less than an hour-that's about 3 -1/2 seconds per piece. It makes you wonder, how could you measure the hardness of a jigsaw puzzle? And what would [...]The post The impossible puzzle first appeared on John D. Cook.

The use of the word moment" in mathematics is related to its use in physics, as in moment arm or moment of inertia. For a non-negative integer n, the nth moment of a function f is the integral of xn f(x) over the function's domain. Uniqueness If two continuous functions f and g have all [...]The post When do moments determine a function? first appeared on John D. Cook.

In the early days of computing hardware (and actually before) mathematicians put a lot of effort into understanding and mitigating the limitations of floating point arithmetic. They would analyze mundane tasks such as adding a list of numbers and think carefully about the best way to carry out such tasks as accurately as possible. Now [...]The post Floating point: Everything old is new again first appeared on John D. Cook.

I've been writing code for the Z3 SMT solver for several months now. Here are my findings. Python is used here as the base language. Python/Z3 feels like a two-layer programming model-declarative code for Z3, imperative code for Python. In this it seems reminiscent of C++/CUDA programming for NVIDIA GPUs-in that case, mixed CPU and [...]The post How hard is constraint programming? first appeared on John D. Cook.

The band-limited expansion of the function f(x) is given by where sinc(x) = sin(x)/x. This is also called the sinc expansion, or the Whittaker cardinal after its discoverer E. T. Whittaker [1]. This is called the band-limited expansion of f because each term in the infinite sum is band-limited, i.e. only has Fourier spectrum within [...]The post Band-limited expansion first appeared on John D. Cook.

Sometimes the future state of a system depends not only on the current state (position, velocity, acceleration, etc.) but also on the previous state. Equations for modeling such systems are known as delay differential equations (DDEs), difference differential equations, retarded equations, etc. In a system with hysteresis, it matters not only where you are but [...]The post Delay differential equations first appeared on John D. Cook.

The way Laplace transforms, as presented in a typical differential equation course, are not very useful. Laplace transformsareuseful, but not as presented. The use of Laplace transforms is presented is as follows: Transform your differential equation into an algebraic equation. Solve the algebraic equation. Invert the transform to obtain your solution. This is correct, but [...]The post Laplace transform inversion theorems first appeared on John D. Cook.

The Mellin transform of a function f is defined as For example, it follows directly from the definition that the gamma function (s) is the Mellin transform of the function e-x. I ran across an exercise that states an impressive-looking theorem about the Mellin transform, namely that where F(s) denotes the Mellin transform of f(x). [...]The post Mellin transform and Riemann zeta first appeared on John D. Cook.

I woke up around 3:00 this morning to some sort of alarm outside. It did not sound like a car alarm; it sounded like a sawtooth wave. The pattern was like a few Morse code O's. Not SOS, or I would have gotten up to see if anyone needed help. Just O's. A sawtooth wave [...]The post Sawtooth waves first appeared on John D. Cook.

A mathematician's reputation rests on the number of bad proofs he has given. (Pioneer work is clumsy.)" - A. S. Besicovitch I'm sure I've written about this quote somewhere, but I can't find where. The quote comes from A Mathematician's Miscellany by J. E. Littlewood, citing Besicovitch. I've more often seen the quote concluding with [...]The post Pioneering work is ugly first appeared on John D. Cook.

Mersenne numbers have the form 2p - 1. A Mersenne prime is a Mersenne number that is also a prime. A new Mersenne prime discovery was announced today: 2p - 1 is prime for p = 136279841. The size of the new Mersenne prime is consistent with what was predicted. For many years now, the [...]The post New Mersenne prime found first appeared on John D. Cook.

Claude Shannon's famous paper A Mathematical Theory of Communication [1] includes an example saying that the channel capacity of a telegraph is log2 W where W is the largest real root of the determinant equation Where in the world did that come from? I'll sketch where the equation above came from, but first let's find [...]The post Channel capacity of a telegraph first appeared on John D. Cook.

I ran across the following theorem in Ross Honsberger's book Mathematical Morsels: Every odd square ends in 1 in base 8, and if you cut off the 1 you have a triangular number. A number is an odd square if and only if it is the square of an odd number, so odd squares have [...]The post Squares, triangles, and octal first appeared on John D. Cook.

Most random number generators are pseudorandom number generators (PRNGs). The distinction may be pedantic or crucial, depending on context. In the context of cryptography, it's critical. For this post, RNG will mean a physical, true random number generator. A PRNG may be suitable for many uses-Monte Carlo simulation, numerical integration, game development, etc.-but not be [...]The post RNG, PRNG, CSPRNG first appeared on John D. Cook.

Let a, b, and c be the sides of a triangle. Let r be the radius of an inscribed circle and R the radius of a circumscribed circle. Finally, let p be the perimeter. Then the previous post said that 2prR = abc. We could rewrite this as 2rR = abc / (a + b [...]The post Triangle circle maximization problem first appeared on John D. Cook.

Let a, b, and c be the sides of a triangle. Let p be perimeter of the triangle. Let r be the radius of the largest circle that can be inscribed in the triangle, and let R be the radius of the circle through the vertices of the triangle. Then all six numbers can be [...]The post Relating six properties of a triangle in one equation first appeared on John D. Cook.

Repetitive text compresses efficiently. Text like the lyrics to Jingle Bells ought to compress more efficiently than ordinary prose, assuming the compression algorithm can exploit the repetition. The idea of the Burrows-Wheeler transform is to permute text in before compressing it. The hope is that the permutation will make the repetition in the text easier [...]The post Preprocessing text to make it more compressible first appeared on John D. Cook.

The original form of radio broadcast was amplitude modulation (AM). With AM radio, the changes in the amplitude of the carrier wave carries the signal you want to broadcast. Frequency modulation (FM) came later. With FM radio, changes to the frequency of the carrier wave carry the signal. I go into the mathematical details of [...]The post Why does FM sound better than AM? first appeared on John D. Cook.

It's interesting to visualize functions of a complex variable, even very simple functions like f(z) = 1/z. The previous post looked at what happens to triangles under the reciprocal map w = 1/z. This post will look at the same map applied to a polar grid, then look at the effect a shift has, replacing [...]The post Shifted reciprocal first appeared on John D. Cook.

The set of functions of the form f(z) = (az + b)/(cz + d) with ad bc are called bilinear transformations or Mobius transformations. These functions have three degrees of freedom-there are four parameters, but multiplying all parameters by a constant defines the same function-and so you can uniquely determine such a function by [...]The post Triangles to Triangles first appeared on John D. Cook.

A golden ellipse is an ellipse whose axes are in golden proportion. That is, the ratio of the major axis length to the minor axis length is the golden ratio = (1 + 5)/2. Draw a golden ellipse and its inscribed and circumscribed circles. In other words draw the largest circle that can fit [...]The post Golden ellipse first appeared on John D. Cook.

Barycentric coordinates are sometimes called area coordinates or areal coordinates in the context of triangle geometry. This is because the barycentric coordinates of a point P inside a triangle ABC correspond to areas of the three triangles PBC, PCA and PAB. (This assumes ABC has unit area. Otherwise divide the area of each of the [...]The post Areal coordinates and ellipse area first appeared on John D. Cook.

Let d(n) be the number of divisors of an integer n. For example, d(12) = 6 because 12 is divisible by 1, 2, 3, 4, 6, and 12. The function d varies erratically as the following plot shows. But if you take the running average of d f(n) = (d(1) + d(2) + d(3) + [...]The post Average number of divisors first appeared on John D. Cook.

Lucas numbers [1] are sometimes called the companions to the Fibonacci numbers. This sequence of numbers satisfies the same recurrence relation as the Fibonacci numbers, Ln+2 = Ln + Ln+1 but with different initial conditions: L0 = 2 and L1 = 1. Lucas numbers are analogous to Fibonacci numbers in many ways, but are also [...]The post Lucas numbers and Lucas pseudoprimes first appeared on John D. Cook.