John D. Cook

If a password of length n is made up of lower case letters and digits, with each symbol appearing with equal probability, how many times would you expect to shift between letters and digits when typing the password? If you’re typing the password on a mobile device, how many times would you change between alphabetic […]The post Runs of letters and digits in passwords first appeared on John D. Cook.

Blog posts about modal logic Typesetting modal logic Modal and temporal logic for security Word problems, modal logic, and regular expressions Axioms for S1 through S5 Naming and numbering modal logic systems Modal logic and science fiction Dual axioms Temporal and polymodal logics I created the image above by asking DALL-E 2 for an image […]The post Modal logic posts first appeared on John D. Cook.

How does the circumference of a circle vary with its radius? What about the area? The answers are simple and familiar in Euclidean geometry, but not as simple or as familiar in non-Euclidean geometry. This post will look at how circumference and area vary as a function of radius in three spaces: a surface with […]The post Radius, circumference, and area in non-Euclidean geometry first appeared on John D. Cook.

A few days ago I wrote that the spherical counterpart of the Pythagorean theorem is cos(c) = cos(a) cos(b) where sides a and b are measured in radians. If we’re on a sphere of radius R and we measure the sides in terms of arc length rather than in radians, the formula becomes cos(c/R) = […]The post Unified Pythagorean Theorem first appeared on John D. Cook.

According to the latest episode of Eclectic Tech, Richard Feynman argued that Brownian motion cannot do work, but researchers at the University of Arkansas have demonstrated that it can by generating an electric current from Brownian motion in a sheet of graphene. You can read more in the physics journal article by the researchers. Unfortunately […]The post Can Brownian motion do work? first appeared on John D. Cook.

This post is the third in a series of posts on spherical trigonometry. We first looked at the analog of the Pythagorean theorem on a sphere, then the analog of the law of cosines on a sphere. Now we look at the analog of the law of sines on a sphere. As before we denote […]The post Direction between two cities first appeared on John D. Cook.

The previous post looked at the analog of the Pythagorean theorem on a sphere. This post looks at the law of cosines on a sphere. Yesterday we looked at triangles on a sphere with sides a and b meeting at a right angle and hypotenuse c. Denote the angle opposite a side with the capital […]The post Law of cosines on a sphere first appeared on John D. Cook.

Suppose you drive a distance a, turn right, then drive a distance b. How far are you from where you started? If a and b are relatively small, then the answer is given by the Pythagorean theorem. Small here means small relative to the radius of the Earth. Any distance you drive is small enough: […]The post Pythagoras on a sphere first appeared on John D. Cook.

I was playing around with DALL-E last night. I pasted the definition of chimera into DALL-E and the results were bizarre. See this Twitter thread for images. I also played around with some more mnemonic images. Students often memorize the sines and cosines of 30°, 45°, and 60°. I thought about making a mnemonic image […]The post Chimera and sine of 60° first appeared on John D. Cook.

I’ve mostly used Windows and Linux for the last several years. When I needed to get a new laptop recently I got a MacBook Pro. I’ve used a Mac before, but it’s been a while, and so I’m starting over. I can move between Windows and Linux and almost forget what OS I’m using because […]The post Cross platform muscle memory first appeared on John D. Cook.

In base 10, four decibel values are approximately integers. Yesterday I explored whether base 10 was unique in this regard. I defined the value of n decibels in base b to be g(n, b) = bn/b. Sticking in 10 for b gives the usual definition of decibel levels. There are two ways to quantify what […]The post More on near-integer decibels first appeared on John D. Cook.

I got a data file from a client recently in “pickle” format. I happen to know that pickle is a binary format for serializing Python objects, but trying to open a pickle file could be a puzzle if you didn’t know this. Be careful There are a couple problems with using pickle files for data […]The post Dump a pickle file to a readable text file first appeared on John D. Cook.

It’s a curious and convenient fact that many decibel values are close to integers [1]: 3 dB ≈ 2 6 dB ≈ 4 7 dB ≈ 5 9 dB ≈ 8 Is base 10 unique in this regard? If we were to look at the analogs of decibels in other bases, would we see a […]The post Duodecibels first appeared on John D. Cook.

With org-mode you can keep data, code, and documentation in one file. Suppose you have an org-mode file containing the following table. #+NAME: mydata | Drug | Patients | |------+----------| | X | 232 | | Y | 351 | | Z | 117 | Note that there cannot be a blank line between the […]The post Keeping data and code together with org-mode first appeared on John D. Cook.

The song YYZ by Rush opens with a theme based on the rhythm of “YYZ” in Morse code: -.-- -.-- --.. YYZ is the designation for the Toronto Pearson International Airport, the main airport serving Toronto. The idea for the song came from hearing the airport identifier in Morse code. However, the song puts no […]The post YYZ and Morse code first appeared on John D. Cook.

A conversation this morning prompted the question of how many Twitter accounts have between 10,000 and 20,000 followers. I hadn’t thought about the distribution numbers of followers in a while and was curious to revisit the topic. Apparently this question was more popular five years ago. When I did a few searches on the topic, […]The post Twitter follower distribution first appeared on John D. Cook.

I ran across a reference to Mahler the other day, not the composer Gustav Mahler but the mathematician Kurt Mahler, and looked into his work a little. A number of things have been named after Kurt Mahler, including Mahler’s inequality. Mahler’s inequality says the geometric mean of a sum bounds the sum of the geometric […]The post Mahler’s inequality first appeared on John D. Cook.

Code katas are programming exercises intended to develop programming skills, analogous to the way katas develop martial art skills. But literal katas are choreographed. They are rituals rather than problem-solving exercises. There may be an element of problem solving, such as figuring how to better execute the prescribed movements, but katas are rehearsal rather than […]The post Code katas taken more literally first appeared on John D. Cook.

Suppose you have a number of seconds n and you want to convert it to hours and seconds. If you divide n by 3600, the quotient is the number of hours and the remainder is the number of seconds. For example, suppose n = 8072022. Here’s the obvious way to do the calculation in Python: […]The post Seconds to hours first appeared on John D. Cook.

Planck’s constant used to be a measured quantity and now it is exact by definition. h = 6.62607015×10−34 J / Hz Rather than the kilogram being implicit in the units used to measure Planck’s constant, the mass of a kilogram is now defined to be whatever it has to be to make Planck’s constant have […]The post Memorizing Planck’s constant with DALL-E first appeared on John D. Cook.

I recently got an account for using OpenAI’s DALL-E 2 image generator. The example images I’ve seen are sorta surreal combinations of common words, and that made me think of the Major memory system. I’ve written about the Major system before. For example, I give an overview here and I describe how to use it […]The post DALL-E 2 and mnemonic images first appeared on John D. Cook.

Given a random variable X, you often want to compute the probability that X will take on a value less than x or greater than x. Define the functions FX(x) = Prob(X ≤ x) and GX(x) = Prob(X > x) What do you call F and G? I tend to call them the CDF (cumulative […]The post Naming probability functions first appeared on John D. Cook.

Let f be a monotone, strictly convex function on a real interval I and let g be its inverse. For example, we could have f(x) = ex and g(x) = log x. Now suppose we round our results to N digits. That is, instead of working with f and g we actually work with fN […]The post Floating point inverses and stability first appeared on John D. Cook.

The previous post discussed how to use org-mode as a notebook. You can have blocks of code and blocks of results, analogous to cells in a Jupyter notebook. The code and the results export as obvious blocks when you export the org file to another format, such as LaTeX or HTML. And that’s fine for […]The post Inline computed content in org-mode first appeared on John D. Cook.

You can think of org-mode as simply a kind of markdown, a plain text file that can be exported to fancier formats such as HTML or PDF. It’s a lot more than that, but that’s a reasonable place to start. Org-mode also integrates with source code. You can embed code in your file and have […]The post Org-mode as a lightweight notebook first appeared on John D. Cook.

James Gregory’s series for π is not so fast. It converges very slowly and so does not provide an efficient way to compute π. After summing half a million terms, we only get five correct decimal places. We can verify this with the following bc code. s = 0 scale = 50 for(k = 1; […]The post Not so fast first appeared on John D. Cook.

The arithmetic-geometric mean (AGM) of two non-negative real numbers a and b is defined as the limit of the iteration starting with a0 = a and b0 = b and an+1 = ½ (an + bn) bn+1 = √(an bn) for n > 0. This sequence converges very quickly and is useful in numerical algorithms. […]The post Complex AGM first appeared on John D. Cook.

My previous post described Galois connections, and how they generalize a pattern first recognized in the context of Galois theory. This pattern can extended far afield of its initial application to fields and their extensions. For example, you could take a random variable X and think of the pair consisting of its distribution function F: […]The post Galois theory without fields first appeared on John D. Cook.

What are Galois connections and what do they have to do with Galois theory? Galois connections are much more general than Galois theory, though Galois theory provided the first and most famous example of what we now call a Galois connection. Galois connections Galois connections are much more approachable than Galois theory. A Galois connection […]The post Galois connections and Galois theory first appeared on John D. Cook.

We finished a bottle of wine this evening, and I blew across the top as I often do. (Don’t worry: I only do this at home. If we’re ever in a restaurant together, I won’t embarrass you by blowing across the neck of an empty bottle.) The pitch sounded lower than I expected, so I […]The post Helmholtz resonator revisited first appeared on John D. Cook.

Given a sequence a1, a2, a3, … let L be the limit of the ratio of consecutive terms: Then the series converges if L < 1 and diverges if L > 1. However, that’s not the full story. Here is an example from Ernesto Cesàro (1859–1906) that shows the ratio test to be more subtle […]The post Ratio test counterexample first appeared on John D. Cook.

Whittaker and Watson’s analysis textbook is a true classic. My only complaint about the book is that the typesetting is poor. I said years ago that I wish someone would redo the book in LaTeX and touch it up a bit. I found out while writing my previous post that in fact someone has done […]The post Whittaker and Watson first appeared on John D. Cook.

The big idea The Cauchy integral theorem says that the integral of a function around a closed path in the complex plane depends only on the poles of the integrand inside the path. You can change the path itself however you like as long as you don’t change which poles are inside. This observation is […]The post Keyhole contour integrals first appeared on John D. Cook.

The previous post listed three posts I’d written before about images on the covers of math books. This post is about the image on the first edition of Dummit and Foote’s Abstract Algebra. Here’s a version of the image on the cover I recreated using LaTeX. The image on the cover appears on page 495 […]The post Galois diagram first appeared on John D. Cook.

When a math book has an intriguing image on the cover, it’s fun to get to the point in the book where the meaning of the image is explained. I have some ideas for book covers I’d like to write about, but here I’d like to point out three such posts I’ve already written. Weierstrass […]The post Book cover posts first appeared on John D. Cook.

When I had a long commute, I listened to everything I could get my hands on. That included a lot of Teaching Company courses from my local library. A couple of the courses I listened to were Elizabeth Vandiver lecturing on classics. One of the things I remember her talking about was aristeia, a character’s […]The post Aristeia first appeared on John D. Cook.

When you solve systems of linear equations, you probably use Gaussian elimination, even if you don’t call it that. You may learn Gaussian elimination before you see it formalized in terms of matrices. So if you’ve had a course in linear algebra, and you sign up for a course in numerical linear algebra, it’s natural […]The post Gaussian elimination first appeared on John D. Cook.

The numbers 152 through 156 have a lot of small prime factors. I’ll be more explicit about that shortly, but take my word for it for now. John Conway [1] took this simple observation and turned it into a technique for mentally factoring integers. Conway’s factoring method To look for factors of a number n, […]The post Conway’s factoring trick first appeared on John D. Cook.

The tent map is a famous example of a chaotic function. We will show how a tiny modification of the tent maps retains continuity of the function but prevents chaos. The tent map is the function f: [0, 1] → [0, 1] defined by This map has an unstable fixed point at x0 = 2/3. […]The post Adding a tiny trap to stop chaos first appeared on John D. Cook.

Pick four integers a, b, c, and d. Now iterate the procedure that takes these four integers to |b – a|, |c – b|, |d – c|, |a – d| You could think of the four integers being arranged clockwise in a circle, taking the absolute value of difference between each number and its neighbor […]The post Ducci sequences first appeared on John D. Cook.

Suppose a triangle has sides a, b, and c. Label the angles opposite these three sides α, β, and γ respectively. Edsger Dijkstra published (EWD975) a note proving the following extension of the Pythagorean theorem: sgn(α + β – γ) = sgn(a² + b² – c²). Here the sgn function is -1, 0, or 1 […]The post Dijkstra extends Pythagoras first appeared on John D. Cook.

I was reading something this afternoon and ran across φ(φ(m)) and thought that was unusual. I often run across φ(m), the number of positive integers less than m and relative prime to m, but don’t often see Euler’s phi function iterated. Application of φ∘φ This section will give an example of a theorem where φ(φ(m)) […]The post Phi Phi first appeared on John D. Cook.

The SI prefixes giga and tera were adopted in 1960. The prefixes eta and peta were adopted in 1975, and zetta and yotta were adopted in 1991. Following this 15-year cadence, we should have adopted a few more prefixes by now. If we ever do introduce new prefixes, what might they be? The latest prefixes […]The post Speculation on new SI prefixes first appeared on John D. Cook.

A couple years ago I wrote about how NASA was interested in regions bounded by curves of the form For example, here’s a plot for A = 2, B = 1, α = 2.5 and β = 6. That post mentioned a technical report from NASA that explains why these shapes are important in application, […]The post NASA and conformal maps first appeared on John D. Cook.

The previous post gave the details of how Möbius transformations m(z) = (az + b)/(cz + d) transform circles. The image of a circle under a Möbius transformation is either a line or a circle, and in our examples the image will always be a circle. We start with an approximation of the Olympic rings […]The post Transformations of Olympic rings first appeared on John D. Cook.

This post will revisit a previous post in more detail. I’ve written before about how Möbius transformations map circles and lines to circles and lines. In this post I’d like to explore how you’d calculate which circle or line a given circle or line goes to. Given an equation of a line or circle, what […]The post Circles and lines under a Möbius transformation first appeared on John D. Cook.

Let C be a circle in the complex plane with center c and radius r. Assume C does not pass through the origin. Let f(z) = 1/z. Then f(C) is also a circle. We will derive the center and radius of f(C) in this post. *** Our circle C is the set of points z satisfying […]The post Reciprocal of a circle first appeared on John D. Cook.

Last year I wrote several posts about computing ζ(3) where ζ is the Riemann zeta function. For example, this post. It happens that ζ can be evaluated in closed form at positive even arguments, but there’s still a lot of mystery about zeta at positive odd arguments. There’s a way to derive ζ(2n) using contour […]The post Computing zeta at even numbers first appeared on John D. Cook.

The previous post concerned the function h(z) = exp(-1/(1 – z² )). We said that the function is badly behaved near -1 and 1. How badly? The function has essential singularities at -1 and 1. This means that not only does h blow up near these points, it blows up spectacularly. Picard’s theorem says that […]The post Constructive Picard first appeared on John D. Cook.

The word “smooth” in mathematics usually means infinitely differentiable. Occasionally the word is used to mean a function has as many derivatives as necessary, but without being specific about how many derivatives that is. A function is analytic if it has a convergent power series representation at every point of its domain. An analytic function […]The post No analytic bump first appeared on John D. Cook.