John D. Cook

The previous post looked at ways of associating a product of four things. This post will look at associating more things, and visualizing the associations. The number of ways to associate a product of n+1 things is the nth Catalan number, defined by The previous post looked at products of 4 things, and there are […]The post Associahedron first appeared on John D. Cook.

There are five ways to parenthesize a product of four things: ((ab)c)d (ab)(cd) (a(b(cd)) (a(bc))d (a((bc)d) In a context where multiplication is not associative, the five products above are not necessarily the same. Maybe all five are different. This post will give two examples where the products above are all different: octonions and matrix multiplication. […]The post Non-associative multiplication first appeared on John D. Cook.

Consider the second order linear differential equation You could think of y as giving the height of a mass attached to a spring at time t where k is the stiffness of the spring. The solutions are sines and cosines with frequency √k. Think about how the solution depends on the spring constant k. If […]The post Slowing oscillations first appeared on John D. Cook.

I’d like to do two things in this post. I’ll present a fun bit of mental math, then use it to illustrate a few deeper points. The story starts with a Twitter thread I wrote on @AlgebraFact. The Strait of Gibraltar is 13 km wide. How wide is that in miles? You can convert km […]The post Mental math trick and deeper issues first appeared on John D. Cook.

Here’s a problem that came up in my work today. Suppose you have a normal random variable X and you observe X in n discrete bins. The first bin is the left α tail, the last bin is the right α tail, and the range between the two tails is divided evenly into n-2 intervals. […]The post Information in a discretized normal first appeared on John D. Cook.

When I took my first abstract algebra course, we had a homework question about subgroups. Someone in the class whined that the professor hadn’t told us yet what a subgroup was. My immediate thought was “I bet you could guess. Sub things are all the same. A sub-X is an X contained inside another X. […]The post A general theory of sub-things first appeared on John D. Cook.

I so despise pop-ups asking me to subscribe to a site’s newsletter that I’ve probably erred on the side of making it too hard to find out how to subscribe to this blog. At the bottom of the “writing” menu is a subscribe link. You can subscribe three different ways: Subscribe to posts via RSS […]The post How to subscribe first appeared on John D. Cook.

Here’s an interesting paragraph from Handbook of Floating Point Arithmetic by Jean-Michel Muller et al. Many common transformations of a code into some naively equivalent code become impossible if one takes into account special cases such as NaNs, signed zeros, and rounding modes. For instance, the expression x + 0 is not equivalent to the expression x […]The post Non-equivalent floating point numbers first appeared on John D. Cook.

My posts on modal logic have mostly been about monomodal logic, logic with one modal operator. This may not seem accurate because I’ve talked about □ (“box”) and ◇ (“diamond”). But these are really just one mode: you can define either in terms of the other. ◇p = ¬ □ ¬p □p = ¬ ◇ […]The post Temporal and polymodal logics first appeared on John D. Cook.

Yesterday I wrote about how you could use the tr utility to find the dual of a modal logic expression. Today I’ll show how you could use tr to work with permutations. This post is a hack, and that’s the fun of it. In the logic post we were using tr to swap characters. More […]The post Permutations at the command line first appeared on John D. Cook.

Fairly often I want to find out whether there is an HTML entity for a given Unicode character, or given an HTML entity I want to look up its Unicode value. For example, ∃ has Unicode value U+2203. I might want to look up whether there is an HTML entity for this. There is, and […]The post Look up HTML entity or Unicode first appeared on John D. Cook.

My recent post Reading plots of a complex function was based on ideas from a book by Elias Wegert. Thanks to a comment on that post I discovered that Wegert and his colleagues have put out a 2022 calendar with images created from plots of complex functions. This year’s calendar, and previous calendars going back […]The post Complex analysis calendar first appeared on John D. Cook.

Axioms in modal logic often say that one sequence of boxes and diamonds in front of a proposition p implies another sequence of boxes and diamonds in front of p. For example, Axiom 4 says □ p → □□ p and Axiom 5 says ◇p → □◇p. Every axiom has a dual form. The dual […]The post Dual axioms in modal logic first appeared on John D. Cook.

Word problems Suppose you have a sequence of symbols and a set of rewriting rules for replacing some patterns of symbols with others. Now you’re given two such sequences. Can you tell whether there’s a way to turn one of them into the other? This is known as the word problem, and in general it’s […]The post Word problems, logic, and regular expressions first appeared on John D. Cook.

The previous post pointed out the analogy between models for modal logic (i.e. Kripke semantics) and science fiction. Rules for relationships between points in a Kripke model are analogous to rules for interplanetary travel in a fictional universe. This post will expand on this last point. The following cube shows the relationships between eight different […]The post Modal axioms and rules for interplanetary travel first appeared on John D. Cook.

Modal logic extends classical logic by adding one or more modes. If there’s only one mode, it’s usually denoted □. Curiously, □ can have a wide variety of interpretations, and different interpretations motivate different axioms for how □ behaves. Modal logic is not one system but an infinite number of systems, depending on your choice […]The post Modal Logic and Science Fiction first appeared on John D. Cook.

This post is about understanding the following plot. If you’d like to read more along these lines, see [1]. The plot was made with the following Mathematica command. ComplexPlot[HankelH1[3, z], {z, -8 - 4 I, 4 + 4 I}, ColorFunction -> "CyclicArg", AspectRatio -> 1/2] The plot uses color to represent the phase of the […]The post Reading plots of a complex function first appeared on John D. Cook.

I recently ran across an article on MathWorld about Glasser’s function. The function is defined by Here’s a plot of G(x) along with its asymptotic value 2√(x/π). This plot was made with the following Mathematica commands. G[x_] := NIntegrate[Sin[t Sin[t]], {t, 0, x}] Plot[{G[x], 2 Sqrt[x/Pi]}, {x, 0, 20}] According to MathWorld, The integral cannot […]The post Glasser’s function first appeared on John D. Cook.

The previous post presented an approximation for the steady-state wait time in queue with a general probability distribution for inter-arrival times and for service times, i.e. a G/G/s queue where s is the number of servers. This post will give an upper bound for the wait time. Let σ²A be the variance on the inter-arrival […]The post Upper bound on wait time for general queues first appeared on John D. Cook.

Textbook presentations of queueing theory start by assuming that the time between customer arrivals and the time to serve a customer are both exponentially distributed. Maybe these assumptions are realistic enough, but maybe not. For example, maybe there’s a cutoff on service time. If someone takes too long, you tell them there’s no more you […]The post Multiserver queues with general probability distributions first appeared on John D. Cook.

If a single server is being pushed to the limit, adding a second server can drastically reduce wait times. This is true whether the server is a human serving food or a computer serving web pages. I first wrote about this here and I give more details here. What if you add an extra server […]The post Queueing and Economies of Scale first appeared on John D. Cook.

Linear logic has connectives not used in classical logic. The connectives ⊗and & are conjunctions, ⊕ and ⅋ are disjunctions, and ! and ? are analogous to the modal operators ◻ and ◇ (necessity and possibility). Another way to classify the connectives is to say ⊕ and & are called additive, ⊗ and ⅋ are […]The post Linear logic arithmetic first appeared on John D. Cook.

The other day I wrote about orthogonal Latin squares. Two Latin squares are orthogonal if the list of pairs of corresponding elements in the two squares contains no repeats. A self-orthogonal Latin square (SOLS) is a Latin square that is orthogonal to its transpose. Here’s an example of a self-orthogonal Latin square: 1 7 6 […]The post Self-Orthogonal Latin Squares first appeared on John D. Cook.

There’s a rule of systems thinking that improving part of a system can often make the system as a whole worse. One example of this is Braess’ Paradox that adding roads can make traffic worse, and closing roads can improve traffic. Another example is the paradox of enrichment: Increasing the food available to an ecosystem […]The post Better parts, worse system first appeared on John D. Cook.

A blog post about queueing theory that I wrote back in 2008 continues to be popular. The post shows that when a system is close to capacity, adding another server dramatically reduces the wait time. In that example, going from one teller to two tellers doesn’t make service twice as fast but 93 times as […]The post Queueing theory equations first appeared on John D. Cook.

Swanson’s rule of thumb [1] says that the mean of a moderately skewed probability distribution can be approximated by the weighted average of the 10th, 50th, and 90th percentile, with weights 0.3, 0.4, and 0.3 respectively. Because it is based on percentiles, the rule is robust to outliers. Swanson’s rule is used in the oil […]The post Swanson’s rule of thumb first appeared on John D. Cook.

I was stunned when my client said that a database query that I asked them to run would cost the company $100,000 per year. I had framed my question in the most natural way, not thinking that at the company’s scale it would be worth spending some time thinking about the query. Things have somewhat […]The post Random sampling to save money first appeared on John D. Cook.

Suppose you create an n × n Latin square filled with the first n letters of the Roman alphabet. This means that each letter appears exactly once in each row and in each column. You could repeat the same exercise only using the Greek alphabet. Is it possible to find two n × n Latin […]The post Greco-Latin squares and magic squares first appeared on John D. Cook.

In a n×n Latin square, each of the numbers 1 through n appears exactly once in each row and column. For example, the 5 × 5 square below is a Latin square. If we placed a rook on each square numbered 1, the rooks would not attack each other since no two rooks would be […]The post Latin squares and 3D chess first appeared on John D. Cook.

Five years ago I recommended the book Learning Base R. Here’s the last paragraph of my review: Now there are more books on R, and some are more approachable to non-statisticians. The most accessible one I’ve seen so far is Learning Base R by Lawrence Leemis. It gets into statistical applications of R—that is ultimately why […]The post New R book first appeared on John D. Cook.

A while back I wrote about computational survivalism, being prepared to work productively in a restricted environment. The first time I ran into computational survivalism was when someone said to me “I prefer Emacs, but I use vi because I only want to use tools I can count on being installed everywhere.” I thought that […]The post Computational asceticism first appeared on John D. Cook.

Here’s a surprising theorem [1]. (Beatty’s theorem) Let a and b be any pair of irrational numbers greater than 1 with 1/a + 1/b = 1. Then the sequences { ⌊na⌋ } and { ⌊nb⌋ } contain every positive integer without repetition. Illustration Here’s a little Python code to play with this theorem.We set a […]The post Beatty’s theorem first appeared on John D. Cook.

In his book Mastering Regular Expressions, Jeffrey Friedl uses corner quotes to delimit regular expressions. Here’s an example I found by opening his book a random: ⌜(\.\d\d[1-9]?)\d*⌟ The upper-left corner at the beginning and the lower-right corner at the end are not part of the regular expression. This particularly comes in handy if a regular […]The post Corner quotes in Unicode first appeared on John D. Cook.

When does the equation x2 + 7 = 2n have integer solutions? This is an interesting question, but why would anyone ask it? This post looks at three paths that have led to this problem. Ramanujan Ramanujan [1] considered this problem in 1913. He found five solutions and conjectured that there were no more. Then […]The post Three paths converge first appeared on John D. Cook.

I was thumbing through a new book on causal inference, The Effect by Nick Huntington-Klein, and the following diagram caught my eye. Then it made my head hurt. It looks like a category theory diagram. What’s that doing in a book on causal inference? And if it is a category theory diagram, something’s wrong. Either […]The post One diagram, two completely different meanings first appeared on John D. Cook.

My favorite part of this post by Ian Miellis the introduction. The article is about shell commands, but the introduction brings up a more general point. … there are thousands of reusable patterns I’ve picked up … Unfortunately, I’ve forgotten about 95% of them. … The point is to reflect on what actually stuck, so […]The post Memorable techniques first appeared on John D. Cook.

I recently received a review copy of Andrew Witt’s new book Formulations: Architecture, Mathematics, and Culture. The Hankel function on the cover is the first clue that this book contains some advanced mathematics. Or rather, it references some advanced mathematics. I’ve only skimmed the book so far, but I didn’t see any equations. Hankel functions […]The post Architecture and Math first appeared on John D. Cook.

I’ve written a couple posts about the Golay problem recently, first here then here. The problem is to find all values of N and n such that is a power of 2, say 2p. Most solutions apparently fall into three categories: n = 0 or n = N, N is odd and n = (N-1)/2, […]The post Turning the Golay problem sideways first appeared on John D. Cook.

I played around with what I’m calling the Golay problem over Christmas and wrote a blog post about it. I rewrote the post as I learned more about the problem due to experimentation and helpful feedback via comments and Twitter. In short, the Golay problem is to classify the values of N and n such […]The post Update on the Golay problem first appeared on John D. Cook.

Stanislaw Ulam once said Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals. There is only one way to be linear, but there are many ways to not be linear. A similar observation applies to non-classical logic. There are many ways to not be […]The post Names and numbers for modal logic axioms first appeared on John D. Cook.

The previous post looked at two of the five Lagrange points of the Sun-Earth system. These points, L1 and L2, are located on either side of Earth along a line between the Earth and the Sun. The third Lagrange point, L3, is located along that same line, but on the opposite side of the Sun. […]The post When do two-body systems have stable Lagrange points? first appeared on John D. Cook.

The James Webb Space Telescope (JWST) is on its way to the Lagrange point L2 of the Sun-Earth system. Objects in this location will orbit the Sun at a fixed distance from Earth. There are five Lagrange points, L1 through L5. The points L1, L2, and L3 are unstable, and points L4 and L5 are […]The post Finding Lagrange points L1 and L2 first appeared on John D. Cook.

These were the most popular posts from 2021 on this site, listed in chronological order. Simple gamma approximation Floor, ceiling, bracket Evolution of random number generators Format text in twitter Where has all the productivity gone? Computing zeta(3) Morse code palindromes Much less than, much greater than Monads and macros Partial functionsThe post Most popular posts of 2021 first appeared on John D. Cook.

Sine and cosine have power series with simple coefficients, but tangent does not. Students are shocked when they see the power series for tangent because there is no simple expression for the power series coefficients unless you introduce Bernoulli numbers, and there’s no simple expression for Bernoulli numbers. The perception is that sine and cosine […]The post The exception case is normal first appeared on John D. Cook.

Marcel Golay noticed that and realized that this suggested it might be possible to create a perfect code of length 23. I wrote a Twitter thread on this that explains how this relates to coding theory (and to sporadic simple groups). This made me wonder how common it is for cumulative sums of binomial coefficients […]The post Binomial sums and powers of 2 first appeared on John D. Cook.

The Christmas hymn “O Come, O Come, Emmanuel” is a summary of the seven “O Antiphons,” sung on December 17 though 23, dating back to the 8th century [1]. The seven antiphons are O Sapientia O Adonai O Radix Jesse O Clavis David O Oriens O Rex Gentium O Emmanuel The corresponding verses of “O […]The post O Come, O Come, Emmanuel: condensing seven hymns into one first appeared on John D. Cook.

Yesterday I wrote about how to multiply octets of real numbers, the octonions. Today I’ll show how to create an error correcting code from the octonions. In fact, we’ll create a perfect code in the sense explained below. We’re going to make a code out of octonions over a binary field. That is, we’re going […]The post Error correcting code from octonions first appeared on John D. Cook.

Yesterday I wrote about the fact that quaternions, unlike complex numbers, can form conjugates via a series of multiplications and additions. This post will show that you can do something similar with octonions. If x is an octonion x = r0 e0 + r1 e1 + … + r7 e7 where all the r‘s are […]The post Conjugate theorem for octonions first appeared on John D. Cook.

This post will present a way of multiplying octonions that’s easy to remember. Please note that there are varying conventions for how to define multiplication for octonions [1]. Octonions The complex numbers have one imaginary unit i, and the quaternions have three: i, j, and k. The octonions have seven, and so it makes sense […]The post How to multiply octonions first appeared on John D. Cook.

The conjugate of a complex number is the complex number Taking the conjugate flips over a complex number, taking its reflection in the real axis. Multiplication stretches and rotates complex numbers, and addition translates complex numbers. You can’t flip the complex plane over by any series of dilatations, rotations, and translations. The situation is different […]The post Complex Conjugates versus Quaternion Conjugates first appeared on John D. Cook.