John D. Cook

Let π(x) be the number of primes less than x. The simplest form of the prime number theorem says that π(x) is asymptotically equal to x/log(x), where log means natural logarithm. That is, This means that in the limit as x goes to infinity, the relative error in approximating π(x) with x/log(x) goes to 0. […]The post Refinements to the prime number theorem first appeared on John D. Cook.

The previous post looked at all Boolean expressions on three or four variables and how much they can be simplified. The number of Boolean expressions on n variables is and so the possibilities explode as n increases. We could do n = 3 and 4, but 5 would be a lot of work, and 6 […]The post Minimizing random Boolean expressions first appeared on John D. Cook.

In the previous post we looked at how to minimize Boolean expressions using a Python module qm. In this post we’d like to look at how much the minimization process shortens expressions. Witn n Boolean variables, you can create 2^n terms that are a product of distinct variables. You can specify a Boolean function by […]The post How much can Boolean expressions be simplified? first appeared on John D. Cook.

This post will look at how to take an expression for a Boolean function and look for a simpler expression that corresponds to the same function. We’ll show how to use a Python implementation of the Quine-McCluskey algorithm. Notation We will write AND like multiplication, OR like addition, and use primes for negation. For example, […]The post Minimizing boolean expressions first appeared on John D. Cook.

Linear logic uses an unusual symbol, an ampersand rotated 180 degrees, for multiplicative disjunction. The symbol is U+214B in Unicode. I was looking into how to produce this character in LaTeX when I found that the package cmll has two commands that produce this character, one semantic and one descriptive: \parr and \invamp [1]. This […]The post Rotating symbols in LaTeX first appeared on John D. Cook.

Suppose you want to find the smallest number with 5 divisors. After thinking about it a little you might come up with 16, because 16 = 24 and the divisors of 16 are 2k where k = 0, 1, 2, 3, or 4. This approach generalizes: For any prime q, the smallest number with q divisors […]The post The smallest number with a given number of divisors first appeared on John D. Cook.

Both Pfizer and Moderna have announced recently that their SARS-COV2 vaccine candidates reduce the rate of infection by over 90% in the active group compared to the control (placebo) group. That’s great news. The vaccines may turn out to be less than 90% effective when all is said and done, but even so they’re likely […]The post Good news from Pfizer and Moderna first appeared on John D. Cook.

The other day I saw an article about some math test and thought “I bet I’d blow that away now.” Anyone who has spent a career using some skill ought to blow away an exam intended for people who have been learning that skill for a semester. However, after thinking about it more, I’m pretty […]The post I think I'll pass first appeared on John D. Cook.

A couple years ago I wrote a blog post looking at how close the quaternions come to commuting. That is, the post looked at the average norm of xy – yx. A related question would be to ask how often quaternions do commute, i.e. the probability that xy – yx = 0 for randomly chosen […]The post Probability of commuting first appeared on John D. Cook.

There are simple rules for telling whether a number is divisible by 2, 3, 4, 5, and 6. A number is divisible by 2 if its last digit is divisible by 2. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 4 if […]The post Test for divisibility by 13 first appeared on John D. Cook.

This evening I ran across a paper on an unusual coordinate system that creates interesting graphs based from simple functions. It’s called “circular coordinates,” but this doesn’t mean polar coordinates; it’s more complicated than that. [1] Here’s a plot reproduced from [1], with some color added (the default colors matplotlib uses for multiple plots). The […]The post Some mathematical art first appeared on John D. Cook.

Let T(N) be the number of distinct (non-congruent) triangles with integer sides and perimeter N. For example, T(12) = 3 because there are three distinct triangles with integer sides and perimeter 12. There’s the equilateral triangle with sides 4 : 4 : 4, and the Pythagorean triangle 3 : 4 : 5. With a little […]The post Counting triangles with integer sides first appeared on John D. Cook.

I ran across a paper [1] this morning on the differential equation y‘ = sin(xy). The authors recommend having students explore numerical solutions to this equation and discover theorems about its solutions. Their paper gives numerous theorems relating solutions and the hyperbolas xy = a: how many times a solution crosses a hyperbola, at what […]The post Ripples and hyperbolas first appeared on John D. Cook.

When the rule for stopping an experiment depends on the data in the experiment, the results could be biased if the stopping rule isn’t taken into account in the analysis [1]. For example, suppose Alice wants to convince Bob that π has a greater proportion of even digits than odd digits. Alice: I’ll show you […]The post Informative stopping first appeared on John D. Cook.

California’s CCPA regulation has been amended to say that data considered deidentified under HIPAA is considered deidentified under CCPA. The amendment was proposed last year and was finally signed into law on September 25, 2020. This is good news because it’s relatively clear what deidentification means under HIPAA compared to CCPA. In particular, HIPAA has […]The post Expert determination for CCPA first appeared on John D. Cook.

I imagine most programmers who develop an interest in category theory do so after hearing about monads. They ask someone what a monad is, and they’re told that if they really want to know, they need to learn category theory. Unfortunately, there are couple unnecessary difficulties anyone wanting to understand monads etc. is likely to […]The post Category theory for programmers made easier first appeared on John D. Cook.

The previous post looked at random Fibonacci sequences. These are defined by f1 = f2 = 1, and fn = fn-1 ± fn-2 for n > 2, where the sign is chosen randomly to be +1 or -1. Conjecture: Every integer can appear in a random Fibonacci sequence. Here’s why I believe this might be […]The post Is every number a random Fibonacci number? first appeared on John D. Cook.

The Fibonacci numbers are defined by F1 = F2 = 1, and for n > 2, Fn = Fn-1 + Fn-2. A random Fibonacci sequence f is defined similarly, except the addition above is replaced with a subtraction with probability 1/2. That is, f1 = f2 = 1, and for n > 2, fn = […]The post Random Fibonacci numbers first appeared on John D. Cook.

I’ve been thinking about Edsger Dijkstra lately because I suspect some of the ideas he developed will be useful for a project I’m working on. While searching for some of Dijkstra’s writings I ran across the article Edsger Dijkstra: The Man Who Carried Computer Science on His Shoulders. It occurred while reading this article that […]The post Edsger Dijkstra, blogger first appeared on John D. Cook.

My wife and I were talking this morning and the phrase”less disingenuous” came up. I thought about how sometimes a positive word fades into obscurity while the negative form lives on. The first example that came to mind is gruntled vs disgruntled. Yes, the former is an English word, but a rare one. Here’s a […]The post Gruntled vs disgruntled first appeared on John D. Cook.

The first time I went a few days without blogging, someone sent me a concerned email asking whether I was OK. And in case anyone has had similar thoughts this week, I’m doing well. I’m busy, though my rate of blogging is fairly independent of how busy I am. Sometimes being busy gives me lots […]The post Doing well first appeared on John D. Cook.

In a comment to my previous post on quatrefoils, Jan Van lint suggested a different equation for quatrefoils: r = a + |cos(2θ)| Here are some examples of how these curves look for varying values of a. As a increases, the curves get rounder. We can quantify this by looking at the angle between the […]The post More fun with quatrefoils first appeared on John D. Cook.

Most people have heard of word problems, but not as many have heard of the word problem. If you’re imagining that the word problem is some superlatively awful word problem, I can assure you it’s not. It’s both simpler and weirder than that. The word problem is essentially about whether you can always apply algebraic […]The post The word problem first appeared on John D. Cook.

There’s an ancient saying “Whom the gods would destroy they first make mad.” (Mad as in crazy, not mad as in angry.) I wrote a variation of this on Twitter: Whom the gods would destroy, they first give real-time analytics. Having more up-to-date information is only valuable up to a point. Past that point, you’re […]The post Real-time analytics first appeared on John D. Cook.

In his book The Algorithm Design Manual, Steven Skiena has several sections called “War Stories” where he talks about his experience designing algorithms for clients. Here’s an excerpt of a story about finding the best airline ticket prices. “Look,” I said at the start of the first meeting. “This can’t be so hard. Consider a […]The post Naive modeling first appeared on John D. Cook.

I often work in a sort of amphibious environment, using Unix software on Windows. As you can well imagine, this causes headaches. But I’ve found such headaches are generally more manageable than the headaches from alternatives I’ve tried. On the Windows command line, you can type the name of a file and Windows will open […]The post Opening Windows files from bash and eshell first appeared on John D. Cook.

A Pythagorean triple is a set of positive integers that can be the lengths of sides of a right triangle, i.e. numbers a, b, and c such that a² + b² = c². A primitive Pythagorean triple (PPT) is a Pythagorean triple whose elements are relatively prime. For example, (50, 120, 130) is a Pythagorean […]The post Generating all primitive Pythagorean triples with linear algebra first appeared on John D. Cook.

A “rose” in mathematics is typically a curve with polar equation r = cos(kθ) where k is a positive integer. If k is odd, the resulting graph has k “petals” and if k is even, the plot has 2k petals. Sometimes the term rose is generalized to the case of non-integer k. This is the […]The post Playing around with a rational rose first appeared on John D. Cook.

I was reading The 99% Invisible City this evening, and there was a section on quatrefoils. Here’s an example of a quatrefoil from Wikipedia. There’s no single shape known as a quatrefoil. It’s a family of shapes that look something like the figure above. I wondered how you might write a fairly simple mathematical equation […]The post Quatrefoils first appeared on John D. Cook.

I’m working on a project these days where I’ve used four different kinds of matrix product, which made me wonder if there’s another kind of product out there that I could find some use for. In the process of looking around for other matrix products, I ran across the Kronecker sum. I’ve seen Kronecker products […]The post Kronecker sum first appeared on John D. Cook.

This post is a follow-on to the previous post on perfectly nonlinear functions. In that post we defined a way to measure the degree of nonlinearity of a function between two Abelian groups. We looked at functions that take sequences of four bits to a single bit. In formal terms, our groups were GF(24) and […]The post Nonlinear mod 5 first appeared on John D. Cook.

The other day I heard someone suggest that a good cocktail party definition of cryptography is puzzle making. Cryptographers create puzzles that are easy to solve given the key, but ideally impossible without the key. Linearity is very useful in solving puzzles, and so a puzzle maker would like to create functions that are as […]The post Perfectly nonlinear functions first appeared on John D. Cook.

Here’s an amusing little Python program to convert integers to Roman numerals: def roman(n): print(chr(0x215F + n)) It only works if n is between 1 and 12. That’s because Unicode contains characters for the Roman numerals I through XII. Here are the characters it produces: Ⅰ Ⅱ Ⅲ Ⅳ Ⅴ Ⅵ Ⅶ Ⅷ Ⅸ Ⅹ […]The post Python one-liner to print Roman numerals first appeared on John D. Cook.

This post will take a look at simple bidirectional text, such as a bit of English inside an Arabic document, or a few words of Hebrew inside a French document. If you want to explore the subject in all its complexity, see Unicode Standard Annex 9. You may not need to do anything special to […]The post Bidirectional text first appeared on John D. Cook.

A simple linear regression model has the form y = μ + βx + ε. This means that the output variable y is a linear function of the input variable x, plus some error term ε that is randomly distributed. There’s a common misunderstanding over whose error the error term is. A naive view is […]The post Understanding statistical error first appeared on John D. Cook.

Alexander Pope famously said A little learning is a dangerous thing; Drink deep, or taste not the Pierian spring: There shallow draughts intoxicate the brain, And drinking largely sobers us again. I’ve been thinking lately about why a little knowledge is often a dangerous thing, and here’s what I’ve come to. Any complex system has […]The post Why a little knowledge is a dangerous thing first appeared on John D. Cook.

Yesterday I blogged about an exercise in the book The Cauchy-Schwarz Master Class. This post is about another exercise from that book, exercise 5.8, which is to prove Kantorovich’s inequality. Assume and for non-negative numbers pi. Then where is the arithmetic mean of m and M and is the geometric mean of m and M. […]The post Expected value of X and 1/X first appeared on John D. Cook.

There’s a theorem that’s often used and assumed to be true but rarely stated explicitly. I’m going to call it “the baseball inequality” for reasons I’ll get to shortly. Suppose you have two lists of k positive numbers each: and Then This says, for example, that the batting average of a baseball team is somewhere […]The post The baseball inequality first appeared on John D. Cook.

A catenary with scale a is the graph of the function f(x; a) = a cosh(x/a) – a. The x and the a are separated by a semicolon rather than a comma to imply that we think of x as the variable and a as a parameter. This graph passes through the origin, i.e. for any […]The post Solving for the catenary scale parameter first appeared on John D. Cook.

ASCII codes may seem arbitrary when you’re looking at decimal values, but they make more sense in hex [1]. For example, the ASCII value for 0 is 48. Why isn’t it zero, or at least a number that ends in zero? Well it is, in hex: 0x30. And the codes are in consecutive order, so […]The post Alphabets and Unicode first appeared on John D. Cook.

There are a couple ways in which a measurement might not be straight. Yesterday I wrote a blog post about not measuring straight toward your target. You’d like to measure from (0, 0) to (x, 0), but something is in the way, and so you measure from (0, 0) to (x, y), where y is […]The post How much does it matter if the measuring tape sags? first appeared on John D. Cook.

This post will start with a motivating example, looking at measuring a room in inches and in feet. Then we will segue into a discussion of contravariance and covariance in the simplest setting. Then we will discuss contravariant and covariant tensors more generally. Using a tape measure In my previous post, I explained why it […]The post From tape measures to tensors first appeared on John D. Cook.

Suppose a contractor is measuring the length of a wall. He starts in one corner of the room, and lets out a tape measure heading for the other end of the wall. But something is in the way, so instead of measuring straight to the corner, he measures to a point near the corner on […]The post It doesn't matter much if the tape is straight first appeared on John D. Cook.

The previous post mentioned a Math Overflow question about unexpected mathematical images, and reproduced one that looks like field grass. This post reproduces another set of images from that post. Start anywhere in the complex plane with integer coordinates and walk west one unit at a time until you run into Gaussian prime [1]. Then […]The post At the next prime, turn left first appeared on John D. Cook.

Math Overflow has an interesting question about unexpected mathematical images. Here’s a response from Payam Seraji that was easy to code up. Here’s the code that produced the image. from numpy import * import matplotlib.pyplot as plt t = linspace(0, 39*pi/2, 1000) x = t*cos(t)**3 y = 9*t*sqrt(abs(cos(t))) + t*sin(0.2*t)*cos(4*t) plt.plot(x, y, c="green") plt.axes().set_aspect(0.3) plt.axis('off')The post Simple equations whose plot looks like field grass first appeared on John D. Cook.

The previous post was a visual introduction to bilinear transformations, a.k.a. Möbius transformations or fractional linear transformations. This post is a short follow-up focused more on calculation. A bilinear transformation f has the form where ad – bc ≠ 0. Inverse The inverse of f is given by The transformation f is defined everywhere except […]The post Constructing bilinear transformations first appeared on John D. Cook.

This post expands on something I said in passing yesterday. I said in the body of the post that … the image of a circle in the complex plane under a Möbius transformation is another circle. and added in a footnote that For this to always be true, you have to include a line as […]The post Circles to Circles first appeared on John D. Cook.

I said something to my wife this evening to the effect that it’s best for employees to have one or at most two projects at a time. Two is good because you can switch off when you’re tired of one project or if you’re waiting on input. But with three or more projects you spend […]The post Simultaneous projects first appeared on John D. Cook.

There are many ways the basic derivative can be generalized: partial derivatives, directional derivatives, covariant derivatives, etc. These all reduce to the basic derivative under the right circumstances. The Schwarzian derivative is not like that. It’s not a generalization of the familiar derivative but rather a differential operator analogous to a derivative. The Schwarzian derivative […]The post Schwarzian derivative first appeared on John D. Cook.

According to the Python Cookbook, “Mixing Unicode and regular expressions is often a good way to make your head explode.” It is thus with fear and trembling that I dip my toe into using Unicode with Greek and Hebrew. I heard recently that there are anomalies in the Hebrew Bible where the final form of […]The post Searching Greek and Hebrew with regular expressions first appeared on John D. Cook.