John D. Cook

For a positive integer n, the nth harmonic number is defined to be the sum of the reciprocals of the first n positive integers: How might we extend this definition so that n does not have to be a positive integer? First approach One way to extend harmonic numbers is as follows. Start with the [...]The post Extending harmonic numbers first appeared on John D. Cook.

Very often when a number is large, and we don't know or care exactly how large it is, we can model it as infinite. This may make no practical difference and can make calculations much easier. I give several examples of this in the article Infinite is easier than big. When you run across a [...]The post A note on Zipf's law first appeared on John D. Cook.

I recently evaluated two software applications designed to find PII (personally identifiable information) in free text using natural language processing. Both failed badly, passing over obvious examples of PII. By contrast, I also tried natural language processing software on a nonsensical poem, it the software did quite well. Doctor's notes It occurred to me later [...]The post Natural language processing and unnatural text first appeared on John D. Cook.

I recently ran across a paper on typesetting rare Chinese characters. From the abstract: Written Chinese has tens of thousands of characters. But most available fonts contain only around 6 to 12 thousand common characters that can meet the needs of everyday users. However, in publications and information exchange in many professional fields, a number [...]The post How rare is it to encounter a rare word? first appeared on John D. Cook.

Machine learning models occasionally memorize training data. Under the right prompt, a model could return portions of the training data verbatim. If a large language model is trained on deidentified medical data, along with data that overlaps with the medical data, it could potentially leak details of a person's medical history. I'm not saying that [...]The post How an LLM might leak medical data first appeared on John D. Cook.

A few days ago I wrote about U-statistics, statistics which can be expressed as the average of a symmetric function over all combinations of elements of a set. V-statistics can be written as an average of over all products of elements of a set. Let S be a statistical sample of size n and let [...]The post V-statistics first appeared on John D. Cook.

Yesterday I wrote about how you could use the spaCy Python library to find proper nouns in a document. Now suppose you want to refine this and find proper nouns that are the subjects of sentences or proper nouns that are direct objects. This post was motivated by a project in which I needed to [...]The post Filtering on how words are being used first appeared on John D. Cook.

I saw a headline saying that donating blood lowers the level of forever chemicals in your body. This post will give a back-of-the-envelope calculation to show that this idea is plausible. Suppose there are chemicals in your bloodstream that do not break down and that your body will not filter out. Suppose you have about [...]The post Forever chemicals and blood donation first appeared on John D. Cook.

Suppose you want to find all the proper nouns in a document. You could grep for every word that starts with a capital letter with something like grep '\b[A-Z]\w+' but this would return the first word of each sentence in addition to the words you're after. You could grep for capitalized words that are not [...]The post Searching for proper nouns first appeared on John D. Cook.

John Tukey developed his so-called g-and-h distribution to be very flexible, having a wide variety of possible values of skewness and kurtosis. Although the reason for the distribution's existence is its range of possible skewness and values, calculating the skewness and kurtosis of the distribution is not simple. Definition Let be the function of [...]The post Moments of Tukey's g-and-h distribution first appeared on John D. Cook.

A symmetric function is a function whose value is unchanged under every permutation of its arguments. The previous post showed how three symmetric functions of the sides of a triangle a + b + c ab + bc + ac abc are related to the perimeter, inner radius, and outer radius. It also mentioned that [...]The post Symmetric functions and U-statistics first appeared on John D. Cook.

Suppose a triangle T has sides a, b, and c. Let s be the semi-perimeter, i.e. half the perimeter. Let r be the inner radius, the radius of the largest circle that can fit inside T. Let R be the outer radius, the radius of the smallest circle that can enclose T. Then three simple [...]The post Relating perimeter, inner radius, outer radius, and sides of a triangle first appeared on John D. Cook.

My two previous posts looked at experiments with ChatGPT and Google Bard. This post will look at redoing the same experiments with Microsoft's Bing Chat: looking for mnemonic encodings and simplifying Boolean expressions. When you open up Bing chat you can select a conversational style: More creative More balanced More precise I chose more precise" [...]The post Experiments with Bing chat first appeared on John D. Cook.

I was curious how well LLMs would do at minimizing a Boolean expression, that is, taking a Boolean expression and producing a smaller equivalent expression. I didn't expect good performance because this problem is more about logic than recall, but sometimes LLMs surprise you, so I wanted to give it a chance. I thought it [...]The post Boolean function minimization with AI first appeared on John D. Cook.

The Major mnemonic system encodes numbers as words in order to make them easier to remember. Digits correspond to consonant sounds (not spellings) as explained here. You can use the system ad hoc, improvising an encoding of a word as needed, or you can memorize canonical encodings of numbers, also known as pegs. Pegs have [...]The post Large language models and mnemonics first appeared on John D. Cook.

Motivating examples The addition theorem for cosine says that and the addition theorem for hyperbolic cosine is analogous, though with a sign change. An addition theorem is a theorem that relates a function's value at x + y to its values at x and at y. The squaring function satisfies a very simple addition theorem [...]The post When does a function have an addition theorem? first appeared on John D. Cook.

In HTML you can mark the language of a piece of text by putting it inside span tags and setting the lang attribute to a two-letter abbreviation. For example, <span lang="fr">Allons enfants de la Patrie, Le jour de gloire est arrive !<span> indicates that the first two lines of the French national anthem are in [...]The post How to mark a language in HTML first appeared on John D. Cook.

I mentioned in the previous post that I had been poking around in HTML entities and noticed symbols for Fourier transforms and such. I also noticed HTML entities for Cyrillic letters. These entities have the form & + transliteration + cy;. For example, the Cyrillic letter is based on the Greek letter and [...]The post Russian transliteration hack first appeared on John D. Cook.

I was looking through HTML entities and ran across &Fouriertrf;. I searched for all entities ending in trf; and also found &Mellintrf;, &Laplacetrf;, and &zeetrf;. Apparently trf" stands transform" and these symbols are intended to be used to represent the Fourier transform, Mellin transform, Laplace transform, and z-transform. You would not know from the Unicode [...]The post Symbols for transforms first appeared on John D. Cook.

The previous post discussed an algorithm developed by Charles Dodgson (better known as Lewis Carroll) for computing determinants. The key identity for proving that Dodgson's algorithm is correct involves the Desnanot-Jacobi identity from 1841. The identity is intimidating in its symbolic form and yet easy to visualize. In algebraic form the identity says Here a [...]The post Visualizing a determinant identity first appeared on John D. Cook.

Charles Dodgson, better known by his pen name Lewis Carroll, discovered a method of calculating determinants now known variously as the method of contractants, Dodgson condensation, or simply condensation. The method was devised for ease of computation by hand, but it has features that make it a practical method for computation by machine. Overview The [...]The post How Lewis Carroll computed determinants first appeared on John D. Cook.

An elegant algebraic identity says If x is the vector [a b] and y is the vector [c d] then this identity can be written where the dot indicates the usual dot product. I posted this on Twitter the other day. Gram matrix Now suppose that x and y are vectors of any length n. [...]The post Gram matrix first appeared on John D. Cook.

This post will explore how the trigonometric functions and the hyperbolic trigonometric functions relate to the Jacobi elliptic functions. There are six circular functions: sin, cos, tan, sec, csc, and cot. There are six hyperbolic functions: just stick an h' on the end of each of the circular functions. There are an infinite number of [...]The post Circular, hyperbolic, and elliptic functions first appeared on John D. Cook.

I posted some notes here about mnemonics for trig identities and hyperbolic trig identities.The post Mnemonic hexagons first appeared on John D. Cook.

Suppose you want to fit a line to three data points. If no line passes exactly through your points, what's the best compromise you could make? Chebyshev suggested the best thing to do is find the minmax line, the line that minimizes the maximum error. That is, for each candidate line, find the vertical distance [...]The post Best line to fit three points first appeared on John D. Cook.

Let A, B, C, and D be positive numbers. Then Lehman's inequality says Proof by circuit This inequality can be proved analytically, but Lehman's proof is interesting because he uses electrical circuits [1]. Let A, B, C, and D be the resistances of resistors arranges as in the circuit on the left. Resistors R1 and [...]The post Lehman's inequality, circuits, and LaTeX first appeared on John D. Cook.

A few weeks ago I wrote about the contraharmonic mean. Given two positive numbers a and b, their contraharmonic mean is This mean has the unusual property that increasing one of the two inputs could decrease the mean. If you take the partial derivative with respect to a you can see that it is zero [...]The post Contraharmonic mean quirk first appeared on John D. Cook.

I had a tweet Twitter go viral yesterday, at least relatively viral. Elon Musk could tweet a punctuation mark and get an orders of magnitude more traffic, but this was viral by my standards [1]. That schadenfreude-like feeling when you realize something you felt you should learn but didn't is now obsolete." Apparently this resonated [...]The post Technological schadenfreude first appeared on John D. Cook.

Differential equations in the complex plane are different from differential equations on the real line. Suppose you have an nth order linear differential equation as follows. The real case If the as are continuous, real-valued functions on an open interval of the real line, then there are n linearly independent solutions over that interval. The [...]The post Complex differential equations first appeared on John D. Cook.

I create nearly all my documents in LaTeX, even documents that might be easier to create in Word. The reason is that even if a particular document would be easier to write in Word, my workflow is more efficient if everything is in LaTeX. LaTeX makes small, plain text files that work well with version [...]The post Convert LaTeX to Microsoft Word first appeared on John D. Cook.

Years ago I wrote about how you can make a decent approximation to the Gaussian density by shifting and scaling the cosine function. This observation dates back at least as far as 1961. Turns out you can do even better by raising the cosine term to a power. Not only can you get a good [...]The post Cosine power approximation to Gaussian density first appeared on John D. Cook.

The two latest blog post have elaborated on the fact that when a function satisfies a second order differential equation, once you've evaluated the function and its derivative you get the second derivative for free, or at least for cheap. Sometimes functions are so closely related that software libraries bundle them together. For example, in [...]The post Buy one, get one free first appeared on John D. Cook.

In an undergraduate course on differential equations, you see an equation, you find a solution. Wax on, wax off. It would be reasonable to assume that the differential equation is simply an obstacle to overcome and that once you have the solution, the differential equation has done its job and can be discarded. It would [...]The post What good is a DE once you have the solution? first appeared on John D. Cook.

Newton's method for solving f(x) = 0 converges quickly once it starts to converge. Specifically, it converges quadratically: the error is roughly squared at each step. This means that the number of correct decimal places doubles at each iteration of Newton's method. Doubling the number of decimal places is nice, but what if we could [...]The post Halley's variation on Newton's method first appeared on John D. Cook.

It's hard to imagine doing anything like Midjourney's image generation without neural networks. The same is true of ChatGPT's text generation. But a lot of business tasks do not require AI, and in fact would be better off not using AI. Three reasons why: Statistical models are easier to interpret. When a model has a [...]The post Three advantages of non-AI models first appeared on John D. Cook.

Suppose you have a bivariate normal distribution with correlation where Then the level sets of the density function are ellipses, and the eccentricity e of the ellipses is related to the correlation by Plots For example, suppose = 0.8. Here's a plot of the density function f(x, y). And here is a [...]The post Eccentricity of bivariate normal level sets first appeared on John D. Cook.

The incomplete beta function is defined by It is called incomplete because the limit of integration doesn't necessarily run all the way up to 1. When z = 1 the incomplete beta function is simply the beta function discussed in the previous post [1]. The incomplete beta function is proportional to the CDF of a [...]The post Bounds on the incomplete beta function (beta CDF) first appeared on John D. Cook.

The beta function B(x, y) is defined by and is the normalizing constant for the beta probability distribution. It is related to the gamma function via The beta function comes up often in applications. It can be challenging to work with, however, and so estimates for the function are welcome. The function 1/xy gives simple [...]The post Upper and lower bounds on the beta function first appeared on John D. Cook.

Last week the function came up in a blog post as the solution to the equation n!=bn. After writing that post I stumbled upon an article [1] that begins by saying, in my notation, The function b(x) ... has many applications in pure and applied mathematics. The article mentions a couple applications but mostly focuses [...]The post nth root of n! first appeared on John D. Cook.

Neural network activation functions transform the output of one layer of the neural net into the input for another layer. These functions are nonlinear because the universal approximation theorem, the theorem that basically says a two-layer neural net can approximate any function, requires these functions to be nonlinear. Activation functions often have two-part definitions, defined [...]The post Activation functions and Iverson brackets first appeared on John D. Cook.

The previous post mentioned the group S4, the group of all permutations of a set with four elements. This post will show a way to visualize this group. The Mathematica command CayleyGraph[ SymmetricGroup[4], VertexLabels -> Placed["Name", Center], VertexSize -> 0.4] generates the graph below. This is an interesting image, but what does it mean? The [...]The post Cayley graphs in Mathematica first appeared on John D. Cook.

I had a fleeting thought about how little I use group theory when I realize I used it just this week. A couple days ago I needed to know which permutations of 4 elements commute with reversal. Ifr takes a sequence and reverses it, I need to find all permutations p such that p [...]The post Permutations and centralizers in Mathematica first appeared on John D. Cook.

I recently stumbled across the identity while reading [1]. Here x is the floor of x, the largest integer not larger than x. My first thought was that this was hard to believe. Although the floor function is very simple, its interactions with other functions tends to be complicated. I was surprised that such a [...]The post Floors and roots first appeared on John D. Cook.

Here's a graphical way to multiply two positive numbers a and b using the parabola y = x^2. Start at the origin, move a units to the left, then go up vertically to the parabola, and draw a point. Go back to the origin, move b units to the right, go up vertically to the [...]The post Multiplication via parabola first appeared on John D. Cook.

When I worked at MD Anderson Cancer Center, we spent a lot of compute cycles evaluating the function g(a, b, c, d), defined as the probability that a sample from a beta(a, b) random variable is larger than a sample from a beta(c, d) random variable. This function was often in the inner loop of [...]The post Beta inequalities and cross ratios first appeared on John D. Cook.

The nth Dedekind number M(n) is the number of monotone Boolean functions of n variables. The 9th Dedekind number was recently computed to be M(9) = 286386577668298411128469151667598498812366. The previous post defines monotone Boolean functions and explicitly enumerates the functions for one, two, or three variables. As that post demonstrates, M(1) = 3, M(2) = 6, [...]The post The 10th Dedekind number first appeared on John D. Cook.

The 9th Dedekind number was recently computed. What is a Dedekind number and why was it a big deal to compute just the 9th one of them? We need to define a couple terms before we can define Dedekind numbers. A Boolean function is a function whose inputs are 0's and 1's and whose output [...]The post Enumerating monotone Boolean functions first appeared on John D. Cook.

The motion of a falling body of mass m is given by where the term -kvr accounts for drag due to air resistance. One can derive r = 2 under simple physical assumptions, but if I remember correctly other values of r may be more realistic in certain circumstances. I don't know much about the [...]The post Drag equation exponent variation first appeared on John D. Cook.

Suppose you select a 100-digit number at random. How many distinct prime factors do you think it would have? The answer is smaller than you might think: most likely between 5 and 6. The function (n) returns the number of distinct prime factors of n [1]. A theorem of Hardy and Ramanujan says that as [...]The post Numbers don't typically have many prime factors first appeared on John D. Cook.

The previous post mentioned that 24! 1024 and 25! 1025. For every n, there is some base b such that n! = bn. For example, 30! 1230. It's easy to find b [1]: What's interesting is that b is very nearly a linear function of n. In hindsight it's clear that this [...]The post Every factorial is a power first appeared on John D. Cook.