John D. Cook

Define Tn to be the Taylor series for exp(x) truncated after n terms: How does this function compare to its limit, exp(x)? We might want to know because it’s often useful to have polynomial upper or lower bounds on exp(x). For x > 0 it’s clear that exp(x) is larger than Tn(x) since the discarded terms […]

Comment from Paul Phillips on making things easy to understand: It’s always been “We can’t do it that way. It would be too slow.” You know what’s slow? Spending all day trying to figure out why it doesn’t work. That’s slow. That’s the slowest thing I know.

In geometry, you’d say that if a square has side x, then it has area x2. In calculus, you’d say more. First you’d say that if a square has side near x, then it has area near x2. That is, area is a continuous function of the length of a side. As the length of the side […]

The hazard function of a probability distribution is the instantaneous probability density of an event given that it hasn’t happened yet. This works out to be the ratio of the PDF (probability density function) to the CCDF (complementary cumulative density function). For the standard normal distribution, the hazard function is and has a surprisingly simple […]

Sam Northshield [1] came up with the following clever proof that there are infinitely many primes. Suppose there are only finitely many primes and let P be their product. Then The original publication gives the calculation above with no explanation. Here’s a little commentary to explain the calculation. Since prime numbers are greater than 1, sin(π/p) is […]

Suppose you have a flat line f(x) = k and an interval [a, b]. Then the area under the line and over the interval is k times the length of the segment of the line. Surprisingly, the same is true for a catenary with scale k. With the flat line, the length of the segment of the graph is […]

Interesting passage from Small is Beautiful: Economics as if People Mattered by E. F. Schumacher: Nature always, so to speak, knows where and when to stop. There is a measure in all natural things—in their size, speed, or violence. As a result, the system of nature, of which man is a part, tends to be […]

Taylor’s theorem says When does the thing on the left equal the thing on the right? A few things could go wrong: Maybe not all the terms on the right side exist, i.e. the function f might not be infinitely differentiable. Maybe f is infinitely differentiable but the series diverges. Maybe f is infinitely differentiable but the […]

Chris Wiggins gave an excellent talk at Rice this afternoon on data science at the New York Times. In the Q&A afterward, someone asked how you would set up a machine learning algorithm where you’re trying to optimize for outcomes and for information. Here’s how I’ve approached this dilemma in the past. Information and outcomes are not […]

At first “discrete logarithm” sounds like a contradiction in terms. Logarithms aren’t discrete, not as we usually think of them. But you can define and compute logarithms in modular arithmetic. What is a logarithm? It’s the solution to an exponential equation. For example, the logarithm base 10 of 2 is the solution to the equation […]

An interim analysis of a clinical trial is an unusual analysis. At the end of the trial you want to estimate how well some treatment X works. For example, you want to how likely is it that treatment X works better than the control treatment Y. But in the middle of the trial you want to know something more subtle. It’s […]

Suppose you have a fraction a/b where 0 < a < b, and a and b are relatively prime integers. The decimal expansion of a/b either terminates or it has an initial non-repeating part followed by a repeating part. How long is the non-repeating part? How long is the period of the repeating part? The answer depends on the prime factorization […]

People often come to me with R code that’s running slower than they’d like. It’s not unusual to make the code 10 or even 100 times faster by rewriting it in C++. Not all that speed improvement comes from changing languages. Some of it comes from better algorithms, eliminating redundancy, etc. Why bother optimizing? If […]

In their book Computer Age Statistical Inference, Brad Efron and Trevor Hastie give a nice description of neutral networks and deep learning. The knee-jerk response [to neural networks] from statisticians was “What’s the big deal? A neural network is just a nonlinear model, not too different from many other generalizations of linear models.” While this […]

The R language is closely tied to statistics. It’s ancestor was named S, because it was a language for Statistics. The open source descendant could have been named ‘T’, but its creators chose to call it’R.’ Most people learn R as they learn statistics: Here’s a statistical concept, and here’s how you can compute it in R. […]

Here’s one of the things about category theory that takes a while to get used to. Mathematical objects are usually defined internally. For example, the Cartesian product P of two sets A and B is defined to be the set of all ordered pairs (a, b) where a comes from A and b comes from B. The definition of P depends on the elements […]

Kevlin Henney’s keynote at GOTO Copenhagen this year discussed how project time varies as a function of the number of people on the project. The most naive assumption is that the time is inversely proportional to the number of people. That is t = W/n where t is the calendar time to completion, W is a measure […]

This week I attended Mads Torgersen’s talk Why you should take another look at C#. Afterward I asked him about the efficiency of C# on Linux. When I last looked into it, it wasn’t good. A few years ago I asked someone on my team to try running some C# software on Linux using Mono. The code worked […]

I gave a talk this morning at GOTO Copenhagen 2016 on ways to mix R with other programming languages: Rcpp, HaskellR, R Markdown, etc. It’s been fun to see some people I haven’t seen since I spoke at the GOTO and YOW conferences four years ago. Photo above by conference photographer Fritz Schumann.

We’re about to see a lot of new, powerful, inexpensive medical devices come out. And to my surprise, I’ve contributed to a few of them. Growing compute power and shrinking sensors open up possibilities we’re only beginning to explore. Even when the things we want to observe elude direct measurement, we may be able to infer them from […]

For an article to be published, it has to be published somewhere. Each journal has a responsibility to select articles relevant to its readership. Articles that make new connections might be unpublishable because they don’t fit into a category. For example, I’ve seen papers rejected by theoretical journals for being too applied, and the same papers […]

One of my lightbulb moments in college was when my professor, Jim Vick, explained the Lagrange multiplier theorem. The way I’d seen it stated in a calculus text gave me no feel for why it should be true, but his explanation made sense immediately. Suppose f(x) is a function of several variables, i.e. x is a vector, and g(x) = c […]

Suppose you did a pilot study with 10 subjects and found a treatment was effective in 7 out of the 10 subjects. With no more information than this, what would you estimate the probability to be that the treatment is effective in the next subject? Easy: 0.7. Now what would you estimate the probability to be […]

I’m starting a new Twitter account for logic and formal methods: @FormalFact. Expect to see tweets about constructive logic, type theory, formal proofs, proof assistants, etc. The image for the account is a bowtie, a pun on formality. It’s also the symbol for natural join in relational algebra.

The number 3435 has the following curious property: 3435 = 33 + 44 + 33 + 55. It is called a Münchausen number, an allusion to fictional Baron Münchausen. When each digit is raised to its own power and summed, you get the original number back. The only other Münchausen number is 1. At least in […]

Beta reduction is essentially function application. If you have a function described by what it does to x and apply it to an argument t, you rewrite the xs as ts. The formal definition of β-reduction is more complicated than this in order to account for free versus bound variables, but this informal description is sufficient […]

I was catching up on Engines of our Ingenuity episodes this evening when the following line jumped out at me: If I flip a coin a million times, I’m virtually certain to get 50 percent heads and 50 percent tails. Depending on how you understand that line, it’s either imprecise or false. The more times you […]

Experience with the normal distribution makes people think all distributions have (useful) sufficient statistics [1]. If you have data from a normal distribution, then the sufficient statistics are the sample mean and sample variance. These statistics are “sufficient” in that the entire data set isn’t any more informative than those two statistics. They effectively condense […]

The other day I found myself saying that I preferred org-mode files to Jupyter notebooks because with org-mode, what you see is what you get. Then I realized I was using “what you see is what you get” (WYSISYG) in exactly the opposite of the usual sense. Jupyter notebooks are WYSIWYG in the same sense […]

Most programmers are at one extreme or another when it comes to floating point arithmetic. Some are blissfully ignorant that anything can go wrong, while others believe that danger lurks around every corner when using floating point. The limitations of floating point arithmetic are something to be aware of, and ignoring these limitations can cause problems, like crashing […]

Here’s an interesting quote omparing writing proofs and writing programs: Building proofs and programs are very similar activities, but there is one important difference: when looking for a proof it is often enough to find one, however complex it is. On the other hand, not all programs satisfying a specification are alike: even if the […]

Statistics can be useful, even if it’s idealizations fall apart on close inspection. For example, take English letter frequencies. These frequencies are fairly well known. E is the most common letter, followed by T, then A, etc. The string of letters “ETAOIN SHRDLU” comes from the days of Linotype when letters were arranged in that order, […]

When people ask me what calculus is, my usual answer is “the mathematics of change,” studying things that change continually. Algebra is essentially static, studying things frozen in time and space. Calculus studies things that move, shapes that bend, etc. Algebra deals with things that are exact and consequently can be fragile. Calculus deals with […]

As systems get larger and more complex, we need new tools to test whether these systems are correctly specified and implemented. These tools may not be new per se, but they may be applied with new urgency. Dimensional analysis is a well-established method of error detection. Simply checking that you’re not doing something like adding […]

This evening I ran across an unexpected reference to spherical trigonometry: Thomas Hales’ lecture on lessons learned from the formal proof of the Kepler conjecture. He mentions at one point a lemma that was awkward to prove in its original form, but that became trivial when he looked at its spherical dual. The sides of […]

The set of primitive recursive (PR) functions is the smallest set of functions of several integer arguments satisfying five axioms: Constant functions are PR. The function that picks the ith element of a list of n arguments is PR. The successor function S(n) = n+1 is PR. PR functions are closed under composition. PR functions are closed under primitive […]

There’s no notion of continuity in linear algebra per se. It’s not part of the definition of a vector space. But a finite dimensional vector space over the reals is isomorphic to a Euclidean space of the same dimension, and so we usually think of such spaces as Euclidean. (We’ll only going to consider real vector spaces […]

The applied math featured here tends to be fairly sophisticated, but there’s a lot you can do with the basics as we’ll see in the following interview with Trevor Dawson of Borrego Solar, a company specializing in grid-connected solar PV systems. JC: Can you say a little about yourself? TD: I’m Trevor Dawson, I’m 25, born in the […]

Let S be the area of triangle T in three-dimensional space. Let A, B, and C be area of the projections of T to the xy, yz, and xz planes respectively. Then S2 = A2 + B2 + C2. There’s an elegant proof of this theorem here using differential forms. Below I’ll sketch a less elegant but more elementary proof. You could prove the identity above by using the fact that the […]

Neal Stephenson coins a useful word Amistics in his novel Seveneves: … it was a question of Amistics, which was a term that had been coined ages ago by a Moiran anthropologist to talk about the choices that different cultures made as to which technologies they would, and would not, make part of their lives. […]

Suppose you have an n by m chessboard. How many ways can you cover the chessboard with dominoes? It turns out there’s a remarkable closed-form solution: Here are some questions you may have. But what if n and m are both odd? You can’t tile such a board with dominoes. Yes, in that case the formula evaluates to […]

Amplitude modulated signals sound rough to the human ear. The perceived roughness increases with modulation frequency, then decreases, and eventually disappears. The point where roughness reaches is maximum depends on the the carrier signal, but for a 1 kHz tone roughness reaches a maximum for modulation at 70 Hz. Roughness also increases as a function […]

There are two uses of the word scalar, one from linear algebra and another from tensor calculus. In linear algebra, vector spaces have a field of scalars. This is where the coefficients in linear combinations come from. For real vector spaces, the scalars are real numbers. For complex vector spaces, the scalars are complex numbers. […]

In the first post in this series I mentioned several apparently unrelated things that are all called tensors, one of these being objects that behave a certain way under changes of coordinates. That’s what we’ll look at this time. In the second post we said that a tensor is a multilinear functional. A k-tensor takes k vectors and […]

In the previous post, we defined the tensor product of two tensors, but you’ll often see tensor products of spaces. How are these tensor products defined? Tensor product splines For example, you may have seen tensor product splines. Suppose you have a function over a rectangle that you’d like to approximate by patching together polynomials so that […]

The simplest definition of a tensor is that it is a multilinear functional, i.e. a function that takes several vectors, returns a number, and is linear in each argument. Tensors over real vector spaces return real numbers, tensors over complex vector spaces return complex numbers, and you could work over other fields if you’d like. A dot product is […]

The word “tensor” is shrouded in mystery. The same term is applied to many different things that don’t appear to have much in common with each other. You might have heared that a tensor is a box of numbers. Just as a matrix is a rectangular collection of numbers, a tensor could be a cube of […]

“Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.” — Joseph Fourier The above quote makes me think of a connection Fourier made between triangles and thermodynamics. Trigonometric functions were first studied because they relate angles in a right triangle to ratios of the lengths of the triangle’s sides. For […]

“Research is what I’m doing when I don’t know what I’m doing.” — Wernher von Braun I find Shinichi Mochizuki’s proof of the abc conjecture fascinating. Not the content of the proof—which I do not understand in the least—but the circumstances of the proof. Most mathematics, no matter how arcane it appears to outsiders, is […]

I had a discussion recently about whether things are really continuous in the real world. Strictly speaking, maybe not, but practically yes. The same is true of all mathematical properties. There are no circles in the real world, not in the Platonic sense of a mathematical circle. But a circle is a very useful abstraction, […]