When I started this blog, almost 17 years ago, I posted nearly every day. The first time I went a couple days without posting I got a message from someone asking if everything was OK.

I’ve slowed down since then, and even more lately. Last week I was busy with professional work, and this week I’ve been busy with personal work.

I’ve never had a schedule for this blog: I write when I feel like writing, which usually means several times a week. I’ve averaged 3 posts every 4 days since I started writing here. Presumably I’ll pick back up next week. We’ll see.

If you’d like to be notified when I write something here, you have several options. You could subscribe via RSS to hear of every post as soon as it is published. If you’d rather get a notification of a few posts at a time, along with a little introduction to each, you could subscribe to my newsletter.

You can also follow me on social media, primarily X but also Mastodon and Bluesky.

I stumbled on a post on X this morning, a commentary on the photo of RFK eating food from McDonalds that has been making the rounds.

This photo divides Puritans from Southerners.

Puritans think because RFK Jr is on the side of health food he can never commit such a “sin.”

Southerners think a rare treat is fine & it’s more important to be gracious in a gathering than to be a stickler.

I can’t read minds, and so I don’t know what RFK’s motives were, but I do believe the post is right about Southern graciousness. Of course this disposition is not limited to the South, nor does everyone in the South live this way.

A puritanical mindset, more typical of metaphorical Puritans than literal Puritans, is flat: all virtues are equally important. A gracious mindset is hierarchical: some virtues take precedent over others. And in a traditional Southern mindset, not offending hosts or guests has high priority.

I’ve had a Bluesky account for over a year, but never posted much on it. Recently I noticed I’d gotten more followers on Bluesky and thought I might try posting there more often.

I am not moving to Bluesky. I have orders of magnitude more followers on X than on Bluesky and so I will focus my effort on X. But I expect to post on Bluesky occasionally.

I learned this morning that there is a bridge that will automatically post your Mastodon content to Bluesky. You should be able to follow

johndcook.mathstodon.xyz.ap.brid.gy

on Bluesky to see the content that I post on my Mastodon account at

johndcook.mathstodon.xyz

Note the server is mathstodon, not mastodon.

It may take some time before the bridge works. I suppose information needs to be propagated analogous to how DNS works. At the time of writing, I can see the bridge account by going to the Bluesky page for the bridge, but I’ve not yet been able to pull up the account in the Bluesky app.

Update: The bridge account is working from the Bluesky app. I’ve posted two articles to Mastodon and verified that they appear in the bridge account.

Just to be clear, we’re talking about two separate Bluesky accounts. My Bluesky account is

johndcook.bsky.social

but my Mastodon posts will be automatically posted to a separate account,

johndcook.mathstodon.xyz.ap.brid.gy.

Here’s a simple calculation that I’ve done often enough that I’d like to save the result for my future reference and for the benefit of anyone searching on this.

A linear combination of sines and cosines

a sin(x) + b cos(x)

can be written as a sine with a phase shift

A sin(x + φ).

Going between {a, b} and {A, φ} is the calculation I’d like to save. For completeness I also include the case

A cos(x + ψ).

Derivation

Define

f(x) = a sin(x) + b cos(x)

and

g(x) = A sin(x + φ).

Both functions satisfy the differential equation

y″ + y = 0

and so f = g if and only if f(0) = g(0) and f′(0) = g′(0).

Setting the values at 0 equal implies

b = A sin(φ)

and setting the derivatives at 0 equal implies

a = A cos(φ).

Taking the ratio of these two equations shows

b/a = tan(φ)

and adding the squares of both equations shows

a² + b² = A².

Equations

First we consider the case

a sin(x) + b cos(x) = A sin(x + φ).

Sine with phase shift

If a and b are given,

A = √(a² + b²)

and

φ = tan⁻¹(b / a).

If A and φ are given,

a = A cos(φ)

and

b = A sin(φ)

from the previous section.

Cosine with phase shift

Now suppose we want

a sin(x) + b cos(x) = A cos(x + ψ)

If a and b are given, then

A = √(a² + b²)

as before and

ψ = − tan⁻¹(a / b).

If A and ψ are given then

a = − A sin(ψ)

and

b = A cos(ψ).

Related posts

• Convert between real and complex Fourier series
• Converting between error function and normal CDF [pdf]
• Converting trig identities to hyperbolic identities
• Technical notes

I recently stumbled upon the Postage Stamp Problem. Given two relatively prime positive numbers a and b, show that any sufficiently large number N, there exists nonnegative integers x and y such that

ax + by = N.

I initially missed the constraint that x and y must be positive, in which result is well known (Bézout’s lemma) and there’s no requirement for N to be large. The positivity constraint makes things more interesting.

The problem is called the Postage Stamp Problem because it says that given any two stamps whose values are relatively prime, say a 5¢ stamp and a 21¢ stamp, you can make any sufficiently large amount of postage using just those two stamps.

A natural question is how large is “sufficiently large,” and the answer turns out to be all integers larger than

ab − a − b.

So in our example, you cannot make 79¢ postage out of 5¢ and 21¢ stamps, but you can make 80¢ or any higher amount.

If you’ve been reading this blog for a while, you may recognize this as a special case of the Chicken McNugget problem, which you can think of as the Postage Stamp problem with possibly more than two stamps.

Related posts

• The Chicken McNugget monad
• Cicadas and chicken nuggets
• Possible and actual football scores

It’s easier to go up a ladder than to come down, literally and metaphorically.

Gian-Carlo Rota made a profound observation on the application of theory.

One frequently notices, however, a wide gap between the bare statement of a principle and the skill required in recognizing that it applies to a particular problem.

This isn’t quite what he said. I made two small edits to generalize his actual statement. He referred specifically to the application of the inclusion-exclusion principle to problems in combinatorics. Here is his actual quote from [1].

One frequently notices, however, a wide gap between the bare statement of the principle and the skill required in recognizing that it applies to a particular combinatorial problem.

This post will expand a little on Rota’s original statement and a couple applications of the more general version of his statement.

The inclusion-exclusion principle

I don’t think Rota would have disagreed with my edited version of his statement, but it’s interesting to think of his original context. The inclusion-exclusion principle seems like a simple problem solving strategy: you may count a set of things by first over-counting, then correcting for the over-counting.

For example, a few days ago I wrote about a graph created by turning left at the nth step if n is divisible by 7 or contains a digit 7. Suppose you wanted to count how many times you turn in the first 100 steps. You could count the number of positive integers up to 100 that are divisible by 7, then the number that contain the digit 7, then subtract the number that both are divisible by 7 and contain a 7.

You can carry this a step further by over-counting, then over-correcting for your over-counting, then correcting for your over-correction. This is the essence of the following probability theorem.

The inclusion-exclusion principle a clever idea, but not that clever. And yet Rota discusses how this idea was developed over decades into the Möbius inversion principle, which has diverse applications, including Euler characteristic and the calculus of finite differences.

Bayes’ theorem

Bayesian statistics is a direct application of Bayes’ theorem. Bayes’ theorem is a fairly simple idea, and yet people make careers out of applying it.

When I started working in the Biostatistics Department at MD Anderson, a bastion of Bayesian statistics, I was surprised how subtle Bayesian statistics is. I probably first saw Bayes’ theorem as a teenager, and yet it was not easy to wrap my head around Bayesian statistics. I would think “This is simple. Why is this hard?” The core principle was simple, but the application was not trivial.

Newtonian mechanics

When I took introductory physics, we would get stuck on homework problems and ask our professor for help. He would almost always begin by saying “Well, F = ma.”

This was infuriating. Yes, we know F = ma. But how does that let us solve this problem?!

There’s more to Newtonian mechanics than Newton’s laws, a lot more. And most of it is never made explicit. You pick it up by osmosis after you’ve worked hundreds of exercises.

Related posts

• There’s more going on here
• Applied abstraction

[1] G. C. Rota. On the Foundations of Combinatorial Theory I. Theory of Möbius Functions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 2 (1964) 340–368.

I learned to use the command line utility make in the context of building C programs. The program make reads an input file to tell it how to make things. To make a C program, you compile the source files into object files, then link the object files together.

You can tell make what depends on what, so it will not do any unnecessary work. If you tell it that a file foo.o depends on a file foo.c, then it will rebuild foo.o if that file is older than the file foo.c that it depends on. Looking at file timestamps allows make to save time by not doing unnecessary work. It also makes it easier to dictate what order things need to be done in.

There is nothing about make that is limited to compiling programs. It simply orchestrates commands in a declarative way. Because you state what the dependencies are, but let it figure if and when each needs to be run, a makefile is more like a Prolog program than a Python script.

I have a client that needs a dozen customized reports, each of which requires a few steps to create, and the order of the steps is important. I created the reports by hand. Then, of course, something changed and all the reports needed to be remade. So then I wrote a Python script to manage the next build. (And the next, and the next, …). But I should have written a makefile. I’m about to have to revise the reports, and this time I’ll probably use a makefile.

Here’s a very simple example of using a makefile to create documents. For more extensive examples of how to use makefiles for a variety of tasks, see How I stopped worrying and loved Makefiles. [1]

Suppose you need to create a PDF and a Microsoft Word version of a report from a LaTeX file [2].

all: foo.pdf foo.docx

foo.pdf: foo.tex
	pdflatex foo.tex

foo.docx: foo.tex
	pandoc foo.tex --from latex --to docx > foo.docx

clean:
	rm foo.aux foo.log

If the file foo.pdf does not exist, or exists but it older than foo.tex, the command make foo.pdf will create the PDF file by running pdflatex. If foo.pdf is newer than foo.tex then running make foo.pdf will return a message

make: 'foo.pdf' is up to date.

If you run make with no argument, it will build both foo.pdf and foo.docx as needed.

The command make clean deletes auxiliary files created by pdflatex. This shows that a make action does have to “make” anything; it simply runs a command.

[1] The blog post title is an allusion to the 1964 movie Dr. Strangelove or: How I Learned to Stop Worrying and Love the Bomb.

[2] I use LaTeX for everything I can. This saves time in the long run, if not in the moment, because of consistency. I used to use Word for proposals and such, and LaTeX for documents requiring equations. My workflow became more efficient when I switched to using LaTeX for everything.