John D. CookModify

2026-03-03 00:08:49

In grad school I specialized in differential equations, but never worked with delay-differential equations, equations specifying that a solution depends not only on its derivatives but also on the state of the function at a previous time. The first time I worked with a delay-differential equation would come a couple decades later when I did some modeling work for a pharmaceutical company.

Large delays can change the qualitative behavior of a differential equation, but it seems plausible that sufficiently delays should not. This is correct, and we will show just how small “sufficiently small” is in a simple special case. We’ll look at the equation

x′(t) = a x(t) + b x(t − τ)

where the coefficients a and b are non-zero real constants and the delay τ is a positive constant. Then [1] proves that the equation above has the same qualitative behavior as the same equation with the delay removed, i.e. with τ = 0, provided τ is small enough. Here “small enough” means

−1/e < bτ exp(−aτ) < e

and

aτ < 1.

There is a further hypothesis for the theorem cited above, a technical condition that holds on a nowhere dense set. The solution to a first order delay-differential like the one we’re looking at here is not determined by an initial condition x(0) = x₀ alone. We have to specify the solution over the interval [−τ, 0]. This can be any function of t, subject only to a technical condition that holds on a nowhere-dense set of initial conditions. See [1] for details.

Example

Let’s look at a specific example,

x′(t) = −3 x(t) + 2 x(t − τ)

with the initial condition x(1) = 1. If there were no delay term τ, the solution would be x(t) = exp(1 − t). In this case the solution monotonically decays to zero.

The theorem above says we should expect the same behavior as long as

−1/e < 2τ exp(3τ) < e

which holds as long as τ < 0.404218.

Let’s solve our equation for the case τ = 0.4 using Mathematica.

tau = 0.4
solution = NDSolveValue[
    {x'[t] == -3 x[t] + 2 x[t - tau], x[t /; t <= 1] == t }, 
    x, {t, 0, 10}]
Plot[solution[t], {t, 0, 10}, PlotRange -> All]

This produces the following plot.

The solution initially ramps up to 1, because that’s what we specified, but it seems that eventually the solution monotonically decays to 0, just as when τ = 0.

When we change the delay to τ = 3 and rerun the code we get oscillations.

[1] R. D. Driver, D. W. Sasser, M. L. Slater. The Equation x’ (t) = ax (t) + bx (t – τ) with “Small” Delay. The American Mathematical Monthly, Vol. 80, No. 9 (Nov., 1973), pp. 990–995

2026-03-02 02:15:16

After writing the previous post, I poked around in the bash shell documentation and found a handy feature I’d never seen before, the shortcut ~-.

I frequently use the command cd - to return to the previous working directory, but didn’t know about ~- as a shotrcut for the shell variable $OLDPWD which contains the name of the previous working directory.

Here’s how I will be using this feature now that I know about it. Fairly often I work in two directories, and moving back and forth between them using cd -, and need to compare files in the two locations. If I have files in both directories with the same name, say notes.org, I can diff them by running

    diff notes.org ~-/notes.org

I was curious why I’d never run into ~- before. Maybe it’s a relatively recent bash feature? No, it’s been there since bash was released in 1989. The feature was part of C shell before that, though not part of Bourne shell.

2026-03-01 02:20:37

I’ve never been good at shell scripting. I’d much rather write scripts in a general purpose language like Python. But occasionally a shell script can do something so simply that it’s worth writing a shell script.

Sometimes a shell scripting feature is terse and cryptic precisely because it solves a common problem succinctly. One example of this is working with file extensions.

For example, maybe you have a script that takes a source file name like foo.java and needs to do something with the class file foo.class. In my case, I had a script that takes a La TeX file name and needs to create the corresponding DVI and SVG file names.

Here’s a little script to create an SVG file from a LaTeX file.

    #!/bin/bash

    latex "$1"
    dvisvgm --no-fonts "${1%.tex}.dvi" -o "${1%.tex}.svg"

The pattern ${parameter%word} is a bash shell parameter expansion that removes the shortest match to the pattern word from the end of the expansion of parameter. So if $1 equals foo.tex, then

    ${1%.tex}

evaluates to

    foo

and so

${1%.tex}.dvi

and

${1%.tex}.svg

expand to foo.dvi and foo.svg.

You can get much fancier with shell parameter expansions if you’d like. See the documentation here.

2026-02-26 09:10:37

The post A curious trig identity contained the theorem that for real x and y,

This theorem also holds when sine is replaced with hyperbolic sine.

The post Trig of inverse trig contained a table summarizing trig functions applied to inverse trig functions. You can make a very similar table for the hyperbolic counterparts.

The following Python code doesn’t prove that the entries in the table are correct, but it likely would catch typos.

    from math import *

    def compare(x, y):
        print(abs(x - y) < 1e-12)

    for x in [2, 3]:
        compare(sinh(acosh(x)), sqrt(x**2 - 1))
        compare(cosh(asinh(x)), sqrt(x**2 + 1))
        compare(tanh(asinh(x)), x/sqrt(x**2 + 1))
        compare(tanh(acosh(x)), sqrt(x**2 - 1)/x)                
    for x in [0.1, -0.2]:
        compare(sinh(atanh(x)), x/sqrt(1 - x**2))
        compare(cosh(atanh(x)), 1/sqrt(1 - x**2)) 

Related post: Rule for converting trig identities into hyperbolic identities

2026-02-25 19:00:54

I ran across an old article [1] that gave a sort of multiplication table for trig functions and inverse trig functions. Here’s my version of the table.

I made a few changes from the original. First, I used LaTeX, which didn’t exist when the article was written in 1957. Second, I only include sin, cos, and tan; the original also included csc, sec, and cot. Third, I reversed the labels of the rows and columns. Each cell represents a trig function applied to an inverse trig function.

The third point requires a little elaboration. The table represents function composition, not multiplication, but is expressed in the format of a multiplication table. For the composition f( g(x) ), do you expect f to be on the side or top? It wouldn’t matter if the functions commuted under composition, but they don’t. I think it feels more conventional to put the outer function on the side; the author make the opposite choice.

The identities in the table are all easy to prove, so the results aren’t interesting so much as the arrangement. I’d never seen these identities arranged into a table before. The matrix of identities is not symmetric, but the 2 by 2 matrix in the upper left corner is because

sin(cos⁻¹(x)) = cos(sin⁻¹(x)).

The entries of the third row and third column are not symmetric, though they do have some similarities.

You can prove the identities in the sin, cos, and tan rows by focusing on the angles θ, φ, and ψ below respectively because θ = sin⁻¹(x), φ = cos⁻¹(x), and ψ = tan⁻¹(x). This shows that the square roots in the table above all fall out of the Pythagorean theorem.

See the next post for the hyperbolic analog of the table above.

[1] G. A. Baker. Multiplication Tables for Trigonometric Operators. The American Mathematical Monthly, Vol. 64, No. 7 (Aug. – Sep., 1957), pp. 502–503.

2026-02-25 07:33:41

Here is an identity that doesn’t look correct but it is. For real x and y,

I found the identity in [1]. The author’s proof is short. First of all,

Then

Taking square roots completes the proof.

Now note that the statement at the top assumed x and y are real. You can see that this assumption is necessary by, for example, setting x = 2 and y = i.

Where does the proof use the assumption that x and y are real? Are there weaker assumptions on x and y that are sufficient?

[1] R. M. Robinson. A curious trigonometric identity. American Mathematical Monthly. Vol 64, No 2. (Feb. 1957). pp 83–85