2025-04-18 22:48:25
This post will venture further into abstract mathematics than most of my posts. If this isn’t what you’re looking for, you might try browsing here for more concrete articles.
Incidentally, although I’m an applied mathematician, I also appreciate pure math. I imagine most applied mathematicians do as well. But what I do not appreciate is pseudo-applied math, pure math that pretends to be more useful than it is. Pure math is elegant. Applied math is useful. The best math is elegant and useful. Pseudo-applied math is the worst because it is neither elegant nor useful [1].
***
A common theme in pure mathematics, and especially the teaching of pure mathematics, is to strip items of interest down to their most basic properties, then add back properties gradually. One motivation for this is to prove theorems assuming no more structure than necessary.
For example, we can think of the Euclidean plane as the vector space ℝ², but we can think of it as having less structure or more structure. If we just think about adding and subtracting vectors, and forget about scalar multiplication for a moment, then ℝ² is an Abelian group. We could ignore the fact that addition is commutative and think of it simply as a group. We could continue to ignore properties and go down to monoids, semigroups, and magmas.
Going the other direction, there is more to the plane than it’s algebraic structure. We can think of ℝ² as a topological space, in fact a Hausdorff space, and in fact a metric space. We could think of the plane as topological vector space, a Banach space, and more specifically a Hilbert space.
In short, there are many ways to classify the plane as a mathematical object, and we can pick the one best suited for a particular purpose, one with enough structure to get done what we want to get done, but one without additional structure that could be distracting or make our results less general.
A topological group is a set with a topological structure, and a group structure. Furthermore, the two structures must play nicely together, i.e. we require the group operations to be continuous.
Unsurprisingly, an Abelian topological group is a topological group whose group structure is Abelian.
Not everything about Abelian topological groups is unsurprising. The motivation for this post is a surprise that we’ll get to shortly.
A category is a collection of objects and structure-preserving maps between those objects. The meaning of “structure-preserving” varies with context.
In the context of vector spaces, maps are linear transformations. In the context of groups, the maps are homomorphisms. In the context of topological spaces, the maps are continuous functions.
In the previous section I mentioned structures playing nicely together. Category theory makes this idea of playing together nicely explicit by requiring maps to have the right structure-preserving properties. So while the category of groups has homomorphisms and the category of topological spaces has continuous functions, the category of topological groups has continuous homomorphisms.
The category of Abelian groups is much nicer than the category of groups. This takes a while to appreciate. Abelian groups are groups after all, so isn’t the category of Abelian groups just a part of the category of groups? No, it’s more subtle than that. Here’s a post that goes into the distinction between the categories of groups and Abelian groups.
The category of Abelian groups is so nice that the term Abelian category was coined to describe categories that are as nice as the category of Abelian groups. To put it another way, the category of Abelian groups is the archetypical Abelian category.
Now here’s the surprise I promised above: the category of topological Abelian groups is not an Abelian category. More on that at nLab.
If you naively think of an Abelian category as a category containing Abelian things, then this makes no sense. If a grocery cart is a cart containing groceries, then you’d think a gluten-free grocery cart is a grocery cart containing gluten-free groceries.
A category is not merely a container, like a shopping cart, but a working context. What makes an Abelian category special is that it has several nice properties as a working context. If that idea is new to you, I’d recommend carefully reading this post.
[1] This reminds me of the quote by William Morris: “Have nothing in your houses that you do not know to be useful or believe to be beautiful.”
The post Topological Abelian Groups first appeared on John D. Cook.2025-04-18 09:20:03
I was poking around Richard Stanley’s site today and found the following problem on his miscellaneous page.
Find a positive integer n < 10,000,000 such that the first four digits (in the decimal expansion) of n1,000,000 are all different. The problem should be solved in your head.
The solution is not unique, but the solution Stanley gives is n = 1,000,001. Why should that work?
Let M = 1,000,000. We will show that the first four digits of (M + 1)M are 2718.
This uses the fact that (1 + 1/n)n → e as n → ∞. If you’re doing this in your head, as Stanley suggests, you’re going to have to take it on faith that setting n = M will give you at least 4 decimals of e, which it does.
If you allow yourself to use a computer, you can use the bounds
to prove that sticking in n = M gives you a value between 2.718280 and 2.718283. So in fact we get 6 correct decimals, and we only needed 4.
There are many solutions to Stanley’s puzzle, the smallest being 4. The first four digits of 4M are 9802. How could you determine this?
You may not be able to compute 4M and look at its first digits, depending on your environment, but you can tell the first few digits of a number from its approximate logarithm.
log10 4M = M log10 4 = 602059.9913279624.
It follows that
4M = 10602059 100.9913279624 = 9.80229937666385 × 10602059.
There are many other solutions: 7, 8, 12, 14, 16, …
2025-04-17 05:26:45
One day Eric Temple Bell (1883–1960) was looking at the power series for the double exponential function, exp(exp(x)) and noticed a similarity to the power series for exp(x). You can find his account in [1]. He would have calculated the series by hand, but we have the advantage of software like Mathematica.
We can get the first five terms of the series, centered at 0, with the command
Series[Exp[Exp[x]], {x, 0, 5}]
This give us
If you pull out the factor of e from each term, and change the denominators to match those in the power series for exp(x) you get
with integers in all the numerators. It’s not obvious a priori that these numbers should even be integers, but they are,
Bell called the sequence numerators the exponential numbers: 1, 1, 2, 5, 15, 52, … The sequence is now known as the Bell numbers despite Bell’s modesty. Bell wasn’t the first to study this sequence of numbers, but he developed their properties more fully.
Bell numbers come up a lot in applications, which is why Bell wasn’t the first to notice them. (He may have been the first to come to them via their exponential generating function.) For example, the nth Bell number Bn is the number of ways to partition a set of n labeled items. Bn is also the nth moment of a Poisson random variable with λ = 1.
There is a construction of Bell numbers analogous to Pascal’s triangle. Charles Sanders Peirce discovered what we now call Bell’s triangle fifty years before Bell discovered the Bell numbers.
To create Bell’s triangle, start with a row containing only 1.
The first number in each successive row is set to the last number in the previous row.
Then fill in the rest of the row by adding the number to the left and the number directly above
1
1 2
2 3 5
5 7 10 15
15 20 27 37 52
…
The numbers in the first column are the Bell numbers.
[1] E. T. Bell. Exponential Numbers. The American Mathematical Monthly, Vol. 41, No. 7 (Aug. – Sep., 1934), pp. 411–419.
The post Mr. Bell and Bell numbers first appeared on John D. Cook.2025-04-16 23:39:24
In this post we want to count the number of ways to divide a regular polygon [1] into triangles by connecting vertices with straight lines that do not cross.
For a square, there are two possibilities: we either connect the NW and SE corners,
or we connect the SW and NE corners.
For a pentagon, we pick one vertex and connect it to both non-adjacent vertices.
We can do this for any vertex, so there are five possible triangulations. All five triangulations are rotations of the same triangulation. What if we consider these rotations as equivalent? We’ll get to that later.
For a hexagon, things are more interesting. We can again pick any vertex and connect it to all non-adjacent vertices, giving six triangulations.
But there are more possibilities. We could connect every other vertex, creating an equilateral triangle inside. We can do this two ways, connecting either the even-numbered vertices or the odd-numbered vertices. Either triangulation is a rotation of the other.
We can also connect the vertices in a zig-zag pattern, creating an N-shaped pattern inside. We could also rotate this triangulation one or two turns. (Three turns gives us the same pattern again.)
Finally, we could also connect the vertices creating a backward N pattern.
So to recap, we have 2 ways to triangulate a square, 5 ways to triangulate a pentagon, and 6 + 2 + 3 + 3 = 14 ways to triangulate a hexagon. Also, there is only 1 way to triangulate a triangle: do nothing.
Let Cn be the number of ways to triangulate a regular (n + 2)-gon. Then we have C1 = 1, C2 = 2, C3 = 5, and C4 = 14.
In general,
which is the nth Catalan number.
Catalan numbers are the answers to a large number of questions. For example, Cn is also the number of ways to fully parenthesize a product of n + 1 terms, and the number of full binary trees with n + 1 nodes.
The Catalan numbers have been very well studied, and we know that asymptotically
so we can estimate Cn for large n. For example, we could use the formula above to estimate the number of ways to triangulate a 100-gon to be 5.84 ×1055. The 98th Catalan number is closer to 5.77 ×1055. Two takeaways: Catalan numbers grow very quickly, and we can estimate them within an order of magnitude using the asymptotic formula.
Now let’s go back and count the number of triangulations again, considering some variations on a triangulation to be the same triangulation.
We’ll consider rotations of the same triangulation to count only once. So, for example, we’ll say there is only one triangulation of a pentagon and four triangulations of a hexagon. If we consider mirror images to be the same triangulation, then there are three triangulations of a hexagon, counting the N pattern and the backward N pattern to be the same.
The number of equivalence classes of n-gon triangulations, grouping rotations together, is OEIS sequence A001683. Note that the sequence starts at 2.
OEIS gives a formula for this sequence:
where Cx is zero when x is not an integer. So a(6) = 4, as expected.
The number of equivalence classes of n-gon triangulations, grouping rotations and reflections together, is OEIS sequence A000207. Note that the sequence starts at 3.
OEIS gives a formula for this sequence as well:
As before, Cx is zero when x is not an integer. This gives a(6) = 3, as expected.
The formula on the OEIS page is a little confusing since it uses C(n) to denote Cn−2 .
[1] Our polygons do not need to be regular, but they do need to be convex.
The post How many ways can you triangulate a regular polygon? first appeared on John D. Cook.2025-04-15 08:59:19
Last week I wrote about a hypothetical radio station that plays the top 100 songs in some genre, with songs being chosen randomly according to Zipf’s law. The nth most popular song is played with probability proportional to 1/n.
This post is a variation on that post looking at text consisting of the the 1,000 most common words in a language, where word frequencies follow Zipf’s law.
How many words of text would you expect to read until you’ve seen all 1000 words at least once? The math is the same as in the radio station post. The simulation code is the same too: I just changed a parameter from 100 to 1,000.
The result of a thousand simulation runs was an average of 41,246 words with a standard deviation of 8,417.
This has pedagogical implications. Say you were learning a foreign language by studying naturally occurring text with a relatively small vocabulary, such as newspaper articles. You might have to read a lot of text before you’ve seen all of the thousand most common words.
On the one hand, it’s satisfying to read natural text. And it’s good to have the most common words reinforced the most. But it might be more effective to have slightly engineered text, text that has been subtly edited to make sure common words have not been left out. Ideally this would be done with such a light touch that it isn’t noticeable, unlike heavy-handed textbook dialogs.
The post 1000 most common words first appeared on John D. Cook.2025-04-14 22:05:16
As longtime readers of this blog have probably noticed, I like to poke around in Unicode occasionally. It’s an endless system of rabbit holes to explore.
This morning I was looking at the Miscellaneous Mathematical Symbols block. These are mostly obscure symbols, though I’m sure for each symbol that I think is obscure, there is someone out there who uses it routinely.
The only common symbol in this block is ⟂ (U+27C2) for perpendicular. Even so, this symbol is a variation on ⊥ (U+22A5). The distinction is semantic rather than visual: U+22A5 is used for the Boolean value “false.”
In addition to using ⟂ to denote perpendicular lines, some (e.g. Donald Knuth) use the symbol to denote that two integers are relatively prime.
The block contains ⟑ (U+27D1) which is used in geometric algebra for the geometric product, a.k.a. the dot-wedge product. The block also contains the symbol for the dual operator ⟇ (U+27c7), the geometric antiproduct. Incidentally, Eric Lengyel’s Projective Geometric Algebra site officially sponsors these two Unicode symbols.
I’m sure these symbols predate Eric Lengyel’s use of them, but I can only recall seeing them used in his work.
Unicode has four symbols for database joins. The bowtie symbol ⨝ (U+2A1D) is used for inner (natural) joins is in another block. The Miscellaneous Mathematical Symbols block has three other symbols for outer joins: left, right, and full. I posted a table of these on @CompSciFact this morning.
— Computer Science (@CompSciFact) April 14, 2025
The Miscellaneous Mathematical Symbols block also has angle brackets: ⟨ (U+27E8) and ⟩ (U+27E9). These correspond to \langle and \rangle in LaTeX. I’ve used the LaTeX commands, but I wasn’t sure whether I’d ever used the Unicode characters. I searched this blog and found that I did indeed use the characters in my post on the Gram matrix.
