Calendar Calculations with Cards
As readers of this previous post will know, I’m rather fond of mental calendar calculations. My friend Al Stanger, with whom I share a passion for recreational mathematics, came up with a remarkable...
View ArticleThe Eudoxus reals
Let’s call a function a near-endomorphism of if there is a constant such that for all . The set of near-endomorphisms of will be denoted by . We put an equivalence relation on by declaring that iff the...
View ArticleCounting with martingales
In this post I will provide a gentle introduction to the theory of martingales (also called “fair games”) by way of a beautiful proof, due to Johan Wästlund, that there are precisely labeled trees on...
View ArticleA Fields Medal for June Huh
Congratulations to all of the winners of the 2022 Fields Medal! The only one I know personally, and whose work I have studied in detail, is June Huh. I’m happy both for June himself and for the field...
View ArticleFinitely generated modules over a P.I.D. and the Smith Normal Form
I’m teaching Graduate Algebra this semester, and I wanted to record here the proof I gave in class of the (existence part of the) structure theorem for finitely generated modules over a PID. It’s a...
View ArticleFitting ideals of modules
In my previous post, I presented a proof of the existence portion of the structure theorem for finitely generated modules over a PID based on the Smith Normal Form of a matrix. In this post, I’d like...
View ArticleLinear algebra over rings
Test your intuition: is the following true or false? Assertion 1: If is a square matrix over a commutative ring , the rows of are linearly independent over if and only if the columns of are linearly...
View ArticleAlgebraic Values of Transcendental Functions at Algebraic Points
In honor of Pi Day 2023, I’d like to discuss Hilbert’s 7th Problem, which in an oversimplified (and rather vague) form asks: under what circumstances can a transcendental function take algebraic values...
View ArticleTorsors as proportion spaces
A torsor (or principal homogeneous space) is, informally speaking, a mathematical structure quite similar to a group, but without a natural identity element. More formally, if is a group, a -torsor is...
View ArticlePi and the AGM
In celebration of Pi Day 2024, I would like to explain how the “Arithmetic-Geometric Mean” of Gauss and Legendre can be used to give a rapid method for computing the digits of . By “rapid” here, I mean...
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