Some time ago, I took an exploratory look at the BLL (Basic Library List) published by the MAA (Mathematical Association of America). In this list, each book is tagged with “topics,” and can have a single topic or multiple topics.
At the time I wrote my previous post, I didn’t know of a good way to visualize the distribution of topics. I could isolate each topic (for example “History of Mathematics”) and count the books that had this topic, but this doesn’t show combinations of topics and how they overlap in a nice way.
It is interesting to look at what happens when you try to “fit” the absolute value function using polynomials. Pick some number \(N\) of points on the absolute value function, and you can find a polynomial of degree \(N - 1\) that goes through those points.
Here is the absolute value function, \(f(x) = |x|\):
For simplicity, I’ll use an odd number of evenly spaced points so that the resulting polynomial will be even, since \(|x|\) is even.
The problem Consider a function \(f: [0, 1] \to \mathbb{R}\) that is continuous with \(f(0) = f(1)\). It is possible to prove that for each \(n \in \mathbb{N}\), there exist \(x_n, y_n \in [0, 1]\) such that \(|x_n - y_n| = 1/n\) and \(f(x_n) = f(y_n)\). This means the set of points at which the function is not one-to-one is at least countably infinite. Providing the proof is part of Exercise 4.
“Code golf” is the practice of trying to solve a specific problem or implement a specific algorithm in the smallest possible number of characters of a given programming language. A related concept is exploring what can be accomplished with a tweet-length program (for example see here), though “tweet-length” can be a moving target. In the spirit of code golf, I propose the following “function golf” challenge.
What is the simplest function (defined on the real numbers) that is continuous everywhere except on a given subset of the reals?
Introduction Many mathematicians have a rough mental list of “desert island” books - a small set of books they would choose if stranded on a desert island, forced to spend the winter in a log cabin, or some similiar scenario. Some have published lists of selected books; for example John Baez’s “How to Learn Math and Physics”, “The Mathematics Autodidact’s Aid” by Kristine Fowler, and the “Chicago undergraduate mathematics bibliography” by Christopher Jeris and other contributers.
Once upon a time, in a small town, there were two carpenters: Bob and Tom. Bob was an honest, hardworking carpenter with a good reputation. Tom was a somewhat incompetent carpenter with shady business practices.
At one point in time, Bob and Tom had each completed 100 carpentry jobs. Bob’s customers were satisfied with 90 out of his 100 jobs. Tom’s customers were satisfied with 95 out of his 100 jobs.
I recently came across a surprising connection between the geometric mean and logarithms. At least, it was surprising to me, but isn’t that surprising once you see the derivation (a lot of things in math are this way).
The connection is this. Take a set of observations \(y_1, y_2, \dots, y_n\). Assume that these are all positive numbers. Then \[ \langle \log(y) \rangle = \log(G).\] Here the angle brackets indicate the mean (expectation value), and \(G\) is the geometric mean of the observations.
In my last post, I discussed the geometric mean and how it relates to the more familiar arithmetic mean. I mentioned that the geometric mean is often useful for estimation in physics.
Lawrence Weinstein, in his book Guesstimation 2.0, gives a mental algorithm for approximating the geometric mean. Given two numbers in scientific notation \[ a \times 10^x \quad \text{and}\quad b \times 10^y, \] where the coefficients \(a\) and \(b\) are both between 1 and 9.
Given two positive numbers \(a\) and \(b\), the most well-known way to average them is to take the sum divided by two. This is often called the average or the mean of \(a\) and \(b\), but to distinguish it from other means it is called the arithmetic mean: \[ \text{arithmetic mean} = \frac{a + b}{2}. \]
Another common, and useful, type of mean is the geometric mean. For two positive numbers \(a\) and \(b\), \[ \text{geometric mean} = \sqrt{ab}.
I was recently flipping through a real analysis textbook, and came across a notation that I had not seen before. First, the set difference of two sets \(A\) and \(B\) (which I had seen before) was defined as \[ A \setminus B = A \cap B^c. \] Here \(B^c\) is the set compliment of \(B\), or all elements not in \(B\). You can equivalently write the set difference as \[ A \setminus B = \{x \, |\, x \in A\, \text{and}\, x \notin B\} \]