|Posts on Regator:||1390|
|Posts / Week:||4.7|
|Archived Since:||April 26, 2011|
If you present calculus students with a definite integral, their Pavlovian response is “Take the anti-derivative, evaluate it at the limits, and subtract.” They think that’s what it means. But it’s not what a definite integral means. It’s how you (usually) calculate its value. This is not a pedantic fine point but a practically important distinction. It pays […]
When I was in high school, one year I made the Region choir. I had no intention of competing at the next level, Area, because I didn’t think I stood a chance of going all the way to State, and because the music was really hard: Stravinsky’s Symphony of Psalms. My choir director persuaded me […]
Francis Su has created an iPhone app MathFeed that gives a stream of new math content: blog posts, book reviews, popular journal articles, and tweets. You can also get the same content via Twitter. Check it out!
Here are a few things I’ve had to figure out in the process of setting up Emacs on a Mac, in particular with getting shell-mode to work as I’d like. Maybe this will save someone else some time if they want to do the same. I’ve used a Mac occasionally since the days of the […]
I don’t have an FAQ page per se, but I’ve written a few blog posts where I answer some questions, and here I’ll answer a few more. Should I get a PhD? See my answer here and take a look at some of the other answers on the same site. Do you have any advice for people […]
For many years, rivals University of Texas and Texas A&M University played each other in football on Thanksgiving. In 1999, the game fell one week after the collapse of the Aggie Bonfire killed 12 A&M students and injured 27. The University of Texas band’s half time show that year was a beautiful tribute to the fallen A&M students.
Yesterday I got a review copy of The Power of Networks. There’s some math inside, but not much, and what’s there is elementary. I’d say it’s not a book about networks per se but a collection of topics associated with networks: cell phone protocols, search engines, auctions, recommendation engines, etc. It would be a good […]
The pinned tweet on my Twitter account at the moment says “Productivity tip: work hard.” It’s gotten a lot of positive feedback, so I assume it has resonated with a few people. Productivity tip: Work hard. — John D. Cook (@JohnDCook) October 8, 2015 I don’t know how people take it, but here’s what I […]
From a NASA page advocating formal methods: We are very good at building complex software systems that work 95% of the time. But we do not know how to build complex software systems that are ultra-reliably safe (i.e. P_f < 10^-7/hour). Emphasis added. Developing medium-reliability and high-reliability software are almost entirely different professions. Using typical […]
I used to wonder why people “convert” from one technology to another. For example, someone might convert from Windows to Linux and put a penguin sticker on their car. Or they might move from Java to Ruby and feel obligated to talk about how terrible Java is. They don’t add a new technology, they switch from […]
Define Tn to be the Taylor series for exp(x) truncated after n terms: How does this function compare to its limit, exp(x)? We might want to know because it’s often useful to have polynomial upper or lower bounds on exp(x). For x > 0 it’s clear that exp(x) is larger than Tn(x) since the discarded terms […]
Comment from Paul Phillips on making things easy to understand: It’s always been “We can’t do it that way. It would be too slow.” You know what’s slow? Spending all day trying to figure out why it doesn’t work. That’s slow. That’s the slowest thing I know.
In geometry, you’d say that if a square has side x, then it has area x2. In calculus, you’d say more. First you’d say that if a square has side near x, then it has area near x2. That is, area is a continuous function of the length of a side. As the length of the side […]
The hazard function of a probability distribution is the instantaneous probability of an event given that it hasn’t happened yet. This works out to be the ratio of the PDF (probability density function) to the CCDF (complementary cumulative density function). For the standard normal distribution, the hazard function is and has a surprisingly simple continued […]
Sam Northshield  came up with the following clever proof that there are infinitely many primes. Suppose there are only finitely many primes and let P be their product. Then The original publication gives the calculation above with no explanation. Here’s a little commentary to explain the calculation. Since prime numbers are greater than 1, sin(?/p) is […]
Suppose you have a flat line f(x) = k and an interval [a, b]. Then the area under the line and over the interval is k times the length of the segment of the line. Surprisingly, the same is true for a catenary with scale k. With the flat line, the length of the segment of the graph is […]
Interesting passage from Small is Beautiful: Economics as if People Mattered by E. F. Schumacher: Nature always, so to speak, knows where and when to stop. There is a measure in all natural things—in their size, speed, or violence. As a result, the system of nature, of which man is a part, tends to be […]
Taylor’s theorem says When does the thing on the left equal the thing on the right? A few things could go wrong: Maybe not all the terms on the right side exist, i.e. the function f might not be infinitely differentiable. Maybe f is infinitely differentiable but the series diverges. Maybe f is infinitely differentiable but the […]
Chris Wiggins gave an excellent talk at Rice this afternoon on data science at the New York Times. In the Q&A afterward, someone asked how you would set up a machine learning algorithm where you’re trying to optimize for outcomes and for information. Here’s how I’ve approached this dilemma in the past. Information and outcomes are not […]
At first “discrete logarithm” sounds like a contradiction in terms. Logarithms aren’t discrete, not as we usually think of them. But you can define and compute logarithms in modular arithmetic. What is a logarithm? It’s the solution to an exponential equation. For example, the logarithm base 10 of 2 is the solution to the equation […]