More Motivated Analysis

As of today the course is up through Part 30, which gets us most of the way through Handout 8. (Note: I edited Handout 8 a bit to streamline it and move some technicalities to the To-Do List–which we will turn to eventually!) The big news is we are now through the proof of (a somewhat restricted version of) Duhamel’s Principle. The proof is almost trivial once we’ve set up the machinery, but in Handout 9 we’ll look carefully at a pretty fundamental issue that we’ve glossed over so far. After a bit more with the story of distributions and convolutions we’ll switch gears a bit to Fourier.

Motivated Analysis–Current Status

The Motivated Analysis course is up to 7 handouts and 27 videos as of September 14. Pretty soon (I hope) the course will get to a few exciting waypoints:

• Proving Duhamel’s Principle in a general setting, using distributions and convolutions
• Pinning down the precise definition of a distribution (partly motivated by what Duhamel will tell us)
• Going back to the original mass-spring system, but this time forcing with sine waves, not hammers, to start the story of Fourier analysis–a story that will get intimately intertwined with distributions.

I’d love any feedback you have if you are following along (even in a non-serious way). In particular, it would be good to know if there is a desire for more practice-type problem sets (in addition to the problems in the handouts, which develop the ideas). Now, I’m not saying I’ll produce reams of extra problems, but I realize that as it is there might not be quite enough practice for a student to really master the ideas as we go.

Motivated Analysis Progress Report

My Motivated Analysis course is up to 13 16 videos and 4 5 handouts. Click on the black bar above for the home page. Feel free to comment here if you have responses, questions, or suggestions; of course you can comment on the video pages as well. You may want to follow this blog if you are trying to do the course (either seriously or informally) and want news about major progress.

Starting an extensive new video series/course

I’m planning to start a new video series, or really, an informal online course, entitled Motivated Analysis. The goal is to bridge the gap between applications-oriented courses in analysis (mostly differential equations) and theoretical courses, and to show how modern theoretical analysis comes out of practical problems.

Here’s the introduction/advertisement/preview video:

https://youtu.be/pLpAYNn2IiQ

The home page for the course is linked in the black bar just above this post.

And here’s a link to the first discovery handout—I’ll be working through this PDF in the first few videos of the course, but most of it should be doable by someone with the appropriate prerequisites.

Adela’s math puzzle

Time to brag on my 7-year-old. Adela has been thinking up math problems for me each night. Tonight it was the following. Start with a number, say 2, and do all four operations on it: 2+2=4, 2*2 = 4, 2-2 = 0, and 2/2 = 1. Now add the results all up, getting 4+4+1+0=9.

If you start with 3, you get 3+3=6, 3*3 = 9, 3-3 = 0, and 3/3 = 1. Adding gives 6+9+1+0=16. Hmm…I think I see a pattern here….

Starting with 4 gets you 8+16+1+0=25, which is 5^2 = (4+1)^2.

Of course this is just a cute way of saying that (n+1)^2 = n^2 + 2*n + 1, or more explicitly

(n+1)^2 = n*n + (n+n) + n/n + (n-n)

which I don’t think I had ever thought of in exactly that way. Hurray for kids!

New math prize announced…is it a good thing?

Courtesy of Not Even Wrong, we have news that in addition to awarding a $3M prize to superstring theory founders Green and Schwarz and giving out a number of similar prizes for biomedical research, Yuri Milner is teaming up with Mark Zuckerberg to fund a $3M prize they are calling the Breakthrough Prize in Mathematics. Whether this will be a good thing is already a source of debate, but it will certainly be more fodder for the small set of people (including me) who like to speculate on who will win prizes like this. Will it make People Magazine? Hard to say.

This will never happen again…

Here’s a quick, fun tidbit about dates. I was thinking about the factorization of 2013 and the next few years, and I noticed something cool. (Note this is good for psyching out math contests, which often have problems with the current year in them.)

2013 = 3 * 11 * 61

2014 = 2 * 19 * 53

2015 = 5 * 13 * 31

Each date factors into primes as (single digit) * (teen) * (double digit bigger than teen). (I’m counting 11 as a teen, i.e. I really mean a prime between 11 and 19.) So I wondered how often you get three such numbers in a row, and I wrote a little Matlab code. It turns out that these three are the only example! In fact there are only three other consecutive pairs: 1221,1222; 1598,1599; and 2821,2822.

I can’t imagine that this kind of factorization has any real importance, but I thought it was cool. The next year, 2016, is quite different, but also cool: it is

2016 = 32 * 63 = 2^5 * 3^2 * 7

which is (power of two) * (power of two – 1), and which is also “7-smooth”, meaning that it has no prime factor larger than 7. (These are sometimes called “highly composite”, but that has a different, and more interesting, meaning, about which I made some videos last year.) The 7-smooth numbers are pretty sparse; the only ones occuring in anyone’s lifetime who is alive right now are 1920, 1944, 1960, 2000, 2016, 2025, 2048, 2058, 2100. Note that 2048 = 2^11, which will be cause for worldwide, year-long celebration, no doubt. :)

Speaking of numbers of the form (power of two) * (power of two – 1), if a number n is n = 2^(k-1) * (2^k – 1), and (2^k – 1) is prime (a Mersenne prime), then n is a perfect number. Now 2016 doesn’t fall into that category, since 63 is not prime, but it is a number with a particularly large sum of all of its factors, called sigma(n) (which, for a perfect number n, would equal exactly 2n). The sum of factors of 2016 is sigma(2016) = 6552, which is higher than for any other number in the years 1900-2099 except for 1980 (which ties it with sigma(1980) = 6552). The year 2040 is very close, with sigma(2040) = 6480, and 2100 beats it with sigma(2100) = 6944.

The story is very similar if you just look at the number of divisors (often called sigma_0(n) or tau(n)). The number n = 2016 again beats everything except 1980 and 2100, which both tie it with tau(n) = 36. So in 2016, let’s make sure to do a lot of division!

If you watch my videos about highly composite numbers (“Why 5040 is cool”) you’ll see that numbers that factor into small primes (but not all 2’s, like 2048, or 2’s and 3’s) tend to have the largest number of divisors and the highest sigma values.

Going one more year out, 2017 is prime. Going back a little, 2012 = 2 * 2 * 503, and 2011 was prime. (Thanks to oeis.org for a lot of the data for all of these statements!)

Hope that was fun.