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- 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.

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Here’s the introduction/advertisement/preview video:

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.

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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!

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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.

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And in a similar vein, you really should take a look at ur-blogger John Baez’s new blog for the AMS, Visual Insight, dedicated to amazing mathematical visualizations.

Enjoy!

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A plane flies horizontally over a radar station at an altitude of 1km at a speed of 500 km/hr. How fast is the distance to the radar station changing when the plane is 2km away from the radar station and heading away from it?

…and replace it with this one:

A plane flies horizontally over a radar station at an altitude of either 1.00m, 1.00km, or 1000km (whichever makes sense) at a speed of either 5.00 km/hr, 500 km/hr, or 50,000 km/hr (whichever makes sense). How fast is the distance to the radar station changing when the plane is either 2.00m, 2.00km, or 20,000km away from the radar station (whichever makes sense) and heading away from it?

Beforedoing any calculation, (a) what units will the answer have, (b) will it be positive, negative, or zero, and (c) is it likely to be in the range [-1000,-500],[-500,0],[0,500],[500,1000], or none of the above? Justify your reasoning.Thenanalyze the problem carefully and give a numerical answer, to appropriate precision.

A more creative option than the “multiple-choice question” format above would be to have every student (or every group) make up their own problem, making sure that they can justify the reasonableness of the numbers they put in. But that might be unwieldy, with many different answers that would be hard to check and hard to have a whole-class discussion about.

We’ll see if I can find time to put this into practice.

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