I found this great article on Digg about moving objects over large distances in space with relatively tiny amounts of energy. I do wish, though, that the article had compared these null cline trajectories with old school gravity assists. Even still, it's a super cool intersection of nonlinear dynamics, engineering, and physics.
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Many modern cryptographic techniques rely on modular powers. In order for these schemes to be secure, the base, exponents, and modulus are often quite large. I'm currently using Matlab in the cryptography course I'm teaching, and double precision overflow was immediately a problem for even relatively small examples. I've coded up an implementation of the left-to-right binary method that allows us to do so more realistic examples: function rem = powMod(base,exponent,modulo) The method itself is really neat. I think it would make a great project for an undergraduate with a little prior coding experience.
Notice that even here we could have double precision overflow if the square of the modulus is too large. I've been thinking about either implementing a variable precision method, or switching out of Matlab entirely the next time I run the course. I think a natural alternative would be Python. If anyone has any thoughts, feel free to post or email. John, Brian, and I recently had a piece of work examining oscillations in simple fluid networks accepted to SIAM Journal on Applied Dynamical Systems. I presented a short version of the work at the Joint Mathematics Meeting this January, and it's looking like I'll give a longer version aimed at undergraduates at Wellesley's mathematics colloquium in April. I'm particularly excited about this last one, because there's so much cool stuff in the project that you can think of as an extension or modification of classic undergraduate topics. Should be fun!
I'm teaching a introductory cryptography and coding theory course this semester in which we're using Matlab to implement a bunch of different cryptosystems. I was a little surprised to see that Matlab doesn't have a built-in function (that I could find, at least) that computes in the inverse of $x$ modulo $n$ if it exists. There's actually a fairly pretty solution. Bezout's identity tells us that for any integers $x$ and $y$ not both zero, there exist integers $a$ and $b$ such that a linear combination gives the greatest common divisor of $x$ and $y$. $$ax + by = gcd(x,y).$$ Matlab actually gives us hooks to compute $a$ and $b$ in it's gcd function: >> help gcd So suppose we want the multiplicative inverse of $x$ modulo $n$. We know (or could show from first principles with relatively little work) that $x$ has a multiplicative inverse modulo $n$ if and only if $gcd(x,n) = 1$. So Bezout's identity tells us that there exist integers $a$ and $b$ such that $$ax + bn = 1.$$ Since $bn \equiv 0 \bmod n$, we're left with $ax \equiv 1$, which shows that $x^{-1} \equiv a \bmod n$. Here's my implementation in Matlab: function xInv = modInv(x,n) As reported by Gil Kalai, some major progress has been made on Steiner systems. The question is: can you find a collection of subsets , each with size $k$, drawn from an ambient set of size $n$, such that each each subset of size $r \leq k$ appears in exactly $\lambda$ subsets in the collection? These designs have a special place in my heart; my thesis work centered on applying them to dynamic rekeying in smart grid systems.
This is problem has been more or less completely open for over 150 years. There have been some good results for $r =2$ and sporadic constructions for $r \geq 3$. Evidently Peter Keevash has cracked the problem wide open by solving the case of general $q$ and $r$. I'm not sure that anyone saw this sort of generalized construction coming. What's more, it seems that there is a new probabilistic construction technique at the heart of the proof. Incredible! Young people doing science is awesome. That is all.
I found a quick and interesting article on /r/math this morning about the Gompentz law of human mortality. The basic idea is that the probability that you will die in a given year doubles roughly every 8 years. This could turn into a grisly but interesting exam question in my first-year applied mathematical methods class.
I just saw an interesting article on /r/math detailing how Polynesians had invented binary notation 600 years ago. I love when these fundamental concepts are rediscovered over and over. I've been reading a lot of Iain Banks' science fiction lately, and one great passage of his says something along the lines of "one of the few things that all sufficiently developed species can agree upon is that integers should be represented in base 2." It wouldn't even require so much conversion: a week becomes 8 days, a month 32 days, a "thousand" becomes 1024, and so on. Somehow I don't think this is going to get much popular support...
I read a great article in Quanta about the recent flurry of progress on the twin prime conjecture. I'm not a number theorist, and the article did a good job of relating the big ideas at a high level for a wide mathematical audience. I absolutely love that Polymath8, and now Polymath8b, have had so much to contribute. It'll be interesting to see how far the current approach can bring down the bound.
It's been forever since my last post! It's been a busy semester so far, with a new curriculum rollout and some good research stuff going on. I wanted to take a quick moment and plug an awesome project that Shivani Janani, an honors program student I'm co-advising with Danielle Krcmar, will be working on next semester and ask for any resources that someone might know of.
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