The Jacobi elliptic functions sn and cn are analogous to the trigonometric functions sine and cosine. They come up in applications such as nonlinear oscillations and conformal mapping. Unfortunately, there are multiple conventions for defining these functions. The purpose of this post is to clear up the confusion around these different conventions.
The image above is a plot of the function sn .
I just ran across
FiniteGroupData and related functions in Mathematica. That would have made some of my earlier posts easier to write had I used this instead of writing my own code.
Here’s something I find interesting. For each n, look at the groups of order at most n and count how many are Abelian versus non-Abelian. At first, there are more Abelian groups, but the non-Abelian groups soon become more numerous. Also, the number of Abelian groups grows smoothly, while the number of non-Abelian groups has big jumps, particularly at powers of 2.
The prime number theorem says that π(x), the number of primes less than or equal to x, is asymptotically x/log x. So it’s easy to estimate the number of primes below some number N. But what if we want to estimate the number of prime powers less than N? This is a question that comes up in finite fields, for example, since there is a finite field with elements if and only if n is a prime power. It’s also important in finite simple groups because these groups are often indexed by prime powers.
Riemann’s prime-power counting function Π(x) counts the number of prime powers less than or equal to x. Clearly Π(x) > π( x) for x ≥ 4 since every prime is a prime power, and 4 is a prime power but not a prime. Is Π(x) much bigger than π(x)? What is its limiting distribution, i.e. what is the analog of the prime number theorem for prime powers?
The day I wrote this was the first day of a new month, which means the exponential sum of the day will be simpler than usual. The exponential sum of the day plots the partial sums of
where m, d, and y are the month, day, and (two-digit) year. The n/ d term is simply n, and integer, when d = 1 and so it has no effect because exp(2π n) = 1. Here’s today’s sum, the plot formed by the partial sums above.