The purpose of this post will be to see if we can implement the algorithm in Julia, and thus leverage Julia's very fast execution time. We are working with polynomials defined on nilpotent variables, which means that the degree of any generator in a polynomial term will be 0 or 1. Assume that our generators are indexed …

Introduction We know that there is no general sub-exponential algorithm for computing the permanent of a square matrix. But we may very reasonably ask -- might there be a faster, possibly even polynomial-time algorithm, for some specific classes of matrices? For example, a sparse matrix will have most terms of the …

As I discussed in my last blog post, the permanent of an \(n\times n\) matrix \(M=m_{ij}\) is defined as \[ \text{per}(M)=\sum_{\sigma\in S_n}\prod_{i=1}^nm_{i,\sigma(i)} \] where the sum is taken over all permutations of the \(n\) numbers \(1,2,\ldots,n\). It differs from the better known determinant in having no sign …

Runge's phenomenon says roughly that a polynomial through equally spaced points over an interval will wobble a lot near the ends. Runge demonstrated this by fitting polynomials through equally spaced point in the interval \([-1,1]\) on the function \[ \frac{1}{1+25x^2} \] and this function is now known as "Runge's …

Introduction and the problem The SIR model for spread of disease was first proposed in 1927 in a collection of three articles in the Proceedings of the Royal Society by Anderson Gray McKendrick and William Ogilvy Kermack; the resulting theory is known as Kermack–McKendrick theory; now considered a subclass of a more …