What's all this, then?
About
Home
Tags
  • Enumerating the rationals

    Jan 18, 2020 mathematics

    The rational numbers are well known to be countable, and one standard method of counting them is to put the positive rationals into an infinite matrix \(M=m_{ij}\), where \(m_{ij}=i/j\) so that you end up with something that looks like this: \[ \left[\begin{array}{ccccc} …

    Read More
  • Fitting the SIR model of disease to data in Julia

    Jan 15, 2020 mathematics julia

    A few posts ago I showed how to do this in Python. Now it's Julia's turn. The data is the same: spread of influenza in a British boarding school with a population of 762. This was reported in the British Medical Journal on March 4, 1978, and you can read the original short article here. As before we use the SIR model, …

    Read More
  • The Butera-Pernici algorithm (2)

    Jan 6, 2020 mathematics computation

    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 …

    Read More
  • The Butera-Pernici algorithm (1)

    Jan 4, 2020 mathematics computation

    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 …

    Read More
  • Permanents and Ryser's algorithm

    Dec 22, 2019 mathematics computation

    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 …

    Read More
  • An interesting sum

    Dec 2, 2019 mathematics analysis

    I am not an analyst, so I find the sums of infinite series quite mysterious. For example, here are three. The first one is the value of \(\zeta(2)\), very well known, sometimes called the "Basel Problem" and first determined by (of course) Euler: \[ \sum_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}{6}. \] Second, …

    Read More
  • Runge's phenomenon in Geogebra

    Sep 15, 2019 mathematics computation geogebra

    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 …

    Read More
  • Fitting the SIR model of disease to data in Python

    Aug 9, 2019 mathematics computation python

    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 …

    Read More
  • Tschirnhausen transformations and the quartic

    May 5, 2019 mathematics algebra

    Here we show how a Tschirnhausen transformation can be used to solve a quartic equation. The steps are: Ensure the quartic is missing the cubic term, and its initial coefficient is 1. We can do this by first dividing by the initial coefficient to obtain an equation \[ x^4+b_3x^3+b_2x^2+b_1x+b_0=0 \] and then replace …

    Read More
  • Tschirnhausen's solution of the cubic

    May 5, 2019 mathematics algebra

    A general cubic polynomial has the form \[ ax^3+bx^2+cx+d \] but a general cubic equation can have the form \[ x^3+ax^2+bx+c=0. \] We can always divide through by the coefficient of \(x^3\) (assuming it to be non-zero) to obtain a monic equation; that is, with leading coefficient of 1. We can now remove the \(x^2\) …

    Read More
    • ««
    • «
    • 1
    • 2
    • »
    • »»

Recent Posts

  • Three-dimensional impossible CAD
  • Voting power (4): Speeding up the computation
  • Voting power (3): The American swing states
  • Voting power (2): computation
  • Voting power
  • Electing a president
  • Enumerating the rationals
  • Fitting the SIR model of disease to data in Julia
  • The Butera-Pernici algorithm (2)
  • The Butera-Pernici algorithm (1)

Tags

MATHEMATICS 14 VOTING 9 PYTHON 8 ALGEBRA 7 COMPUTATION 5 JULIA 4 CAD 3 GEOMETRY 3 GIS 3 JSXGRAPH 3 LINEAR-PROGRAMMING 3 ANALYSIS 1 ASTRONOMY 1 CRYPTOGRAPHY 1 GEOGEBRA 1 HASKELL 1 HISTORY 1 HUGO 1 HUMOUR 1 IMAGE-PROCESSING 1 MUSIC 1 ORG 1 PROGRAMMING 1 SCIENCE 1

Copyright WHAT'S ALL THIS, THEN?. All Rights Reserved