Mathematics on What's all this, then?
https://numbersandshapes.net/tags/mathematics/
Recent content in Mathematics on What's all this, then?Hugo -- gohugo.ioen-usSun, 09 Sep 2018 00:00:00 +1000Householder's methods
https://numbersandshapes.net/post/householders_methods/
Sun, 09 Sep 2018 00:00:00 +1000https://numbersandshapes.net/post/householders_methods/These are a class of root-finding methods; that is, for the numerical solution of a single nonlinear equation, developed by Alston Scott Householder in 1970. They may be considered a generalisation of the well known Newton-Raphson method (also known more simply as Newton’s method) defined by
\[ x\leftarrow x-\frac{f(x)}{f’(x)}. \]
where the equation to be solved is \(f(x)=0\).
From a starting value \(x_0\) a sequence of iterates can be generated byThe Joukowsky Transform
https://numbersandshapes.net/post/joukowsky-transform/
Fri, 24 Aug 2018 00:00:00 +1000https://numbersandshapes.net/post/joukowsky-transform/The Joukowksy Transform is an elegant and simple way to create an airfoil shape.
Let \(C\) be a circle in the complex plane that passes through the point \(z=1\) and encompasses the point \(z=-1\). The transform is defined as
\[ \zeta=z+\frac{1}{z}. \]
We can explore the transform by looking at the circles centred at \((-r,0)\) with \(r<0\) and with radius \(1+r\):
\[ \|z-r\|=1+r \]
or in cartesian coordinates with parameter \(t\):The trinomial theorem
https://numbersandshapes.net/post/trinomial_theorem/
Thu, 05 Apr 2018 00:00:00 +1000https://numbersandshapes.net/post/trinomial_theorem/When I was teaching the binomial theorem (or, to be more accurate, the binomial expansion) to my long-suffering students, one of them asked me if there was a trinomial theorem. Well, of course there is, although in fact expanding sums of greater than two terms is generally not classed as a theorem described by the number of terms. The general result is
\[ (x_1+x_2+\cdots+x_k)^n=\sum_{a_1+a_2+\cdots+a_k=n} {n\choose a_1,a_2,\ldots,a_k}x_1^{a_1}x_2^{a_2}\cdots x_k^{a_k} \]
so in particular a “trinomial theorem” would be