Show Me the Lambda

A Glitch In The Matrix.

This is the first in a series of posts that will eventually get round to implementing a Kalman Filter (or filters) in Haskell.  It’s not my intention to explain Kalman filtering in any detail – just sufficiently to complelent the Haskell code. There’s a plethora of tutorials and videos about Kalman Filtering and one of the better series is here.

As will be seen, the computations involved in…

Haskell and F# – Advent of Code Day 1.

This is the first in a series of occasional posts comparing Haskell and F# when solving the 2018 Advent of Code puzzles  (AoC). One puzzle is available on each of the 25 days of December leading up to the 25th. Each puzzle is in two parts and the second part is only available once the first part has been solved!

I’ve been learning Haskell for a few years now – I ‘got’ monads about a year ago then…

It’s just fun playing around with Stern-Brocot trees and a bit of Haskell…!

For any  fraction on the Stern-Brocot tree there is just one path to it from the root. That path can written as a sequence of left or right ‘turns’ at each node depending on whether the target node is smaller or larger than the current node. This leads to the

Stern Path

For example in this small tree

the path to, say,…

The purpose of this post is to show how to generate a visual representation of the Stern-Brocot tree, like this: SBTree the rendering of which uses Northwood’s GoJS Javascript library.

To create to a rendered Stern-Brocot tree we need to:

Create a binary tree datatype with supporting functions.

Generate a specific type of binary tree – i.e. a Stern-Brocot tree.

Examine the GoJS model details and…

A Stern View…

[latexpage]

    We all know the Fibonacci sequence and how it is generated…

1, 1, 2, 3, 5, 8,…

We just add the ‘previous’ pair to get the next number and then move along to the next pair. An interesting variation on this is to add adjacent pairs and write down their sum as before but, after appending the sum, also append the second digit of the pair just added and this gives…

1, 1, 2, 1, 3, 2,…

[latexpage]

    Not too complicated and just a different view of Ford’s circles and a way of morphing them along with a bit of animation.

http://gitcommit.co.uk/wp-content/uploads/2018/06/rotated-1.mp4

It’s a continuation of the previous post and there are two parts to it – the real bit and the imaginary part.

The Real Part

To start with we take fractions not between 0 and 1 but rather between…

Ford and his Circles.

[latexpage]

    A Ford circle  is a circle derived from a pair of numbers that are co-prime, i.e. they have no common factors. For  a pair of co-prime integers p and q the Ford circle has radius r and centre at a point

P(x, y) where \begin{align*} r=\frac{1}{2q^2}\\ P=(\frac{p}{q}\ , r)\\ \end{align*}

No matter what co-prime numbers, p and q, are used to create Ford circles the circles never…

[latexpage] This is the first in a series of posts about sequences of fractions, circles, trees of fractions, binary search trees and ways of representing rational numbers as paths through these trees . The idea is to enjoy a bit of recreational mathematics and to use Haskell to express some of the notions in its own succinct  way.

One Lot Of Fractions

First we’ll look at all unique and simplified…