Mathematics
Bayesian statistics for dummiesAn explanation from first principles of this much-misunderstood principle of statical inference.
Categories: mathematics
The peculiar Benford distributionBenford's law or the first digit rule states that in many data sets, the first digit of each value tends to be small. This is not at all obvious, but it has significant implications.
Categories: mathematics
Calculus of variations: a lunchbreak guideThe calculus of variations is a branch of mathematics that deals with finding functions that optimize systems. Although calculus of variations has its origins in the 18th century, it is still not widely taught, either to science or engineering students. This brief guide intends to explain the basic principles, with a fully-worked example, in something that can be read in about an hour.
Categories: mathematics
Why is covid-19 testing so unreliable? A pictorial viewThe UK Government's response to the covid-19 is to 'test, test, test'. This article demonstrates in pictorial terms why this strategy will catastrophically overestimate the number of people actually infected, leading to widespread disruption.
Categories: science and technology, mathematics
Why you can't play Bach on a pianoWell, perhaps you can play Bach on a piano, if you have the skill and the patience; but you can't hear Bach's music the way he and his contemporaries would have heard it because, in general, we don't using the same tuning schemes. This is an article about the mathematics of music tuning, from Pythagoras to the modern equally-tempered scale.
Categories: mathematics, music
An overview of finite fieldsThis article describes the concepts of finite fields, from the ground up, to a level at which it is possible to understand modern encryption algorithms. I assume only a high-school understanding of math.
Categories: mathematics
Gâteaux differentials and Euler-Lagrange equations using MapleMy article on calculus of variations assumed without explanation the basic form of the Euler-Lagrange equation. This article attempts to demonstrate how an Euler-Lagrange equation arises, by solving a variational problem algebraically using Maple.
Categories: mathematics
Understanding the Hamming 'sphere packing' bound in coding theoryThe Hamming bound is an important metric in the evaluation of information coding schemes. It sets an upper limit on the number of distinct codewords that an error-correcting block code can provide, for a given block length. The Hamming bound is a benchmark for the evaluation of error correction schemes used in digital data storage and communication -- a scheme that meets the Hamming bound is described as perfect. In this article I try to describe, from first principles, how the Hamming bound is calculated.
Categories: mathematics, general computing
Integrating factors from the ground upThe use of integrating factors is a well-known method for turning an intractable differential equation into an integration problem (which may itself turn out to be intractable, but we live in hope). The method is usually described in textbooks and lectures with a fair amount of hand-waving. This article attempts to describe it in detail, with none of the difficulties left out.
Categories: mathematics
An introduction to joint, marginal, and conditional probabilitiesAn explanation from first principles of the concepts of probability applied to two random discrete variables.
Categories: mathematics
The tabular method for integration by partsThere is a tabular method for organizing integration computations which require repeated application of the integration-by-parts formula. It is seldom taught, which is a shame: it hugely reduces the algebraic complexity of the problem.
Categories: mathematics
Why pitch is not the same as frequencyIn many cases, pitch is just the musician's way of saying frequency. However, pitch has a perceptual element to it, and the ear can be fooled. This article explains one of the ways in which this can happen, and why it isn't correct to use the terms pitch and frequency interchangeably in music.
Categories: mathematics, music
Solving a quadratic equation in a prime (finite) fieldThis article describes how to do arithmetic in a small finite (prime) field. Being able to do this is essential for the implementation of cryptography, among other things. The arithmetic isn't particularly difficult -- particularly if we use software that specifically supports such operations. However, it's illustrative to do it manually, at least for a small field. I will demonstrate simple arithmetic in the field GF(5), by solving a quadratic equation.
Categories: mathematics, general computing
Converting a floating-point number to a fraction (approximately) using continued fraction expansionA detailed description of a method for performing this common numerical conversion, with C source code.
Categories: mathematics, C
Did aliens really talk to us in binary code at Rendelsham Forest?It isn't often that coding theory can be used to evaluate a claim of a UFO encounter. Here is one instance where it can.
Categories: science and technology, mathematics
Differential equations: how does separation of variables really work?'Separation of variables' is one of the first methods usually taught to math and engineering students for solving differential equations, and yet the method as taught is mathematically very sloppy. That's not to say it doesn't work, but rather that it isn't made clear why it works, and whether similar methods will work elsewhere. This article attempts to treat the subject with a bit of mathematical rigour. We end up exactly where we started but, perhaps, with a bit more insight.
Categories: mathematics
They don't make them like that any more: the slide ruleThis simple analog computer has served humanity for more than four centuries. Shouldn't we try to preserve it?
Categories: mathematics, TDMTLTAM
Night of the living differential equations -- modeling the zombie apocalyseUsing a simple application of Euler's method to estimate the solutions of non-linear differential equations, and work out your chances of surviving the zombie apocalypse. No, really.
Categories: mathematics
Night of the living drag coefficients -- ballistics applied to zombie controlAn introduction to the science of exterior ballistics -- tracking the flight of a projectile under the influence of gravity and drag.
Categories: mathematics
