Mathematics

My 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

AC circuit analysis relies heavily on complex numbers, and the concept of 'complex impedance'. But why? What actually is a complex impedance?

Categories: mathematics, electronics

'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

This simple analog computer has served humanity for more than four centuries. Shouldn't we try to preserve it?

Categories: mathematics, TDMTLTAM

The 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

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

Benford'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

A detailed description of a method for performing this common numerical conversion, with C source code.

Categories: mathematics, C

This 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

There 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

The 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

In 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

The 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

The 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

This 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

An explanation from first principles of the concepts of probability applied to two random discrete variables.

Categories: mathematics

An explanation from first principles of this much-misunderstood principle of statical inference.

Categories: mathematics

Using 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

An introduction to the science of exterior ballistics -- tracking the flight of a projectile under the influence of gravity and drag.

Categories: mathematics

Well, 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