In post 24.3 we explored a model of how a population could control itself. This model was based on a differential equation – the most common way to describe how things change with time. This approach was used to describe exponential growth (post 18.15), waves (post 19.12), diffusion (posts 19.15 and 19.16), the behaviour of… Continue reading 24.4 Logistic difference equation: self-controlled growth and chaos

Before you read this, I suggest you read post 18.15. In post 22.17 we saw the effect of predators on the population of a species: the example I used was the natural control of a rabbit population by foxes. But, in post 16.5, we saw that a population will control itself, without the need for… Continue reading 24.3 Limits to growth: the logistic differential equation

In post 23.4, we saw that the interpretation of x-ray diffraction patterns from crystals involves a temperature factor because atoms are displaced from their ideal lattice points by thermal vibrations. We have seen that heat is kinetic energy of atomic molecular motion (post 16.35). In a crystalline solid, the atoms must be oscillating about their… Continue reading 24.2 Thermal vibration of an atom in a crystal: the temperature factor in x-ray crystallography

I am going to calculate the Fourier transform of a Gaussian function because I want to use the result in a later post. I can define the Gaussian function, g(x) as an exponential function of the form g(x) = exp(-ax2). Here, exp(y) is another way of writing ey where e a number with the approximate… Continue reading 24.1 Fourier transform of a Gaussian function

Before you read this, I suggest you read section 10 of post 22.10. In post 22.10 we saw that integration reverses the process of differentiation. Suppose f and f’ are functions of x only and that f’ = df/dx.                    (2) To evaluate the integral we recognise the function f that satisfies equation 2. Then Sometimes… Continue reading 23.11 Integration by parts

At the beginning of post 23.4, I described two routes to understanding x-ray crystallography. If you prefer the second, less mathematical, route – ignore this post! Before you read this, I suggest you read post 23.8. In post 22.14, we saw that the x-ray diffraction pattern of an object is given by the Fourier transform… Continue reading 23.10 Frequency analysis and x-ray crystallography

Before you read this, I suggest you read post 23.8. Let’s think about a wave or signal that looks like the picture above. It might be the result of an observation – for example, the electrical signal that results from recording a sound using a microphone. In this example we might want to know the… Continue reading 23.9 Discrete Fourier Transform

Let’s think about a wave, that we can represent by a wave function ψ(t) where t represents time, that has a time-period is T. An example is the time-dependent electrical potential, shown above (see also post 18.14) that has T = 0.25 ms. The Fourier transform of this one-dimensional function is defined by This is… Continue reading 23.8 Frequency analysis

Before you read this, I suggest you read post 21.8. We usually think that when individual animals can breed, to produce fertile offspring, that they belong to the same species. If they can’t – they belong to different species (see post 23.5). So where do individual species come from? The English biologist Charles Darwin (1809-1882)… Continue reading 23.7 Darwin’s theory of evolution

Before you read this, I suggest you read post 21.8. In my previous posts (21.11, 21.14 and 21.23), it may appear that genes are constant sources of information that never change. This idea explains most of what we know about why individuals have a different genotype and, therefore, a different phenotype to their parents. But… Continue reading 23.6 Mutation