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

Before you read this, I suggest you read post 16.18. I have written this post not because I know a lot about the subject but because I find some aspects of it interesting. “The naming of cats is a difficult matter.” T.S. Elliot (American/British poet, 1888-1965) Old Possum’s Book of Practical Cats. So let’s see… Continue reading 23.5 Biological classification – taxonomy

If you want to understand the background to this post, I suggest you read posts 22.12, 22.14, 22.15, 22.20, 22.21, 22.23, 22.24, 23.1 and 23.3 (route 1). But, for a less mathematical introduction, you can read posts 18.10, 18.16, 22.21 (but not the last paragraph) and 23.2 (route 2). Most of the statements that appear… Continue reading 23.4 X-ray crystallography and molecular structure

Before you read this, I suggest you read posts 22.14 and 22.21. In post 22.21, we saw that an ideal crystal is an arrangement of atoms (that may be bonded together in molecules) or ions in a regularly repeating three-dimensional pattern. And, in post 22.14, we saw that the x-ray diffracted by an object could… Continue reading 23.3 X-ray diffraction by a crystal