Archive for the mathematics Category

R.I.P. Sir David Cox (1924-2022)

Posted in Biographical, mathematics, The Universe and Stuff with tags , , , , on January 21, 2022 by telescoper

I was saddened to hear a few days ago that the eminent statistician David Cox has passed away at the age of 97. I didn’t know Professor Cox personally – I met him only once, at a joint astronomy-statistics meeting at (I think) the Royal Astronomical Society back in the day – but I learnt a huge amount from books he co-wrote, despite the fact that he was of the frequentist persuasion. Three examples from my bookshelf are shown above.

I started my PhD DPhil in 1985 with virtually no formal study of statistics under my belt so I had to follow a steep learning curve and I was helped enormously by these books. I bought the book on Point Processes so as to understand some of the ideas being applied to galaxy clustering. It’s only a short book but it’s crammed with interesting ideas. Cox & Miller on Stochastic Processes is likewise a classic.

I know I’m not the only person in astrophysics whose career has been influenced by David Cox and I’m sure there are many other disciplines who have benefitted from his knowledge.

Among many other awards, David Cox was elected a Fellow of the Royal Society in 1973 and knighted in 1985.

Rest in peace Sir David Cox (1924-2022)

The Coronavirus Vaccine Effect

Posted in Covid-19, mathematics with tags , , , , , , on December 12, 2021 by telescoper

When I was updating my Covid-19 page today I thought I would try something a bit different. Here are the cases and deaths (in the form of 7-day rolling averages) as I usually plot them:

You can see a slight recent downturn – the latest 7-day average of new cases is 4214.3; it has been falling for a few days. A log plot like this shows up the changing ratio between deaths and cases quite well, as in l if you multiply a quantity by a factor that manifests itself as a constant shift upwards or downwards. There is clearly a bigger shift between the orange and blue curves after 500 days than there is, say, between, 300 and 400.

(I don’t think you can read much into the gap between the curves at the beginning (up to around 100 days in) as testing coverage was very poor then so cases were significantly underestimated.

Anyway, to look at this a bit more clearly I plotted the ratio of daily reported deaths to daily confirmed cases over the course of the pandemic. This is the result:

The sharp downward glitches occur whenever the number of reported deaths is zero, as log of zero is minus infinity. The broader downward feature after about 300 days represents the period in January 2021 when cases were climbing but deaths had not caught up. To deal with that I tried plotting the deaths recorded at a particular time divided by the cases two weeks earlier. This is that result:

The spike is still there, but is much decreased in size, suggesting that a two week lag between cases and deaths is a more useful ratio to look at. Note the ratio of deaths to cases is significantly lower from 500 days onwards than it was between 200 and 400 (say), by a factor a bit less than ten.

This obviously doesn’t translate into a direct measure of the efficacy of vaccines (not least because many of the recent cases and deaths are among the minority of unvaccinated people in Ireland) but it does demonstrate that there is a vaccine effect. Without them we would be having death rates up to ten times the current level for the same number of daily cases or, more likely, we would be in a strict lockdown.

On the other hand if cases do surge over the Christmas period there will still be a huge problem – 10 % of a large number is not zero.

Solar Corona?

Posted in Bad Statistics, Covid-19, mathematics, The Universe and Stuff on December 8, 2021 by telescoper

A colleague pointed out to me yesterday that  evidence is emerging of a four-month periodicity in the number of Covid-19 cases worldwide:

The above graph shows a smoothed version of the data. The raw data also show a clear 7-day periodicity owing to the fact that reporting is reduced at weekends:

I’ll leave it as an exercise for the student to perform a Fourier-transform of the data to demonstrate these effects more convincingly.

Said colleague also pointed out this paper which has the title New indications of the 4-month oscillation in solar activity, atmospheric circulation and Earth’s rotation and the abstract:

The 4-month oscillation, detected earlier by the same authors in geophysical and solar data series, is now confirmed by the analysis of other observations. In the present results the 4-month oscillation is better emphasized than in previous results, and the analysis of the new series confirms that the solar activity contribution to the global atmospheric circulation and consequently to the Earth’s rotation is not negligeable. It is shown that in the effective atmospheric angular momentum and Earth’s rotation, its amplitude is slightly above the amplitude of the oscillation known as the Madden-Julian cycle.

I wonder if these could, by any chance, be related?

P.S. Before I get thrown into social media prison let me make it clear that I am not proposing this as a serious theory!

The Omicron Variant

Posted in Covid-19, Crosswords, mathematics on November 30, 2021 by telescoper

As a theoretical physicist I use Greek characters all the time in mathematical work but, being very slow on the uptake, I only just realized a few days ago that the name of the Greek letter ‘omicron’ (ο) is derived from the Greek meaning ‘little-o’ while the name ‘omega’ means ‘big o’.

More recently still a Greek friend of mine pointed out that the lower-case symbol for omega (ω) was originally formed as ‘oo’, i.e. double-o.

In modern Greek ο and ω are pronounced the same but in ancient Greek the vocalisation of ω was longer than that of ο, suggesting that οmicron is more like short ‘o’ than little ‘o’ while omega is long `o’ rather than big ‘o’.

Incidentally, I was brought up to pronounce π like “pie” but in most of Europe (including Greece) it is pronounced “pee”. It is in fact the Greek letter ‘p’. I feel I’ve been delta very weak hand when it comes to Greek pronunciation and I’ll beta majority of theoretical physicists feel the same. I think we need to take a nu approach in schools, and rho back from the old ways. Anyway I’m going home now to eta bit of curry for supper…

On Fourier Series

Posted in mathematics, Maynooth, The Universe and Stuff with tags , , , , , , on November 30, 2021 by telescoper

So here we are, in the antepenultimate week of the Autumn Semester, and once again I find myself limbering up for the “and” bit of my second-year module on Vector Calculus and Fourier Series, i.e. Fourier Series.

As I have observed periodically, I don’t like to present the two topics mentioned in the title of this module as completely disconnected, so I linked them in a lecture in which I used the divergence theorem of vector calculus to derive the heat equation, the solution of which led Joseph Fourier to devise his series in Mémoire sur la propagation de la chaleur dans les corps solides (1807), a truly remarkable work for its time that inspired so many subsequent developments.

 

Anyway I was looking for nice demonstrations of Fourier series to help my class get to grips with them when I remembered this little video recommended to me some time ago by esteemed Professor George Ellis. It’s a nice illustration of the principles of Fourier series, by which any periodic function can be decomposed into a series of sine and cosine functions.

This reminds me of a point I’ve made a few times in popular talks about astronomy. It’s a common view that Kepler’s laws of planetary motion according to which which the planets move in elliptical motion around the Sun, is a completely different formulation from the previous Ptolemaic system which involved epicycles and deferents and which is generally held to have been much more complicated.

The video demonstrates however that epicycles and deferents can be viewed as the elements used in the construction of a Fourier series. Since elliptical orbits are periodic, it is perfectly valid to present them in the form a Fourier series. Therefore, in a sense, there’s nothing so very wrong with epicycles. I admit, however, that a closed-form expression for such an orbit is considerably more compact and elegant than a Fourier representation, and also encapsulates a deeper level of physical understanding.

The Hardest Problem

Posted in Cute Problems, Education, mathematics with tags , , , on November 19, 2021 by telescoper

The following Question, 16(b), is deemed to have been the hardest problem on the Maths Extension 2 paper of this year’s HSC (Higher School Certificate), which I think is the Australian Equivalent of the Leaving Certificate. You might find a question like this in the Applied Mathematics paper in the Leaving Certificate actually. Since it covers topics I’ve been teaching here in Maynooth for first-year students I thought I’d share it here.

I don’t think it’s all that hard really, probably because it’s really a physics problem (which I am supposed to know how to solve), but it does cover topics that tend to be treated separately in school maths (vectors and mechanics) which may be the reason it was found to be difficult.

Anyway, answers through the comments box please. Your time starts now.

Newton’s Fractal

Posted in mathematics with tags , , on October 30, 2021 by telescoper

Here’s another thing I bookmarked recently and then forgot about. It’s about a fractal structure that arises when using Newton’s Method (aka the Newton-Raphson Method) to find the roots of a real-valued function. I hope I remember this when I’m teaching root-finding in Computational Physics next term!

Del or Nabla?

Posted in Biographical, mathematics with tags , , on October 12, 2021 by telescoper

I am today preoccupied with vector calculus so, following on from yesterday’s notational rant, I am wondering about the relative frequency of usage of names for this symbol, commonly used in math to represent the gradient of a function ∇f:

To write this in Tex or Latex you use “\nabla” which is, or so I am told, so called because the symbol looks like a harp and the Greek word for the Hebrew or Egyptian form of a harp is “nabla”:

When I was being taught vector calculus many moons ago, however, the name always used was “del”. That may be a British – or even a Cambridge – thing. Here is an example of that usage a century ago.

Anyway, I am interested to know the relative frequency of the usage of “nabla” and “del” so here’s a poll.

There may be other terms, of course. Please enlighten me through the comments box if you know of any…

Writing Vectors

Posted in mathematics, The Universe and Stuff with tags , , , on October 11, 2021 by telescoper

Once again it’s time to introduce first-year Mathematical Physics students to the joy of vectors, or specifically Euclidean vectors. Some of my students have seen them before, but probably aren’t aware of how much we use them theoretical physics. Obviously we introduce the idea of a vector in the simplest way possible, as a directed line segment. It’s only later on, in the second year, that we explain how there’s much more to vectors than that and explain their relationship to matrices and tensors.

Although I enjoy teaching this subject I always have to grit my teeth when I write them in the form that seems obligatory these days.

You see, when I was a lad, I was taught to write a geometric vector in the following fashion:

\vec{r} =\left(\begin{array}{c} x \\ y \\ z \end{array} \right).

This is a simple column vector, where x,y,z are the components in a three-dimensional cartesian coordinate system. Other kinds of vector, such as those representing states in quantum mechanics, or anywhere else where linear algebra is used, can easily be represented in a similar fashion.

This notation is great because it’s very easy to calculate the scalar (dot) and vector (cross) products of two such objects by writing them in column form next to each other and performing a simple bit of manipulation. For example, the scalar product of the two vectors

\vec{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \vec{v}=\left(\begin{array}{c} 1 \\ 1 \\ -2 \end{array} \right)

can easily be found by multiplying the corresponding elements of each together and totting them up:

\vec{u} \cdot \vec{v} = (1 \times 1) + (1 \times 1) + (1\times -2) =0,

showing immediately that these two vectors are orthogonal. In normalised form, these two particular vectors appear in other contexts in physics, where they have a more abstract interpretation than simple geometry, such as in the representation of the gluon in particle physics.

Moreover, writing vectors like this makes it a lot easier to transform them via the action of a matrix, by multipying rows in the usual fashion, e.g.
\left(\begin{array}{ccc} \cos \theta & \sin\theta & 0 \\ -\sin\theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array} \right) \left(\begin{array}{c} x \\ y \\ z \end{array} \right) = \left(\begin{array}{c} x\cos \theta + y\sin\theta \\ -x \sin \theta + y\cos \theta \\ z \end{array} \right)
which corresponds to a rotation of the vector in the x-y plane. Transposing a column vector into a row vector is easy too.

Well, that’s how I was taught to do it.

However, somebody, sometime, decided that, in Britain at least, this concise and computationally helpful notation had to be jettisoned and students instead must be forced to write a vector laboriously in terms of base vectors:

\vec{r} = x\hat{\imath} + y \hat{\jmath} + z \hat{k}

Some of you may even be used to doing it that way yourself. Why is this awful? For a start, it’s incredibly clumsy. It is less intuitive, doesn’t lend itself to easy operations on the vectors like I described above, doesn’t translate easily into the more general case of a matrix, and is generally just …well… awful. The only amusing thing about this is that you get to tell students not to put a dot on the “i” or the “j” – it always gets a laugh when you point out that these little dots are called “tittles“.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \hat{\imath} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \vec{k} and it gets even worse, especially when you want to write \exp(i\vec{k}\cdot \vec{x}), and if you want the answer to be the current density \vec{j} then you’re in big trouble!

Call me old-fashioned, but I’ll take the row and column notation any day!

(Actually it’s better still just to use the index notation, a_i which generalises easily to a_{ij} and, for that matter, a^{i}.)

Or perhaps being here in Ireland we should, in honour of Hamilton, do everything in quaternions.

Romanesco and the Golden Spiral

Posted in mathematics, The Universe and Stuff with tags , , , on October 7, 2021 by telescoper

This week’s veggie box included the following beauty

The vegetable in the picture is called Romanesco. I’ve always thought of it as a cauliflower but I’ve more recently learned that it’s more closely related to broccoli. It doesn’t really matter because both broccoli and cauliflower are forms of brassica, which term also covers things like cabbages, kale and spinach. All are very high in vitamins and are also very tasty if cooked appropriately. Incidentally, the leaves of broccoli and cauliflower are perfectly edible (as are those of Romanesco) like those of cabbage, it’s just that we’re more used to eating the flower (or at least the bud).

A while ago, inspired by a piece in Physics World,  I wrote an item about  Romanesco, which points out that a “head” of Romanesco displays a form of self-similarity, in that each floret is a smaller version of the whole bud and also displays structures that are smaller versions of itself. That fractal behaviour is immediately obvious if you take a close look. Here’s a blow-up so you can see more clearly:
romanesco-broccoli2-550x412

There is another remarkable aspect to the pattern of florets, in that they form an almost perfect golden spiral. This is a form of logarithmic spiral that grows every quarter-turn by a factor of the golden ratio:

\phi = \frac{1+\sqrt{5}}{2}.

Logarithmic, or at least approximately logarithmic, spirals occur naturally in a number of settings. Examples include spiral galaxies, various forms of shell, such as that of the nautilus and in the phenomenon of phyllotaxis in plant growth (of which Romanesco is a special case). It would seem that the reason for the occurrence of logarithmic spirals  in living creatures is that such a shape allows them to grow without any change in shape.

Although it is rather beautiful, the main attraction of Romanesco is that it is really delicious. It can be eaten like cauliflower (e.g. in a delicious variation of cauliflower cheese) but my favourite way of cooking it is to roast it with a bit of olive oil, lemon juice and garlic. Yum!