Archive for the mathematics Category

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!

How to fit any dataset with a single parameter

Posted in mathematics with tags , , , , on October 1, 2021 by telescoper

There’s a famous quote that physicist Enrico Fermi attributed to John von Neumann that goes something like

With four parameters I can fit an elephant and with five I can make him wiggle his trunk.

Well, there’s a fun paper by Laurent Boué on the arXiv with the title Real numbers, data science and chaos: How to fit any dataset with a single parameter. If von Neumann were alive today he’d be turning in his grave!

The abstract of the paper, which is not new but which I only came across recently, reads

We show how any dataset of any modality (time-series, images, sound…) can be approximated by a well-behaved (continuous, differentiable…) scalar function with a single real-valued parameter. Building upon elementary concepts from chaos theory, we adopt a pedagogical approach demonstrating how to adjust this parameter in order to achieve arbitrary precision fit to all samples of the data. Targeting an audience of data scientists with a taste for the curious and unusual, the results presented here expand on previous similar observations regarding expressiveness power and generalization of machine learning models.

The function of which this claim is made is

(which actually has two parameters but τ is “a constant that basically controls the desired level of accuracy”). Here some examples of the datasets that can be fitted for various values of α:

If you want to know more, read the paper!

 

Ireland’s Covid-19 Models

Posted in Covid-19, mathematics, Maynooth with tags , , , , , on July 1, 2021 by telescoper

Yesterday the Chair of the National Public Health Emergency Team (NPHET), who also happens to be the President of Maynooth University, Professor Philip Nolan published a lengthy but interesting Twitter thread (which you can find unrolled here). In these tweets he explained the reason behind NPHET’s recommendation to pause the process of relaxing Covid-19 restrictions, postponing the next phase which was due to begin on 5th July with indoor dining.

The basic reason for this is obvious. When restrictions were lifted last summer the reproduction number increased to a value in the range 1.4 to 1.6 but the infection rate was then just a handful per day (on July 1st 2020 the number of new cases reported was 6). Now the figures are orders of magnitude higher (yesterday saw 452 new cases). A period of exponential growth starting from such a high base would be catastrophic. It was bad enough last year starting from much lower levels and the Delta variant currently in circulation is more transmissable. Vaccination obviously helps, but only about 40% of the Irish population is fully immunized.

Incidentally the target earlier this year was that 82% of the adult population should have received one jab. We are missing detailed numbers because of the recent ransomware attack on the HSE system, but it is clear that number has been missed by a considerable margin. The correct figure is more like 67%. Moreover, one dose does not provide adequate protection against the Delta variant so we’re really not in a good position this summer. In fact I think there’s a strong possibility that we’ll be starting the 2021/22 academic year in worse shape than we did last year.

In general think the Government’s decision was entirely reasonable, though it obviously didn’t go down well with the hospitality sector and others. What does not seem reasonable to me is the suggestion that restaurants should be open for indoor dining only for people who are fully vaccinated. This would not only be very difficult to police, but also ignores the fact that the vast majority of people serving food in such environments would not be vaccinated and are therefore at high risk.

As things stand, I think it highly unlikely that campuses will be open in September. Rapidly growing pockets of Delta variant have already been seeded in Ireland (and elsewhere in Europe). It seems much more likely to me that September will see us yet again in a hard lockdown with all teaching online.

But the main reason for writing this post is that the thread I mentioned above includes a link to a paper on the arXiv (by Gleeson et al.) that describes the model used to describe the pandemic here in Ireland. Here is the abstract:

We describe the population-based SEIR (susceptible, exposed, infected, removed) model developed by the Irish Epidemiological Modelling Advisory Group (IEMAG), which advises the Irish government on COVID-19 responses. The model assumes a time-varying effective contact rate (equivalently, a time-varying reproduction number) to model the effect of non-pharmaceutical interventions. A crucial technical challenge in applying such models is their accurate calibration to observed data, e.g., to the daily number of confirmed new cases, as the past history of the disease strongly affects predictions of future scenarios. We demonstrate an approach based on inversion of the SEIR equations in conjunction with statistical modelling and spline-fitting of the data, to produce a robust methodology for calibration of a wide class of models of this type.

You can download a PDF of the paper here.

This model is a more complicated variation of the standard compartment-based models described here. Here’s a schematic of the structure:

This model that makes a number of simplifying assumptions but it does capture the main features of the growth of the pandemic reasonably well.

Coincidentally I set a Computational Physics project this year that involved developing a Python code that does numerical solutions of this model. It’s not physics of course, but the network of equations is similar to what you mind find in physical systems – it’s basically just a set of coupled ODEs- and I thought it would be interesting because it was topical. The main point is that if you study Theoretical Physics you can apply the knowledge and skills you obtain in a huge range of fields and disciplines. Developing the model does of course require domain-specific epidemiological knowledge but the general task of modelling complex time-evolving systems is definitely something physicists should be adept at doing. Transferable skills is the name of the game!

P.S. It came as no surprise to learn that the first author of the modelling paper, Prof. James Gleeson of the University of Limerick, has an MSc in Mathematical Physics.

Grand National Takeaway

Posted in mathematics, Sport with tags , , , on April 11, 2021 by telescoper

Congratulations to Rachael Blackmore for becoming the first female jockey ever to ride the winner Minella Times of the Grand National yesterday. It was a good race for Ireland generally as the top five were all Irish horses.

The race was led for a long time by 80-1 outsider Jett who at one point was about 10 lengths clear of the field but you could see that about three fences from home the horse was very tired, fading badly over the final stages of the race to finish in eighth place.

At 11-1, Minella Times would have netted quite a few people a good return on their investment. I wasn’t so lucky but had a modest success. After studying the form carefully (i.e. sticking a pin in the list of runners), I settled on Any Second Now, also at 11-1, betting €25 each way. I was pleased yesterday to see the odds shortening to 15/2 at the start, which meant quite a lot of people were backing the same horse.

In the event Any Second Now finished 3rd which was a great result given that it was badly hampered by a faller (Double Shuffle) at the 12th fence. A thing like that is normally difficult to recover from but jockey Mark Walsh did well to get back in contention, though he was too far back and too tired to catch the winner, who ran a perfect race.

The Grand National is one race where I think an each-way bet is a sensible strategy. As a handicap with 40 runners (and a very tough race for which the probability of a horse not making it to the finish line is quite high) the odds are usually pretty long even on the favourite, and most bookies pay out for a place down to sixth. I bet €25 each way, which means €25 to win outright at 11-1 and €25 for a place at one-fifth the odds, i.e. 2.2 to 1. I lost the first €25 but won €55 on the place (plus the stake). My net result was therefore €50 staked for an €80, more than enough profit to pay for last night’s takeaway dinner.

The point is that if you want the place to cover the loss on the win the starting price has to be good. If the odds are N:1 they will only cover the loss if N/5 ≥1 with the equality meaning that you break even. In a race in which the odds are much shorter the place bet is usually not worth very much at all. In yesterday’s Grand National the favourite was 5-1.

Decimal Day – 50 Years On!

Posted in Biographical, History, mathematics with tags , , , , , on February 15, 2021 by telescoper

The old half-crown coin (2/6)

People of a certain age will remember that fifty years ago today, on 15th February 1971, it was Decimal Day. That was the day that the United Kingdom finally switched completely to the “new money”. Ireland made a similar switch on the same day. Out went old shillings and pennies and in came “new pence”. Old pennies were always abbreviated as `d’ but the new ones were `p’.

In the old system there were 12 pennies in a shilling and 20 shillings in a pound. The pound was therefore 240 old pennies while in the new money it became 100 new pence.

It was not only shillings that disappeared in the process of decimalization. The old ten-bob note (10 shillings) made way for what is now the 50p piece. The shilling coin became 5p. The sixpence was no longer minted after 1970 but stayed in circulation until 1980, worth 2½p.

The crown (5 shillings) and half-crown (two shillings and sixpence, written 2s 6d or 2/6) disappeared, as did the threepenny bit. For a personal story about the latter, see here.

The old penny was a very large and heavy coin, whereas the new one was much smaller despite being worth more. If you had an old penny in your pocket you felt you had something substantial where as one new “pee” seemed insignificant. Even the ha’penny was quite a big piece.

At first, to echo the old ha’penny, there was a ½p coin but that was discontinued in 1984. The old farthing (a quarter of an old penny) had long since ceased to be legal tender (in 1960) although we still had some in the house for some reason.

I was just 7 on Decimal Day but I remember some things about it rather well. There were jingles on the radio announcing Decimal Day and at Junior School we played “Decimal Bingo” to get used to the new money. I remember taking our elderly neighbour’s ten-bob notes to the Post Office to change them into the new coins, though this would have been before Decimal Day as the ten-bob note was phased out in 1970. I remember my Grandad being convinced that the Government had stolen 140 pennies out of every pound he owned…

Youngsters probably find the old system incredibly cumbersome and archaic, which in some ways it was, but at least it got us doing arithmetic in different bases (i.e. base 12 and based 20). The advantage of base 12 is that it has prime factors 2, 3, 4, and 6 so is relatively easier to divide into equal shares; base 10 only has 2 and 5.

Imperial weights and measures also included base 3 (feet in a yard), 8 (pints in a gallon), 14 (pounds in a stone) and 16 (ounces in a pound). I have to admit that to this day when I follow a cookery recipe if it says “100 g” of something, I have to convert that to ounces before I can visualize what it is!

NUI Dr Éamon De Valera Post-Doctoral Fellowship in Mathematical Sciences

Posted in History, mathematics, Maynooth, The Universe and Stuff with tags , , , on January 21, 2021 by telescoper

I found out yesterday that the National University of Ireland is commemorating the centenary of the election of Éamon de Valera as its Chancellor. To mark this occasion, NUI will offer a special NUI Dr Éamon De Valera Post-Doctoral Fellowship in Mathematical Sciences. This post is in addition to the regular NUI awards, which include a position for Science & Engineering.

Éamon de Valera, photographed sometime during the 1920s.

Éamon de Valera, founder of Fianna Fáil (formerly one of the two largest political parties in Ireland) and architect of the Irish constitution. De Valera (nickname `Dev’) is an enigmatic figure, who was a Commandant in the Irish Republican Army during the 1916 Easter Rising, who subsequently became Taoiseach  and then President of the Irish Republic.

You may or may not know that de Valera was a mathematics graduate, and for a short time (1912-13) he was Head of the Department of Mathematics and Mathematical Physics at St Patrick’s College, Maynooth,  a recognized college of the National University of Ireland. The Department became incorporated in Maynooth University, when it was created in 1997.Mathematical Physics is no longer a part of the Mathematics Department at Maynooth, having become a Department in its own right and it recently changed its name to the Department of Theoretical Physics.

Anyway, the Fellowship will be awarded on the basis of a common competition open to NUI graduates in all branches of the Mathematical Sciences. All branches of the Mathematical Sciences will be deemed as including, but not limited to, all academic disciplines within Applied Mathematics, Pure Mathematics, Mathematical Physics and Statistics and Probability.

You can find more details of the position here. I should say however that it is open to NUI graduates only, though it can be held at any of the constituent colleges of the National University of Ireland. Given the de Valera connection with Maynooth, it would be fitting if it were held here!
The deadline for applications is February 9th.

Calculations, Calculations…

Posted in Biographical, mathematics, Politics on November 6, 2020 by telescoper

So it’s past 1pm GMT on Friday 6th November and the USA is still trying to work out who will be its next President after the elections that took place on Tuesday. The process is taking so long I wonder if Americans might be starting to appreciate the nature of Test Match cricket?

In the meantime I’ve been occupying myself with some simpler calculations for my second-year vector calculus module:

Standing Up for Online Lectures

Posted in Covid-19, Education, mathematics, Maynooth with tags , , , , on November 3, 2020 by telescoper

I have a break of an hour between my last lecture on Vector Calculus (during which I introduced and did some applications of Green’s Theorem) and my next one on Mechanics & Special Relativity (during which I’m doing projectile motion), so I thought I’d share a couple of thoughts about online teaching.

I started the term by doing my lectures in the form of webcasts live from lecture theatres but since we returned from the Study Break on Monday I’ve been doing them remotely from the comfort of my office at home, which is equipped with a blackboard (installed, I might add, at my own expense….)

I still do these teaching sessions “live”, though, rather than recording them all offline. I toyed with the idea of doing the latter but decided that the former works better for me. Not surprisingly I don’t get full attendance at the live sessions, but I do get around half the registered students. The others can watch the recordings at their own convenience. Perhaps those who do take the live webcasts appreciate the structure that a regular time gives to their study. Even if that’s not the reason for them, I certainly prefer working around a stable framework of teaching sessions.

“Why am I still using a blackboard?” I hear you ask. It’s not just because I’m an old fogey (although I am that). It’s because I’m used to pacing myself that way, using the physical effort of writing on the blackboard to slow myself down. I know some lecturers are delivering material on slides using, e.g., Powerpoint, but I have never felt comfortable using that medium for mathematical work. Aside from the temptation to go too fast, I think it encourages students to see the subject as a finished thing to be memorized rather than a process happening in front of them.

I did acquire some drawing tablets for staff to enable them to write mathematical work out, which is useful for short things like tutorial questions, but frankly they aren’t very good and I wouldn’t want to use them to give an hour long lecture.

In addition to these considerations, my decision to record videos in front of a blackboard was informed by something I’ve learnt about myself, namely that I find I am much more comfortable talking in this way when I’m standing up than sitting down. In particular, I find it far easier to communicate enthusiasm, make gestures, and generally produce a reasonable performance if I’m standing up. I know several colleagues who do theirs sitting down talking to a laptop camera, but I find that very difficult. Maybe I’m just weird. Who else prefers to do it standing up?