Archive for the mathematics Category

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?

Thought for the Day

Posted in mathematics, Maynooth on October 15, 2020 by telescoper

Littlewood on `the real point’ of lectures

Posted in Education, mathematics, The Universe and Stuff with tags , , , on September 3, 2020 by telescoper

We’re often challenged these days to defend the educational value of the lecture as opposed to other forms of delivery, especially with the restrictions on large lectures imposed by Covid-19. But this is not a new debate. The mathematician J.E. Littlewood felt necessary to defend the lecture as a medium of instruction (in the context of advanced mathematics) way back in 1926 in the Introduction to his book The Elements of the Theory of Real Functions.

(as quoted by G. Temple in his Inaugural Lecture as Sedleian Professor of Natural Philosophy at the University of Oxford in 1954 “The Classic and Romantic in Natural Philosophy”.)

Temple concluded his lecture with:

Classic perfection should be reserved for the monograph: the successful lecture is almost inevitably a romantic adventure. It is at once the grandeur and misery of a scientific classic that it says the last word: it is the charm of a scientific romance that it utters the first word, and thus opens the windows on a new world.

Modern textbooks do try to be more user-friendly than perhaps they were in Littlewood’s day, and they aren’t always “complete and accurate” either, but I think Littlewood is right in pointing out that they do often hide `the real point’ so students sometimes can’t see the wood for the trees. The value of lectures is not in trying to deliver masses of detail but to point out the important bits.

It seems apt to mention that the things I remember best from my undergraduate lectures at Cambridge are not what’s in my lecture notes – most of which I still have, incidentally – but some of the asides made by the lectures. In particular I remember Peter Scheuer who taught Electrodynamics & Relativity talking about his first experience of radio astronomy. He didn’t like electronics at all and wasn’t sure radio astronomy was for him, but someone – possibly Martin Ryle – reassured him by saying “All you need to know in order to do this is Ohm’s Law. But you need to know it bloody well.”

On Grinds

Posted in Literature, mathematics with tags , , , , on July 24, 2020 by telescoper

When I moved to Ireland a couple of years ago, one of the words I discovered had a usage with which I was unfamiliar was grind. My first encounter with this word was after a lecture on vector calculus when a student asked if I knew of anyone who could offer him grinds. I didn’t know what he meant but was sure it wasn’t the meaning that sprang first into my mind so I just said no, I had just arrived in Ireland so didn’t know of anyone. I resisted the temptation to suggest he try finding an appropriate person via Grindr.

I only found out later that grinds are a form of private tuition and they are quite a big industry in Ireland, particularly at secondary school level. School students whose parents can afford it often take grinds in particular subjects to improve their performance on the Leaving Certificate. It seems to be less common for third level students to pay for grinds, but it does happen. More frequently university students actually offer grinds to local schoolkids as a kind of part-time employment to help them through college.

The word grind can also refer to a private tutor, i.e. you can have a maths grind. It can also be used as a verb, in which sense it means `to instil or teach by persistent repetition’.

This sense of the word grind may be in use in the United Kingdom but I have never come across it before, and it seems to me to be specific to Ireland.

All of which brings me back to vector calculus, via Charles Dickens.

In Hard Times by Charles Dickens there is a character by the name of Mr Thomas Gradgrind, a grimly utilitarain school superintendent who insisted on teaching only facts.

Thomas Gradgrind (engraving by Sol Eytinge, 1867).

If there is a Mr Gradgrind, why is there neither a Mr Divgrind nor a Mr Curlgrind?

Straight from Ireland

Posted in mathematics with tags on June 15, 2020 by telescoper

I came across this the other day. I think it’s fun because it’s a bit counterintuitive and it has generated quite a lot of discussion so I thought I would share it here. Two things are worth amplifying:

  1. By “in a straight line” I assume it means “along a great circle“.
  2. As it states in the small print on the diagram all lines originate at the geographical centre of Ireland which apparently lies at a place called Carnagh East, close to the border between County Roscommon and County Westmeath.

The main bone of contention is why the USA looks so small, in the matter of which I direct you to this reddit thread. The answer is clear when you look at what a great circle from Ireland to the USA looks like: most great circles from Ireland to the Eastern seaboard pass over Canada: