Archive for the mathematics Category

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Lá Saoire i mí Lúnasa

Posted in Biographical, Education, mathematics, Maynooth with tags , , , , on August 3, 2026 by telescoper

Today, Monday 3rd August 2026, being the first Monday in August, is a Bank Holiday in Ireland. This holiday was created by the Bank Holiday Act of 1871 when Ireland was under British rule. While the August Bank Holiday was subsequently moved to the end of August in England and Wales, it has remained at the start of August in Ireland. It has been a nice morning weather-wise, but in true Bank Holiday fashion we’re promised heavy rain later on, with a chance of thunderstorms.

As I mentioned on Saturday, the first day of August marks the old pagan festival of Lughnasadh, named after the God Lugh, on which is celebrated the beginning of the harvest season. This coincides with the English Lammas Day, one of many Christian festivals with pagan origins. Traditionally this is the start of the harvest season and is celebrated accordingly, with rites involving the first fruit and bread baked from flour obtained from the first corn. It is also one of the cross-quarter days, lying roughly half-way between the Summer Solstice and the Autumnal Equinox (in the Northern Hemisphere).

Two years ago at this time I was still on sabbatical. That seems like ages ago! When I resumed teaching, in September 2024, I had to teach two modules I’d never taught before: a fourth-year Mathematical physics course on Differential Equations and Complex Analysis and a second-year Engineering Mathematics course. I am doing both of these again so resumption of teaching in September won’t be as challenging, although I will be changing a few things in the light of last year’s experience. Lectures start on Monday 22nd September.

After I return to work tomorrow the next big item on the agenda is the repeat examination period, which starts on Wednesday August 5th. The first paper for which I have responsibility is on Friday 7th, but I won’t be able to get the scripts until Monday 10th so I have a bit of time to do other stuff before I start marking. I am not expecting many scripts, actually, so it shouldn’t be too onerous a task to get them done promptly.

Next week also sees the deadline for the dissertations for our MSc course. I have been supervising a student this year and will probably be asked to give comments on the final draft of his report. Then the MSc students have their presentations on Friday 14th August. They have all completed the taught component of the course so the only thing left to assess are these projects. Students will get their final results later this month, along with the resitting undergraduate students.

In between all this we have our Eclipse Event on August 12th. This sold out very quickly so we expect a good turnout. The event was advertised quite a bit in the local media, e.g. here and here:

The viewing of the eclipse will of course depend on the weather, but we have talks lined up indoors and a pop-up planetarium so I hope people will turn up even if the weather isn’t great. Although it is officially full, there may be a few returns, so it is worth checking on the day to see if we can fit you in.

Anyway, after the repeat examinations and project assessments are done, the marks uploaded, and the Undergraduate and Postgraduate Examination Boards have done their work, the next job will be to prepare for the new intake of students. This year’s Leaving Certificate results will be announced on Friday 21st August, at which point we’ll see how many students (if any) we have studying Physics next academic year (which, if all goes to plan, will be my penultimate…)

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Composition Derived from the Equation y= −ax²+bx−18 with Green, Orange, Violet (Black) – Georges Vantongerloo

Posted in Art, mathematics with tags , , , on July 22, 2026 by telescoper

Composition Derived from the Equation y=−ax2+bx+cy=-ax^2+bx+c with Green, Orange, Violet (Black) by Georges Vantongerloo (1930, Oil on Canvas, 119.4 x 68.2 cm, Guggenheim Museum, New York)

I leave it as an exercise to the reader to determine how this painting is derived from the stated equation.

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Euclid Update – Revised Timeline

Posted in Euclid, mathematics, The Universe and Stuff with tags , , , , , , , , on June 16, 2026 by telescoper

In a post last week I hinted that I hoped soon to be able to provide an update on progress with the European Space Agency’s Euclid Mission. People within the Euclid Consortium have known for some time, but it has now been officially announced by ESA, that the timeline for the first full Data Release (DR1; originally scheduled for October 2026) has now been revised.

The plan now is for DR1 to happen in stages, with a first tranche to occur in November 2026 (precise date yet to be announced). The complete data release will take place in mid-2027 (probably in June). The sky area covered by DR1 will be about covering a large sky area of about 1900 deg².

To give a little more background, the data products from the Euclid survey are divided into three levels of data processing function:

  • LE1 – the “raw” data frames, prior to calibration, generated at the Science Operations Centre from the time-ordered data and telemetry received from the spacecraft
  • LE2 – the calibrated and corrected data for the two instruments (VIS and NISP) – images, spectra, catalogues of point sources, etc
  • LE3 – the high-level data products (galaxy catalogues, cosmic shear maps, etc) designed for cosmological analysis

The first release will comprise LE1 and LE2 only. This will be called DR-Foundation. The LE3 data will be added next year to make the full DR1. Since LE3 is required for the cosmological analysis that is the prime motivation for the Euclid mission, it follows that there will be no official cosmology results from Euclid DR1 until mid-2027 at the earliest. Other results based on “Foundation” data may of course emerge before then.

I hope this clarifies the situation.

P.S. Irish physicists and astronomers will be particularly interested to know that the LE1 data includes information about the spacecraft and instrument pointing orientations, which is stored in the form of quaternions…

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The 2026 Leaving Certificate Mathematics Papers

Posted in Education, mathematics with tags , , on June 8, 2026 by telescoper

As I mentioned a few days ago, examinations for the 2026 school Leaving Certificate are under way. One of the interesting things about the Irish system is that the examination papers are put up online immediately after the examinations. Students took their first paper in Mathematics (either Ordinary or Higher level) on Friday and the second was this morning. There has been some reaction in the news here and here.

Anyway, I thought I’d share the Mathematics papers here so you can see what you think of them.

Here are the two Higher Mathematics papers:

LC003ALP100EVDownload
LC003ALP200EV (2)Download

The Ordinary Level papers are here:

LC003GLP100EV (2)Download
LC003GLP200EV (1)Download

They look reasonable to me. The thing that strikes me about them is that they are much more structured than the A-level mathematics examinations I took way back in 1981.

Comments are welcome through the box below.

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Newtonian Mechanics is Simple…

Posted in Education, mathematics with tags , on May 29, 2026 by telescoper


People keep telling me that Newtonian Mechanics is simple. Here’s a demonstration of the apparently simple Double Pendulum that demonstrates very complicated behaviour.

P.S. I did not make the video – it can be found all over the interrnet. I am sharing it here for the education and entertainment of my readers, both of them.

P.P.S. The music is a track called “Swing Theory”…

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That Scottish Higher Maths Paper 1…

Posted in Education, mathematics with tags , , , on May 15, 2026 by telescoper

Talking of Examinations, I saw an article on the BBC website about a recent Higher Maths paper in Scotland which has generated complaints and a petition because it was allegedly unfair. The introduction to the petition states:

This is not a complaint that the paper was too hard. Students expect to be challenged. The problem is that the 2026 Higher Maths Paper 1 used language and phrasing that was confusing, ambiguous, and inconsistent with every past paper students had revised from. Questions were not simply difficult — they were worded in ways that made it genuinely unclear what was being asked.

Past SQA Higher Maths papers have followed a recognisable style: clear command words, standard notation, and questions that test understanding rather than the ability to decode unusual phrasing. The 2026 Paper 1 departed from this in ways that penalised well-prepared students simply because the wording did not match the conventions they had been taught to expect.

Numerous other news outlets have covered the story too. It is frustrating that most of the pieces focus on what people said about the paper but don’t actually include a link to the paper itself, making it impossible to make your own mind up.

So you can make your own mind up here is a scan of the actual paper (obtained from here):

2026_Higher_Maths_Paper_1_ScanDownload

Bear in mind that the Scottish examination system is not the same as in England & Wales – the “Highers” are not as advanced as A-levels and are more similar to the Irish Leaving Certificate.

My opinion, for what it’s worth having neither taught nor studied in the Scottish system, is that there is nothing out of the ordinary with this paper. There is a lot to do in just 75 minutes – for 12 questions that’s just over 6 minutes a question. I don’t like examinations that are speed tests.

That said, the questions look well structured and the “command words” are without exception on the list here. Some questions are easy and others harder: I think Question 12 is the most difficult, but I think that’s intentional – to stretch the stronger students. The only thing I would quibble with is the wording of 11(a) (ii):

The second sentence is redundant. How can one possibly “explain why” without giving “a reason”? The reason is basically that the quadratic remaining after you have taken out the factor (x+2) does not factorize.

I looked at the 2025 Paper 1 and it seems a similar level, though the questions are phrased in a terser fashion. Here it is for reference:

NH_Mathematics_Paper1-Non-calculator_2025Download

There may well be context that I’m missing, however, so I’d welcome comments on the diffculty and/or fairness through the box below.

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John Venn’s Bowling Machine

Posted in Cricket, mathematics with tags , , on April 14, 2026 by telescoper

Most readers of this blog will be aware of the existence of Venn Diagrams, but not a lot of people know that their creator, John Venn, invented a mechanical bowling machine for use by cricketers in batting practice. Recently some folks at Cambridge University built a replica of this machine, presumably with the aid of some sort of diagram, and here’s a fascinating video about it.

Modern bowling machines fire the ball at you like a cannon, but this machine has the virtue of the batter being able to see the arm coming over which must help a little. This creation is also able to impart quite a bit of spin, in either direction but it looks like it should be quite easy to pick. The examples shown in the video also demonstratte that batting is only partly about hand-eye coordination – a key element is how to move your feet!

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Ahead of a Four-Day Week…

Posted in Artificial Intelligence, Biographical, Education, mathematics, Maynooth with tags , , , , , , on March 29, 2026 by telescoper

It’s Sunday 29th March – Palm Sunday in fact – and Friday 3rd April is Good Friday, which is followed by a break of a week for Easter, so I’ve been looking at what I have to do in the four days between Monday and Thursday.

On Tuesday afternoon part of my 3rd Year Computational Physics class will be doing a supervised test in the computer lab. I (foolishly) promised to ensure they would get their grades before Easter, so I’m going to have to mark them straight away. This is a larger group than usual because some students who would normally be in the lab on Thursdays swtiched to Tuesday so they could go on a trip to Armagh. Anyway, this is the third lab test and at least I have graded the first two tests for all groups in time for the arrival of the new ones.

There will be one more of these lab tests after the Easter break but after that the students will be working full-time for 3 weeks or so on mini-projects. That is the part they usually enjoy most and I’m very happy to see that some have already started work.

Then, on Wednesday I have the second class test for my 4th year Particle Physics module. This is the second such test, and it will be held during a tutorial session. This is a pen-and-paper test rather than a coding test to be done in the lab. For such tests I allow students to bring whatever they like on paper but phones, laptops and tablets are banned. This is the easiest way I could think of to avoid students using AI to solve the problems. In previous years I gave take-home assignments for this module, and I still hand out exercise sheets to be gone over in tutorials, but these are for formative purposes only. The summative assessments are the class tests. There will be three of those, which means they will have to endure one more after Easter. In a normal week I would have a Particle lecture on Friday, but that won’t happen because it’s Good Friday and my lectures apparently aren’t good enough to happen on that day.

As well as the Computational Physics lab test and the Particle Physics class test, next I have two lectures, both at 9am – one on Tuesday and one on Thursday – and another lab session on Thursday which is not a test, but a practical session about solving ODEs.

Then it will be the Easter Break. After that, according to my calculations, there will be four more teaching weeks before the examination period. The last day of teaching is May 8th. Between that and the examinations there is a gap of a week during which I will have to mark all the completed Computational Physics project reports, as well as giving some revision classes if there is demand for them.

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A Quarter Term

Posted in Biographical, Education, mathematics, The Universe and Stuff with tags , , on February 20, 2026 by telescoper
I searched for a free stock image using “quarks” as a search term and found this photo by Anh Nguyen on Pexels.com.

On most Friday afternoons there is a seminar in the Physics Department at Maynooth University, and I got it into my mind that there was one this afternoon and set aside an hour to attend it. It turns out that there isn’t a seminar so I have time to write a quick blog post before I head to Dublin for a concert.

We’ve (almost) reached the end of Week 3 of the Semester which means we’re about a quarter of the way through the term. Yesterday we did the first Lab Test of four in Computational Physics, which went off without any major problems. The first class test in Particle Physics will be next week and I hope that goes equally well.

Today’s the day students officially received their provisional first semester results. When I arrived at my particle physics lecture this morning the students were discussing their exam marks. Most seemed relatively happy, which is good because this is the final year for most of these students so their grades matter more now than in previous years.

This morning’s lecture was quite amusing. I was discussing electrostatic interactions between quarks in nucleons and in the course of that I asked the class to calculate (2/3)×(-1/3) +(-1/3)×(-1/3)+(-1/3)×(2/3). It took a surprisingly long time to arrive at the right answer! To make matters worse, when I announced the correct answer I got the sign wrong*.

It’s been a long week.

Anyway, next week I’ll be starting on the Dirac Equation, and on the basis of today’s events I wonder about the wisdom of having a lecturer who can’t do minus signs teach relativistic quantum mechanics to students who struggle to do simple arithmetic with fractions!

*ANS=-1/3

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The Next Semester

Posted in Artificial Intelligence, Education, mathematics, Maynooth with tags , , , , , , , on January 26, 2026 by telescoper

There’s just a week to go before the next Semester at Maynooth University so I’ve been looking at my calendar for the weeks ahead. Actually, I won’t start teaching again until Tuesday 3rd February, because Monday 2nd February is a national holiday. As it turns out, however, I don’t have any lectures, labs or tutorials on Mondays anyway so I won’t be missing a session either on February 2nd or on May 4th, another holiday. I will have to miss one on Friday 3rd April (Good Friday), though.

The Timetable has given me two 9 o’clock lectures a week for the forthcoming Semester, one on Tuesdays and the other on Thursdays. I don’t think the students like 9am lectures very much, but I don’t mind them at all. I find it quite agreeable to have accomplished something concrete by 10am, which I don’t always do. This schedule might mean that I defer publishing papers at the Open Journal of Astrophysics on those days. I usually do this before breakfast, but I might not have time if I have to be on campus and ready to teach for 9am.

As usual, Semester 2 is a stop-start affair. We have six weeks until the Study Break, which includes the St Patrick’s Day holiday, then we’re back for two weeks (minus Good Friday) before another week off for Easter. We return on Monday April 13th to complete the Semester; the last lectures are on Friday 8th May and exams start a week later. This arrangement creates no problems for lecture-based teaching, but it takes some planning to organize labs and project deadlines around the breaks. I’ll have to think about that for my Computational Physics module.

A more serious issue for Computational Physics is how to deal with the use of Generative AI. I’ve written about this before, in general terms, but now it’s time to write down some specific rules for a specific module. A default position favoured by some in the Department is that students should not use GenAI at all. I think that would be silly. Graduates will definitely be using CoPilot or equivalent if they write code in the world outside university so we should teach them how to use it properly and effectively.

In particular, such methods usually produce a plausible answer, but how can a student be sure it is correct? It seems to me that we should place an emphasis on what steps a student has taken to check an answer, which of course they should do whether they used GenAI or did it themselves. If it’s a piece of code to do a numerical integration of a differential equation, for example, the student should test it using known analytic solutions to check it gets them right. If it’s the answer to a mathematical problem, one can check whether it does indeed solve the original equation (with the appropriate boundary conditions).

If anyone out there reading this blog has any advice to share, or even a link to their own Department’s policy on the use of GenAI in computational physics for me to copy adapt for use in Maynooth, I’d be very grateful!

(My backup plan is to ask ChatGPT to generate an appropriate policy…)

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