Archive for the mathematics Category
A Formula for the New Year
Posted in mathematics on January 4, 2024 by telescoper
Marking Scheme
Posted in mathematics with tags assessment, Examinations, marking schemes, mathematics on December 8, 2023 by telescoperWith Christmas looming and the January examination period getting closer, I thought I’d help (?) those involved in such assessments by sharing this model of an elegant marking scheme from a Mathematics examination.
What could be simpler?
Do Black Holes have Singularities?
Posted in mathematics, The Universe and Stuff with tags arXiv:2312.00841, black holes, Roy Kerr, singularity on December 7, 2023 by telescoper
A paper appeared on arXiv this week that has ruffled a few feathers. It’s by Roy Kerr (yes, him) and it has the abstract:
There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking then asserted that these must end in actual singularities. When they could not prove this they decreed it to be self evident. It is shown that there are counterexamples through every point in the Kerr metric. These are asymptotic to at least one event horizon and do not end in singularities.
arXiv:2312.00841
I don’t think this paper is as controversial as some people seem to find it. I think most of us have doubts that singularities – specifically curvature singularities – are physically real rather than manifestations of gaps in our understanding. On the other hand, this paper focusses on an interesting technical question and provides a concrete counterexample. The point is that the famous Penrose-Hawking singularity theorems don’t actually prove the existence of singularities; they prove geodesic incompleteness, i.e. that there are geodesics that can only be extended for a finite time as measured by an observer travelling along one. Geodesic incompleteness does imply the existence of some sort of boundary, often termed a trapped surface, but not necessarily that anything physical diverges there at that boundary. Though a singularity will result in geodesic incompleteness, the assertion that geodesic incompleteness necessarily implies the existence of a singularity is really just a conjecture.
For more details, read the paper. It’s technical, of course, but well written and actually not all difficult to understand.
New Publication at the Open Journal of Astrophysics
Posted in mathematics, OJAp Papers, Open Access, The Universe and Stuff with tags arXiv:2309.00948, Bayesian Methods, Curve-fitting, Instrumentation and Methods for Astrophysics on November 17, 2023 by telescoperIt’s Friday afternoon but before I collapse, exhausted, into the arms of the weekend I’ll take the opportunity to announce yet another new paper at the Open Journal of Astrophysics.
The latest paper is the 45th so far in Volume 6 (2023) – just five to go for a half-century – and it’s the 110th altogether. This one was actually published on Tuesday November 14th.
The title is “Marginalised Normal Regression: Unbiased curve fitting in the presence of x-errors” and it’s by Deaglan J. Bartlett (Institut d’Astrophysique de Paris, France) and Harry Desmond (Portsmouth, UK). It sounds like a statistical methods paper, and indeed it is, but remember that there’s a very long historical connection between astronomy and the development of statistical methods for data analysis, and this paper tackles a very longstanding issue: how best to fit curves in the presence of noisy data. This paper presents a new method for doing this, together with applications to cosmological and astrophysical data, and accompanying software. It is in the folder marked Instrumentation and Methods for Astrophysics.
Here is the overlay of the paper containing the abstract:
You can click on the image of the overlay to make it larger should you wish to do so. You can find the officially accepted version of the paper on the arXiv here.
The Returning and The Leaving
Posted in Biographical, Education, mathematics, Maynooth, Politics with tags CAO points, Higher Mathematics, Leaving Certificate on August 26, 2023 by telescoperI got back to Maynooth last night after a pleasantly uneventful train journey. Just for the record both outward and return trips were perfectly on schedule. In fact it has been a very pleasant couple of days. Congratulations to the organizers for running the meeting so well and to all the speakers for delivering such an interesting programme. Next year’s INAM will be in Galway. I’m looking forward to it already!
Anyway, now I’m back I should mentioned that the 2023 Leaving Certificate results were released to students yesterday; the first round of CAO offers will go out on Wednesday 30th. Soon after that we will find out how many students we’ll have next year. Student enrolment begins on 11th September; Orientation Week for new students starts on Monday 18th September; and lectures start the following Monday (25th). I am on sabbatical for a year from next Friday (1st September) so I won’t be teaching the new students, but I know they’ll be in capable hands.
There’s a lot of discussion – much of it poorly informed – in the media about grade inflation in the Leaving Certificate (e.g. here). This happens every year (as it does with A-levels in the UK), and its very sad that people use this occasion to publicly disparage the accomplishments of students. The students can only take the examinations that are put in front of them. Any problems with the system are not their fault at all.
This year the problem stems from a decision by Minister for Education Norma Foley to impose a condition that overall grades this year would not be lower than last year. This has led to the deployment of scaling which has resulted in an uplift of around 8%. The Higher Mathematics Leaving Certificate results also benefitted from an alteration of the marking scheme because one of the papers was deemed to be too hard. Despite this, the number of students receiving the top grade of H1 fell this year from 18% to under 11%. One might argue that this disadvantages students applying to courses that actually require mathematics compared to those that don’t.
There seems to be a widespread misunderstanding that the CAO points required for a course is somehow a measure of the level of difficulty of that course. In most cases this is not the case: having a high points threshold is basically just a way of controlling the number of students allowed in. I find the connection that has been made between grade inflation and drop-out rates extremely unconvincing. High drop-out rates in recent years are probably dominated by the pandemic, housing crisis and cost of living increases, leading to many students struggling to study effectively.
During the pandemic years, grades were inflated by including coursework rather than examinations, a change enforced because of public health restrictions. The main argument for deliberate grade inflation this year was to prevent this year’s LC students being disadvantaged with respect to last year’s. It doesn’t seem to have occurred to the Government that the same argument could be used next year, and indeed forever. Fairly typically for a politician, kicking the can down the road for the next government to deal with seems to be strategy.
As a final thought, I find myself wondering what will happen to admissions at Maynooth this year. Will the decision by The Management to scrap the promised Student Centre have a big effect? And what about the further reputational harm caused by the recent furore over the Governing Authority? I suppose we’ll find out next week!
The 2023 Irish National Astronomy Meeting
Posted in Biographical, History, mathematics with tags Cork, Eduroam, INAM 2023, Seán Mac an tSíthigh, University College Cork on August 24, 2023 by telescoperIt’s been a busy day at the Irish National Astronomy Meeting, culminating with a fascinating public lecture by journalist Seán Mac an tSíthigh – bonus marks for getting the pronunciation right – and a very enjoyable shindig involving pizza and beer. There was a strong Maynooth contribution today, with excellent talks by students Noah, Saoirse, Joe & Hannah and postdocs Lewis & John. My contribution was limited to chairing a session, though I will be giving a talk tomorrow.
The only problem today was that I couldn’t get eduroam to work on the UCC campus so have only just managed to connect after getting back to my hotel, so am a bit late posting this. Anyway, here are some snaps I took on the way this morning, on the campus including a bust of George Boole.
ps. I’m sure to blog again about the public talk, but that will have to wait until I get home at the weekend.
Talking about the Leaving Certificate
Posted in Literature, mathematics, Maynooth, Poetry with tags English Paper 1, English Paper 2, Higher Mathematics, Leaving Cerfificate, Leaving Certificate 2023, Mathematics Paper 1, Mathematics Paper 2 on June 10, 2023 by telescoper
One thing that I forgot to mention in my post about examinations a few days ago is that students at Irish schools all sit exactly the same examination papers at the same time. This is very different from the UK where there are several different Exam Boards that have different syllabuses and set different papers. One consequence of the Irish system is immediately an exam is over, there is a national discussion of the students’ and teachers’ reaction to it. The examination papers are posted online after the examination too – you can find them here – so that everyone can join in the discussion.
I have to admit that when I was a student I was never one for talking about examinations after I had taken them. While most of my peers stood around outside the Exam Hall conducting a post mortem on the paper, I usually just went home. I always figured that there was nothing I could do about the results then so it was best to put it behind me and focus on the next thing. That’s what I’ve recommended to students throughout my career too: don’t look back, look forward.
Anyway, the first Leaving Certificate examination this year (on Wednesday) was English Paper 1, followed by Paper 2 on Thursday. Both seem to have been received relatively favourably by students; see some discussion here and here. Paper 1 is really an English Language Examination, with exercises on comprehension and composition while Paper 2 focuses on literature. Every year summer I look at the set books and poems for the English Leaving Certificate Paper 2 and they’re usually an interesting mix. This year the novels included Mary Shelley’s Frankenstein, Raymond Chandler’s The Big Sleep, Oscar Wilde’s The Picture of Dorian Gray, and Margaret Atwood’s The Handmaid’s Tale. The list of poets for the Higher examination was Elizabeth Bishop, Emily Dickinson, John Donne, Patrick Kavanagh, Derek Mahon, Paula Meehan, Adrienne Rich, and W.B. Yeats. Not all the texts come up in the examination. In the case of the poets, for example, Mahon, Kavanagh, Meehan, Donne and Rich appeared on Paper 2 but there was no Dickinson, Donne or Bishop.
While I have a personal interest in English literature, the English examinations are not relevant to me in a professional capacity. On the other hand, the Leaving Certificate papers in Mathematics are of direct relevance to me as a Professor in the Department of Theoretical Physics because they indicate the level of mathematical preparation of students likely to come in next academic year.
General reaction to Higher Mathematics Paper 1 seems to have been much more mixed than for the English papers, with many students taking to social media to express shock that it was so difficult: the hashtag #MathsPaper1 is still trending on Irish Twitter; you can also find some reaction here.
I have looked at the paper but can’t really comment on the level of difficulty because I haven’s studied previous years examinations in detail but I will say that (a) there’s quite a lot to do in the 150 minutes allowed and (b) there’s nowhere near as much calculus as in my A-level Mathematics over 40 years ago (though remember that Irish students do more subjects in the LC than UK students who do A-levels). Note also that because of the pandemic, this would have been the first state examination taken in Mathematics by many students.
The Leaving Certificate Higher Mathematics examination is split into two sections of equal weight. Section A (‘Concepts and Skills’) requires students to answer 5 questions from 6 (each split into parts); Section B (‘Contexts and Applications’) gives a choice of 3 out of 4 longer questions. That’s less choice than I expected; students have to answer 8 out of 10 questions. The Ordinary Level Examination has the same structure, but the questions are much more straightforward.
Mathematics Paper 2 is on Monday, so I’ll update this post then.
Update: Mathematics Paper 2 seems to have gone down much better than Paper 1. You can find it, along with some reaction, here.
Examinations, Past and Future
Posted in Biographical, Education, mathematics, Maynooth with tags Examinations, Leaving Certificate, Leaving Certificate 2023, mathematics, Maynooth University, Physics on June 7, 2023 by telescoperNo sooner is yesterday’s departmental Examination Board done and dusted (after just two and a half hours) when attention switches to school examinations. The Junior Certificate and Leaving Certificate examinations both start today, so the first thing I need to do is wish everyone taking examinations the very best of luck!
Among other things, the results of the leaving certificate examinations are important for next year’s University admissions. As we gradually dispense with the restrictions imposed during the pandemic, it seems this year we just might have the results before the start of teaching at the end of September. That will make a nice change!
In the system operating in England and Wales the standard qualification for entry is the GCE A-level. Most students take A-levels in three subjects, which gives them a relatively narrow focus although the range of subjects to choose from is rather large. In Ireland the standard qualification is the Leaving Certificate, which comprises a minimum of six subjects, giving students a broader range of knowledge at the sacrifice (perhaps) of a certain amount of depth; it has been decreed for entry into this system that an Irish Leaving Certificate subject counts as about 2/3 of an A-level subject for admissions purposes, so Irish students do the equivalent of at least four A-levels, and many do more than this. It’s also worth noting that all students have to take Mathematics at Leaving Certificate level.
Overall I prefer the Leaving Certificate over the UK system of A-levels, as the former gives the students a broader range of subjects than the latter (as does the International Baccalaureate). I would have liked to have been allowed to take at least one arts subject past O-level, for example.
For University admissions points are awarded for each paper according to the marks obtained and then aggregated into a total CAO points, CAO being the Central Applications Office, the equivalent of the UK’s UCAS. This means, for example, that our main Science pathway at Maynooth allows students to study Physics without having done it at Leaving Certificate level. This obviously means that the first year has to be taught at a fairly elementary level, but it has the enormous benefit of allowing us to recruit students whose schools do not offer Physics.
As much as I like the Leaving Certificate, I have concerns about using a simple CAO points count for determining entry into third-level courses. My main concern about is with Mathematics. Since the pandemic struck, students have been able to choose to questions from just six out of ten sections. That means that students can get very high grades despite knowing nothing about 40% of the syllabus. That matters most for subjects that require students to have certain skills and knowledge for entry into University, such as Physics.
I’ve been teaching the first year Mathematical Physics course in Maynooth for about 5 years. At the start of the module I put up a questionnaire asking the students about various mathematical concepts and asking them how comfortable they feel with them. It’s been noticeable how the fraction that are comfortable with basic differentiation and integration has been falling. That’s not a reflection on the ability of the students, just on the way they have been taught. As well as making adjustments during the pandemic for online teaching, etc, I have changed various things about the teaching, in particular adjusting the way I have introduced calculus into the module. Another problem is that we have been forced to start teaching first-years a week late because of delays to the CAO process caused by the pandemic.
I’ll be on sabbatical next academic year so I won’t be teaching the first-years (or anyone else) in September. It’s time to hand these challenges on to someone else!
A Question of Images
Posted in Cute Problems, mathematics, The Universe and Stuff with tags electrostatics, method of images on May 10, 2023 by telescoperToday I gave a revision lecture/tutorial for my module Advanced Electromagnetism. With the Examination Period starting on Friday, that was the last class I will do for that. One of the topics I’ve been asked to cover in revision was the Method of Images for electrostatics. Preparing for the class I came across this cute problem which I thought I’d share here:
The question concerns a charge +q placed at a distance d as shown above an infinite earthed conducting plane distorted by the presence of a hemispherical bulge with radius R.
- Using the method of images, or otherwise, calculate the potential at an arbitrary point above the conducting surface. (HINT: you need three image charges)
- Find the magnitude and direction of the electrostatic force on the charge.
If you’re feeling keen you might also find what fraction of the total induced on the conductor is on the hemispherical part.
Answers through the comments box please!
Well, nobody posted an answer so here’s an outline solution.
To solve this problem you need three image charges: one is of charge – q at z=-d to make the plane an equipotential. For an isolated sphere you need a charge of -qR/d at z=-R^2/d (the inverse point of the sphere). But this charge also has an effect on the plane, which you need to correct by placing another image charge of +qR/d at z=-R^2/d. That is, the solution for the potential is due to the original charge plus three image charges. Then the potential is just the sum of four point charges.
You can differentiate the answer to the first bit to get the force, or you could work out the force on the original charge directly by adding the forces in the z-direction from the three image charges, it being obvious by symmetry that there is no other component of the force. For d>R this results in a force which is downward, so the charge is pulled towards the conductor. I’ll leave that as an exercise!
In the Dark 