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:

The Ordinary Level papers are here:

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.

Nine Purple Dots

Posted in Uncategorized with tags , , , on June 8, 2026 by telescoper

I posted this little optical illusion yesterday on BlueSky and it proved so popular I thought I’d repeat it here.Whichever of the purple dots you look directly at turns more purple:

I find the effect is stronger on a smaller screen (e.g. on a phone).

In a nutshell, this happens because your eyes have reduced sensitivity to blue light at the centre of your field of view so the purple (which is a mixture of red and blue) changes hue if you look directly at it. For more details see here and the following blog post:

I’m interested to know whether or not this works for people who are colour blind, so if that is you please let me know!

June – Francis Ledwidge

Posted in Poetry with tags , , , on June 7, 2026 by telescoper
Wilted Roses
Broom out the floor now, lay the fender by,
And plant this bee-sucked bough of woodbine there,
And let the window down. The butterfly
Floats in upon the sunbeam, and the fair
Tanned face of June, the nomad gipsy, laughts
Above her widespread wares,the while she tells
The farmer’s fortunes in the fields, and quaffs
The water from the spider-peopled wells.

The hedges are all drowned in green grass seas,
And bobbing poppies flare like Elmo’s light
While siren-like the pollen-stained bees
Drone in the clover depths. And up the height
The cuckoo’s voice is hoarse and broke with joy.
And on the lowland crops the crows make raid,
Nor fear the clappers of the farmer’s boy,
Who sleeps, like drunken Noah, in the shade.

And loop this red rose in that hazel ring
That snares your little ear, for June is short
And we must joy in it and dance and sing,
And from her bounty draw her rosy worth.
Ay! soon the swallows will be flying south,
The wind wheel north to gather in the snow
Even the roses spilt on youth’s red mouth
Will soon blow down the road all roses go.

by Francis Ledwidge (1887-1917)

Ledwidge was born in Slane, County Meath, in Ireland. He served in the British Army in the First World War and was killed at Passchendaele during the Third Battle of Ypres, just a few weeks before his 30th birthday. I’ve posted this poem before but was reminded of it when I saw some roses in my garden had died while I was away last week.

Weekly Update from the Open Journal of Astrophysics – 06/06/2026

Posted in OJAp Papers, Open Access, The Universe and Stuff with tags , , , , , , , , , , , , , , , , , , , , , , on June 6, 2026 by telescoper

Another Saturday, another update of activity at the Open Journal of Astrophysics. Since the last update we have published a further five papers, bringing the number in Volume 9 (2026) to 119 and the total so far published by OJAp up to 567.

I will continue to include the posts made on our Mastodon account (on Fediscience); these announcements also show the DOI for each paper.

The first paper to report this week, published on Tuesday 2nd June, is “The impact of the formation channel on gravitational-wave-galaxy cross-correlations” by Kabir Chakravarti (Chennai Mathematical Institute, India) and Federico R Urban (CEICO-FZU, Czech Republic). This article, published in the folder Cosmology and Nongalactic Astrophysics, explores how uncertainties in binary formation affect the cross-correlation signal between gravitational wave events and galaxy catalogues, finding that time-delay distribution significantly impacts the signal.

The overlay for this paper is here

You can find the officially accepted version on arXiv here and the announcement on Fediverse here:

Open Journal of Astrophysics

New Publication at the Open Journal of Astrophysics: "The impact of the formation channel on gravitational-wave-galaxy cross-correlations" by Kabir Chakravarti (Chennai Mathematical Institute, India) and Federico R Urban (CEICO-FZU, Czech Republic)

doi.org/10.33232/001c.162783

June 2, 2026, 7:22 am 1 boosts 2 favorites

The second paper for this week, also published on Tuesday 2nd June but in the folder High-Energy Astrophysical Phenomena, is “Transient X-ray Sources as Extremely Eccentric Mass-Transfer Binaries with Compact Companions” by Jonathan I Katz and Michael A Nowak (Washington University, St Louis, USA). This article suggests that X-ray transients, similar to tidal disruption events, are produced in eccentric stellar-compact object binaries, with their frequency gradually increasing over time.

The overlay for this one looks like this:

The official version of the paper can be found on arXiv here and the Fediverse announcement here:

Open Journal of Astrophysics

New Publication at the Open Journal of Astrophysics: "Transient X-ray Sources as Extremely Eccentric Mass-Transfer Binaries with Compact Companions" by Jonathan I Katz and Michael A Nowak (Washington University, USA)

doi.org/10.33232/001c.162784

June 2, 2026, 7:44 am 1 boosts 1 favorites

Next one up, the third paper of the week, also published on Tuesday 2nd June in the folder High-Energy Astrophysical Phenomena is “Resolving the (Debate About) Nozzle Shocks in Tidal Disruption Events” by Zachary L. Andalman & Eliot Quataert (Princeton U., USA), Eric R. Coughlin (Syracuse U. USA) and C. J. Nixon (U. Leeds, UK). This paper presents a model to understand the role of nozzle shocks in the circularization of stellar debris during a tidal disruption event when a star approaches a supermassive black hole (SMBH)

The overlay for this one is here:

The final, accepted version can be found on arXiv here and the Mastodon announcement is here:

Open Journal of Astrophysics

New Publication at the Open Journal of Astrophysics: "Resolving the (Debate About) Nozzle Shocks in Tidal Disruption Events" by Zachary L. Andalman & Eliot Quataert (Princeton U., USA), Eric R. Coughlin (Syracuse U. USA) and C. J. Nixon (U. Leeds, UK)

doi.org/10.33232/001c.162785

June 2, 2026, 7:52 am 1 boosts 1 favorites

The fourth paper this week, published on Wednesday 3rd June in the folder Cosmology and Nongalactic Astrophysics, is “Validating Digital Twins of the Local Universe with the Thermal Sunyaev-Zel’dovich Signal” by Richard Stiskalek (University of Oxford, UK) and Harry Desmond (University of Portsmouth, UK). The thermal Sunyaev-Zel’dovich effect and constrained simulations are used to analyze the thermal pressure of ionized gas in galaxy clusters and produce a set of digital twins for cosmological study.

The overlay is here:

The officially accepted version can be found on arXiv here and here is the Mastodon announcement:

Open Journal of Astrophysics

New Publication at the Open Journal of Astrophysics: "Validating Digital Twins of the Local Universe with the Thermal Sunyaev-Zel’dovich Signal" by Richard Stiskalek (U. Oxford, UK) and Harry Desmond (U. Portsmouth, UK)

doi.org/10.33232/001c.162811

June 3, 2026, 5:15 am 1 boosts 1 favorites

The fifth and final paper this week, published on Thursday 4th June in the folder Cosmology and Nongalactic Astrophysics is “Photon (Non)Conservation in the Reduced Speed of Light Approximation and How to (Almost) Fix It” by Nickolay Y. Gnedin (University of Chicago, USA). The “Reduced Speed of Light” approximation in cosmological simulations can lead to photon non-conservation, and while some missing photons can be counted, adding them back is challenging.

The overlay for this one is here:

The officially accepted version of this paper can be found on arXiv here and Mastodon announcement here

Open Journal of Astrophysics

New Publication at the Open Journal of Astrophysics: "Photon (Non)Conservation in the Reduced Speed of Light Approximation and How to (Almost) Fix It" by Nickolay Y. Gnedin (U. Chicago, USA)

doi.org/10.33232/001c.162879

June 4, 2026, 11:20 am 2 boosts 1 favorites

And that concludes this week’s update. I’ll do another one next Saturday.

What Leads to Administrative Bloat?

Posted in Finance with tags , , , , , on June 5, 2026 by telescoper

I’ve commented many times before on this blog about runaway expenditure on management resulting in the diversion of resources away from the core missions of a university, i.e. teaching and research, while producing no significant improvement in the efficiency, and indeed often a deterioration, of administrative processes. David Graeber wrote a book called Bullshit Jobs about this phenomenon. Management response to this is generally to assert that this administrative bloat is a response to regulatory burden. Many of us working in higher education would instead argue that the entire sector has been hijacked by self-serving parasites who are deliberately sucking the lifeblood out of the system.

I just came across a paper on the Physics and Society section of arXiv that tries to explain management bloat from the point of view of systems theory. The title is What Leads to Administrative Bloat? A Dynamic Model of Administrative Cost and Waste, the authors are Vicky Chuqiao Yang and Levi Grenier of MIT and the abstract is here:

The functioning of complex systems depends on the coordination of diverse components, often supported by regulatory structures that incur costs. In human organizations, such costs manifest as administrative burden, which has been rising despite often reducing efficiency. Classic explanations point to bureaucrat self-interest or regulation, yet they do not explain variation across organizations or clarify how this burden can be reduced. Here, we develop a dynamical model of administrative growth that integrates known behavioral mechanisms of process creation, obsolescence, and removal. The model conceptualizes processes as developed for problem solving, but becoming obsolete as conditions change, while continuing to consume resources until actively pruned. This interplay generates two long-term outcomes: stable equilibrium or run-away growth. The threshold separating these outcomes is shaped by organizations’ propensity to create new processes when faced with problems, and their propensity to prune obsolete ones in response to administrative burden. Importantly, their effects are asymmetric: sufficiently high creation propensity leads to bloat regardless of pruning propensity. Faster environmental change shifts this threshold, making bloat more likely. Simulations of interventions show that lasting reductions in administrative costs and waste require permanent shifts in priorities and investments in distinguishing obsolete from useful processes. Temporary efforts or indiscriminate cuts provide only short-lived relief, and counterintuitively, prioritizing direct production can increase waste. Our work highlights a general mechanism by which well-intentioned problem-solving can create self-reinforcing inefficiencies in complex systems, offering insights possibly generalizable to broader applications, such as legal, policy, and software systems where obsolete elements accumulate.

Here’s a a figure from the paper that provides ample illustration of the problem:

You will find a similar phenomenon on display at universities across the world. In my view this is a large part of the crisis engulfing higher education in the United Kingdom.

It’s an interesting paper, based on a very simple model. The authors also suggest various ways in which this burden could be reduced. The problem with that is that there is no incentive at all for The Management (who hold all the power) to improve the situation, as that would involve eliminating the bullshit jobs held by many of their cronies. With university governance structures notoriously weak and compliant, who manages the Managers? The most likely response from my University would be to appoint a new Vice-President for Self-reinforcing Inefficiency…

INO Norma at the National Opera House

Posted in Opera with tags , , , , , , , , on June 4, 2026 by telescoper

I had seen the by Vincenzo Bellini‘s opera Norma twice (before yesterday evening) but both times were before I starting blogging so I was very pleased to see that Irish National Opera were doing a new production this year. It had a run at the Gaiety Theatre in Dublin, but I took the chance to see it National Opera House in Wexford. The first time I went there was last year and was to see a very different opera, although that one and Norma are both generally categorized by the term bel canto. I was so excited by the prospect of seeing this production that I made myself a Bellini on Saturday, only to discover that the cocktail is named after a different Bellini…

Norma is not performed all that often, largely because it is difficult to find a singer capable of doing justice to the title role which is acknowledged as being one of the most vocally demanding roles in all opera. It is a role that the great Maria Callas made her own in the 1950s which places the additional burden on the the singer of stepping out of the shadow of such a legend. I was delighted that in Salome Jicia INO found an artist who could meet the challenges of the part. She was stunning.

I think the part of Norma is difficult not only because of the vocal range of the part, from the lyrical beauty of the famous aria Casta Diva to the agility needed to perform the coloratura passages, but also for the sheer stamina required. There really is a lot of singing for her to do in this opera! Jicia took it in her stride. Bellini rarely gets the orchestra to double the vocal line, so the voice of the singer is very exposed. At times it’s like watching acrobatics knowing that there’s no safety net!

Anyway, to the plot.

Norma is a tragedia lirica in two Acts, set in ancient Gaul which is under occupation by the Romans. The eponymous Norma is the high priestess of the native Druids. She is also a complex chartacter, not least because she is in a relationship with a high-ranking Roman, Pollione, with whom she has had two kids. It turns out that Pollione is tiring of Norma and has turned his attention to her friend Adalgisa, who does not know about Pollione’s involvement with Norma.

The Opera begins with the heavily armed Gauls, led by Norma’s father Oroveso, planning to rise up against the Roman occupiers. But only Norma can sanction an armed rebellion and she says no. The aria Casta Diva expresses Norma’s desire for peace, but this is not just because she feels the Romans have greater military strength and would crush the rising, but because of her thing with Pollione. Over the course of the Opera we find out about Pollione’s infatuation with Adalgisa and their planned elopement to Rome which sends Norma into such a rage she threatens to kill her own children. Eventually Norma decides to change her mind about the uprising and calls for it to go ahead. Protocol requires a human sacrifice to initiate such a move, so the question is who is for the chop? Will it be Pollione, or Adalgisa? Norma surprises the assembled Gauls by confessing her sacreligious relationship with Pollione and declaring herself to be the sacrifice. Pollione is overcome by remorse at what he has done, and decides to join Norma in death.

Well, what did you expect in an Opera, a happy ending?

This production eschews the Normal (geddit?) setting of forest groves and scared shrines and places it in a sort of modern post-apocalyptic dystopia. The stage is dominated by barricades apparently hastily constructed from broken furniture and scrap metal. Lighting is sombre and claustrophobic. This jars with Bellini’s music when it evokes pastoral beauty – the music is too light and pretty for such a gloomy setting.

Gone are the swords, spears and druidic robes and in come AK-47s, pistols and somewhat scruffy modern-looking outfits. Even Norma’s dress is nothing fancy. The Gauls look like a ramshackle but heavily armed paramilitary group and are indistinguishable in costume from the Romans, except that the latter have special haircuts – mohawks died bright red on top. I found this a bit confusing and felt that it negated the theme of “occupiers versus occupied” which runs throughout the piece. This choice is probably intended to show the struggle as one between two rival groups in a polarized community. In the hands of the chorus the guns create a very menacing backgroup to several scenes, and of course the ending ends not by ritual burning, as in the original, but by firing squad.

The musical pyrotechnics do pose challenges for the staging, however, because an opera is not just a concert. It doesn’t work as music drama if the singers are just standing there belting out tunes. They also need to employ gestures and facial expressions to match the emotions expressed by the score and it can’t be too static. In this production the set is rather simple, and when I first saw it I assumed that it would be moved around a lot to create different locations, but that was only done to a limited extent. To compensate for the inflexible scenery, the chorus often provides a moving backdrop to the action. In the hands of the chorus the proliferation of guns create a very menacing backgroup to several scenes, and of course the ending ends not by ritual burning, as in the original, but by firing squad. The chorus was used very imaginatively, I’d say, not only to make up the numbers on stage, but also in their singing. I liked the idea of the chorus being split for some passages, leading one’s attention around the stage as the different groups gave voice.

I already mentioned Salome Jicia, but the other principals were also very good. Mario Chang (tenor) was a fine Pollione, William Guanbo Su a towering Oroveso with a rich bass and Siobhan Stagge as Adalgisa sang and acted beautifully. Despite some reservations about the staging, I thought this was a hugely enjoyable performance. Bellini has his critics, but I think the last part of Act II, the principals and chorus singing their hearts out to wonderful music, as the drama moves inexorably to its tragic conclusion, is one of the most intensely moving experiences in the entire operatic repertoire.

P.S. If you have 20 minutes to spare, listen to the Act II finale as performed by Callas here.

Leaving Again

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

It is an important day for many young people in Ireland because the Junior Certificate and Leaving Certificate examinations both start today, so the first thing I need to do is wish everyone starting their examinations the very best of luck!

Among other things, the results of the leaving certificate examinations are important for September’s university admissions. 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, with many students taking more than this. This gives 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. One can choose to do Leaving Certificate subjects at Ordinary or Higher level and there’s quite a big difference between the two, especially in Mathematics.

Overall I prefer the Leaving Certificate over the UK system of A-levels, partly the former gives the students a broader range of subjects than the latter. I would have liked to have been allowed to take at least one arts subject past O-level, for example. Another reason is that all students across the country take the same examination, so the difficulty or otherwise of the papers is a topic of national conversation. There will no doubt be many reactions in the media over the forthcoming days!

Anyway, tomorrow is a significant date for me (in more ways than one) so I’m up early this morning before heading off on a short journey to mark the occasion in style.

Eclipse Event – a Date for your Diaries!

Posted in Maynooth, The Universe and Stuff with tags , , , , , on June 2, 2026 by telescoper

Here’s an invitation to people interested in astronomy to join staff from the Department of Physics at Maynooth University on Wednesday, August 12th 2026 from 5:30 pm to 8:30 pm for a special public celebration of one of the deepest solar eclipses visible from Ireland in decades. The next eclipse with > 90% obscuration of the Sun by the Moon won’t be seen again in Ireland until 2090, by which time I will have retired.

Experience the eclipse safely through:

  • eclipse viewing glasses
  • solar telescopes
  • live astronomy demonstrations

List of Solar Talks during the evening:

  • Dr. Emma Whelan – The Story of the Sun
  • Dr. Joshuah Heath – The Quantum Sun
  • Dr. Marcin Gradziel – (Electric) Power from the Sun. The good, the bad, and the glinty!
  • Dr. Michelle McCrystall – The star of the show: How the Sun drives our climate
  • Prof. Peter Coles – Einstein and the Eclipse (Who He? Ed.)
  • Dr. Patrick Kavanagh – Will our Sun go Supernova?

The evening will also include:

  • an immersive* astronomy show in our inflatable planetarium
  • hands-on arts and crafts activities for children

*especially if it rains

Maynooth University Staff can reserve a place here until June 14th, after which booking will be opened to the general public – I’ll repost this invitation with a link at that time.

Lá Saoire i mí Mheitheamh

Posted in Biographical, Irish Language, LGBTQ+ on June 1, 2026 by telescoper

Today, 1st June, has been (and indeed continues to be for a few hours) the June Bank Holiday (Lá Saoire i mí Mheitheamh) in Ireland. It is the equivalent of the usual May Bank Holiday in the UK in that both have their origin in the old festival of Whitsuntide (or Pentecost) which falls on the 7th Sunday after Easter. Because the date of Easter moves around in the calendar so does Whit Sunday, but it is usually in late May or early June. Pentecost was actually on Sunday 24th May this year. Here in Ireland the Bank Holiday is always on the first Monday in June whereas on the other side of the Irish Sea it is on the last Monday in May. This year the weather was better last Monday, but you can’t have everything.

Although I’m only at beginners’ level in Irish, the phrase Lá Saoire i mí Mheitheamh gives me another chance to bore you about it. It’s actually quite a straightforward phrase until you reach the last word. “Lá” means “day” and “Saoire” means “leave” or “vacation” so “Lá Saoire” means “holiday”; “i” is a prepositional pronoun meaning “in” and “mí” means “month”. So far so good.

The word for June, however, is Meitheamh (at least when it is in the nominative singular case). Irish is an inflected language, which means that words change form according to their grammatical function. As an Indo-European language, Irish is distantly related to Latin which has six grammatical cases for nouns (actually seven if you count the rarely used locative case). Irish has only four cases – there’s no ablative and, curiously, no distinction between nominative and accusative. That leaves nominative, dative, genitive, and vocative. The dative – used after simple prepositions – is only rarely distinct from the nominative so basically the ones you have to learn are the genitive and the vocative.

In Latin cases are indicated by changes to the end of a word, but in Irish they involve initial mutations. In the example of “mí Mheitheamh” meaning “month of June”, requiring the genitive form of “June”, the initial consonant “M” undergoes lenition (softening) to sound more like a “v”. In old Irish texts this would be indicated by a dot over the M but in modern orthography it is indicated by writing an “h” after the consonant. This is called a séimhiú (pronounced “shay-voo” ). Note the softened m in the middle of that word too but it’s not a mutation – it’s just part of the regular spelling of the word, as is the -mh at the end of Meitheamh. There’s also a softened “t” in the middle of Meitheamh which makes it vrtually disappear in pronunciation. Meitheamh is thus pronounced something like “Meh-hiv” whereas “Mheitheamh” is something like “Veh-hiv”.

As well as being Lá Saoire i mí Mheitheamh, today is also the start of Pride Month, so let me take the opportunity to say Bród sona daoibh a chairde!

An Employment Tribunal

Posted in Harassment Bullying etc with tags , , , , , on June 1, 2026 by telescoper

I only have time for a brief post here to point out that today saw the start of an Employment Tribunal in Bury St Edmunds in a case brought by Neil Wyn Evans relating to the Institute of Astronomy at Cambridge University. I gather the proceedings will take up to four weeks. For more details see here, where updates will be provided as the case continues.

I can’t comment further because the matter is sub judice.