Archive for the mathematics Category

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:

Math versus Maths

Posted in mathematics, Pedantry with tags , on June 8, 2020 by telescoper

I was amused by this discussion on Dictionary.Com of the different abbreviations of mathematics..

I’d like to think that ending is deliberate!

Job Alert!

Posted in mathematics, Maynooth on May 21, 2020 by telescoper

It occurred to me that there might be among the readers of this blog people interested in a job opportunity just announced at the Hamilton Institute at Maynooth University, which exists to promote interdisciplinary research spanning applied mathematics, computer science, engineering, and statistics. Applicants from any of those areas are welcome.

There is a lot more detail including instructions on how to apply here. The deadline is at the end of September 2020.

Calculating the UK COVID Alert Level

Posted in Covid-19, mathematics, Politics with tags , , , on May 11, 2020 by telescoper

I didn’t watch yesterday’s broadcast by the UK’s Clown-in-Chief Bozo Johnson but I gather that he delivered an address that was every bit as coherent and lucid as one might have expected.

I for one am delighted that at last there is some clarity in the UK Government’s position and that they have applied the necessary level of mathematical rigour to their treatment of the Covid-19 Pandemic.

Catching up on these pronouncements via Twitter I was impressed to see, for example, to see this precise formulation of the calculation required to establish the COVID Alert Level.

Let me take you through a detailed calculation using this important formula.

As far as I know the best estimate of the basic reproduction number R in the UK is around 0.8.

As of this morning (11th May) the number of confirmed Covid-19 infections in the UK is 219,183.

Applying the formula I obtain a value

COVID Alert Level = 219183.8

That seems a big number. I thought it should only go up to 11. Have I slipped up somewhere?

Testing YouTube

Posted in Biographical, mathematics, YouTube with tags , , on May 3, 2020 by telescoper

To do something a bit different during this Covid-19 lockdown I decided to set up my very own YouTube channel to which you may (or may not) wish to subscribe.

I’m new to this so I posted a short video to test how it works. It’s a little video explainer about Cramer’s Rule in linear algebra I made using Screencast-o-matic. I’ve done a lot of these over the past few weeks but they’re not what the channel is about: I posted this example is just to try out the system (mainly to see how long the upload would take).

I put this up yesterday and I’ve already amassed five subscribers so I’m well on the way to becoming a YouTube sensation. I may even become viral so please ensure that you practice social distancing while watching the videos.

A Virus Testing Probability Puzzle

Posted in Cute Problems, mathematics with tags , on April 13, 2020 by telescoper

Here is a topical puzzle for you.

A test is designed to show whether or not a person is carrying a particular virus.

The test has only two possible outcomes, positive or negative.

If the person is carrying the virus the test has a 95% probability of giving a positive result.

If the person is not carrying the virus the test has a 95% probability of giving a negative result.

A given individual, selected at random, is tested and obtains a positive result. What is the probability that they are carrying the virus?

Update 1: the comments so far have correctly established that the answer is not what you might naively think (ie 95%) and that it depends on the fraction of people in the population actually carrying the virus. Suppose this is f. Now what is the answer?

Update 2: OK so we now have the probability for a fixed value of f. Suppose we know nothing about f in advance. Can we still answer the question?

Answers and/or comments through the comments box please.

R. I. P. John Conway (1937-2020)

Posted in Biographical, mathematics with tags , , , on April 12, 2020 by telescoper

I’ve just heard the sad news that that mathematician John Horton Conway has passed away at the age of 82.

John Conway made very distinguished contributions to many areas of mathematics, especially topology and knot theory, but to many of us he’ll be remembered as the inventor of the Game Of Life. I’ll remember him for that because one of the very first computer programs I ever wrote (in BASIC) was an implementation of that game.

It’s a great illustration of how simple rules can lead to complex structures and it paved the way to a huge increase in interest in cellular automata.

I think he got a bit fed up with people just associating him with a computer game and neglecting his deeper work, but he deserves great credit for directly or indirectly inspiring future scientists.

Rest in peace John Conway (1937-2020).

Coronavirus in Ireland – the Latest!

Posted in Covid-19, mathematics on March 28, 2020 by telescoper

Just a quick post to point out that I’ve set up a page on which I’m tracking the number of cases of Covid-19 in the Republic of Ireland.

I intend to keep the data on that page up to date as information is announced by the HSE and won’t do lots of updates as posts. I thought I’d show the latest figures here though.

The second (log-linear) plot is perhaps more informative. It shows some evidence of a flattening compared to an exponential curve. The plot in green is an exponential with a time constant of 3.5 days; it’s not a fit to the data, it’s just there to show what exponential growth would look like on such a plot (ie a straight line).