Archive for the mathematics Category

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).

Clifford’s `Space-Theory of Matter’

Posted in Beards, History, mathematics, The Universe and Stuff with tags , , , on February 26, 2020 by telescoper

Well, here’s another thing I didn’t know until I was informed by Twitter.

Way back in 1876 –  forty years before Einstein presented his Theory of General Relativity – the mathematician W.K. Clifford (who is most famous nowadays for the Clifford Algebra) presented a short paper in the Proceedings of the Cambridge Philosophical Society in which he speculated that space might be described by Riemannian rather than Euclidean Geometry.

Here are a couple of excerpts:

and

The paper does not contain any actual equations, and his concentration on small scales rather than large was misguided, but it is quite remarkable that he was thinking about such matters such a long time ago!

Unfortunately Clifford died very young, in 1879, at the age of 33, tuberculosis. Had he lived longer he might have been able to develop these ideas a bit further.

As a postscript I should mention that Clifford had an impressive beard.

What’s the Vector, Victor?

Posted in Film, mathematics, The Universe and Stuff with tags , , , on February 18, 2020 by telescoper

Following on from Sunday’s post about the trials and tribulations caused by Storm Dennis, here is a clip of a plane (an Airbus 380)  landing at Heathrow airport on Saturday.

There are other clips of this same event on Youtube and some of them describe this landing as `dangerous’. Although it undoubtedly involved skill and concentration by the pilot it’s not actually dangerous. Aircrew are trained to land in windy weather like this, and it’s fairly routine. My plane to Dublin (an Airbus 320) landed like this on Saturday evening and, although the pilot got a well-deserved round of applause on landing, nobody was ever really at risk.

As it happens, this week I start teaching vector algebra to my first-year Engineering students, so the weekend’s weather events have  given me a good illustration of vector addition. The plane has to have a velocity vector relative to the air such that the sum of it and the wind vector adds to a resultant vector directed along the runway. Lots of people seem to think this is just guesswork but it isn’t. It’s applied mathematics.

This is in principle simple as long as the crosswind is steady, but obviously the pilot needs to be alert to gusting and make adjustments along the way. When the plane has slowed down enough to land in normal conditions, the wind over the wings is still causing a bit of extra lift. You can see that in the last moments before touchdown this aircraft is gliding because of this effect. I’m told that because of this, in windy conditions planes usually descend at a steeper angle than usual.

The interesting bit for me is that the plane touches down in such a way that its body is at an angle to the runway. As soon as it has landed it has to correct this and point along the runway. I think this is done with the rudder rather than the undercarriage, but I don’t know. Perhaps any experienced pilots that happen to be reading this could give more details through the comments box?

P.S. The title of this post is a reference to the film Airplane!