Archive for the mathematics Category

Additional Mathematics O-level 1979

Posted in Biographical, Education, mathematics with tags , , , on July 1, 2019 by telescoper

Yesterday a comment appeared on an old post of mine about the O-level Examination I took in Mathematics when I was at School. With a shock that reminded me that it was FORTY years ago this summer that I was taking my O-levels at the Royal Grammar School in Newcastle. That’s a memory lane down which I wasn’t anxious to take a trip.

For any youngsters reading this, the GCE (General Certificate of Education) Ordinary Level Examinations O-levels were taken at age sixteen in the United Kingdom back in the day; they were replaced during the 1980s by the modern GCSE Examination. For readers in Ireland the O-levels were roughly equivalent to the Junior Certificate, just as A-levels are roughly equivalent to the Leaving Certificate.

Anyway, that also reminded me that I never got round to posting the other O-level I took in Mathematics that summer, in Additional Mathematics. I thought I’d remedy that failing now, so here are the two papers I took (on Tuesday 26 June 1979 and Thursday 5 July respectively.

I had forgotten that there was so much mechanics in this actually (Section C of each paper). Is that different from equivalent papers nowadays? In fact I’d be interested in comments about the content and level of difficulty of this compared to modern examinations in mathematics via the box below.

P.S. I did ten O-levels that summer of ’79: Mathematics; Additional Mathematics; Combined Science (2); English Language; English Literature; French; Latin; History; and Geography. I still have all the papers and have only posted a subset. If anyone has requests for any others please let me know and I’ll scan them.

Advertisements

The First Bookie

Posted in Football, mathematics, Sport with tags , , , , , , on April 24, 2019 by telescoper

I read an interesting piece in Sunday’s Observer which is mainly about the challenges facing the modern sports betting industry but which also included some interesting historical snippets about the history of gambling.

One thing that I didn’t know before reading this article was that it is generally accepted that the first ever bookmaker was a chap called Harry Ogden who started business in the late 18th century on Newmarket Heath. Organized horse-racing had been going on for over a century by then, and gambling had co-existed with it, not always legally. Before Harry Ogden, however, the types of wager were very different from what we have nowadays. For one thing bets would generally be offered on one particular horse (the Favourite), against the field. There being only two outcomes these were generally even-money bets, and the wagers were made between individuals rather than being administered by a `turf accountant’.

Then up stepped Harry Ogden, who introduced the innovation of laying odds on every horse in a race. He set the odds based on his knowledge of the form of the different horses (i.e. on their results in previous races), using this data to estimate probabilities of success for each one. This kind of `book’, listing odds for all the runners in a race, rapidly became very popular and is still with us today. The way of specifying odds as fractions (e.g. 6/1 against, 7/1 on) derives from this period.

Ogden wasn’t interested in merely facilitating other people’s wagers: he wanted to make a profit out of this process and the system he put in place to achieve this survives to this day. In particular he introduced a version of the overround, which works as follows. I’ll use a simple example from football rather than horse-racing because I was thinking about it the other day while I was looking at the bookies odds on relegation from the Premiership.

Suppose there is a football match, which can result either in a HOME win, an AWAY win or a DRAW. Suppose the bookmaker’s expert analysts – modern bookmakers employ huge teams of these – judge the odds of these three outcomes to be: 1-1 (evens) on a HOME win, 2-1 against the DRAW and 5-1 against the AWAY win. The corresponding probabilities are: 1/2 for the HOME win, 1/3 for the DRAW and 1/6 for the AWAY win. Note that these add up to 100%, as they are meant to be probabilities and these are the only three possible outcomes. These are `true odds’.

Offering these probabilities as odds to punters would not guarantee a return for the bookie, who would instead change the odds so they add up to more than 100%. In the case above the bookie’s odds might be: 4-6 for the HOME win; 6-4 for the DRAW and 4-1 against the AWAY win. The implied probabilities here are 3/5, 2/5 and 1/5 respectively, which adds up to 120%, not 100%. The excess is the overround or `bookmaker’s margin’ – in this case 20%.

This is quite the opposite to the Dutch Book case I discussed here.

Harry Ogden applied his method to horse races with many more possible outcomes, but the principle is the same: work out your best estimate of the true odds then apply your margin to calculate the odds offered to the punter.

One thing this means is that you have to be careful f you want to estimate the probability of an event from a bookie’s odds. If they offer you even money then that does not mean they you have a 50-50 chance!

On the alleged socialist dominance in academia

Posted in mathematics, The Universe and Stuff with tags , , , , on February 11, 2019 by telescoper

Various forms of Turning Point

Yesterday there came to my attention to a tweet from an organization called `Turning Point’. Disappointingly this is not as its name suggests, something to do with differential calculus, but a far-right propaganda organization which, among other things, is bemoaning the `socialist dominance in academia’.

Left-wing infiltration of university education would be a very serious matter if it existed, so to allay the fears of my readership that this is not really a problem, in the following I am going to list a few physics topics I will be teaching this week to make it clear that they can not possibly be accused of being influenced by political bias.

  • Mathematical Physics. I will be explaining how Laplace Transformations can be used to solve ordinary differential equations by seizing the means of production.
  • Quantum Mechanics. I will be demonstrating how the path integral formalism allows the result of a quantum mechanical calculation to be obtained by considering the sum over all historical class struggles.
  • Electrostatics. I will be discussing  why some substances are insulators rather than conductors using the theory of dielectrical materialism.
  • Optics. The topic here is Snell’s Law, which relates the Engels of incidence and refraction for light of a given colour and for given pair of media.

It goes without saying that students will not pass the examination on these topics unless they get enough Marx.

I hope this clarifies the situation.

Hyperbolic sine, shine or sinsh (or sinch)?

Posted in Education, mathematics with tags , , , on February 6, 2019 by telescoper

An important coffee-time discussion just revealed a significant cultural difference between members of staff here in the Department of Theoretical Physics at Maynooth University, which I wish to explore further via a completely scientific poll.

The hyperbolic sine, usually written sinh, is well known to be defined in thuswise fashion:

But the question is how do you pronounce it? Since my schooldays I have always pronounced it like `shine’ whereas I’m told others have pronounced it more like `sinch’. Yet others simply call it hyperbolic sine. What is your pronunciation?

This also gives me the excuse to tell a little story of when I was at school. One morning, which happened to be April 1st, our maths teacher started on the subject of hyperbolic functions, talking about `shine’ and `cosh’ and `tanch’. We all assumed it was an April Fool’s joke and although it was very clever it wasn’t all that funny, particularly as it went on for the whole class. We only realised it wasn’t a joke when he carried on in the same vein the following day…

And another thing, for bonus marks. In all European languages with which I am familiar (which is by no means all), the trigonometric function sin is pronounced `sinus’ not `sine’. Is English the only language to depart from sinus?

Answers through the comments box!

Ahead of Teaching

Posted in Biographical, Education, mathematics, Maynooth, Music with tags , , , , on February 3, 2019 by telescoper

It’s 3rd February 2019, which means that today is two days after Imbolc, a Gaelic festival marking the point halfway between the winter solstice and vernal equinox. This either happens 1st or 2nd February, and this year it was former, i.e. last Friday In Ireland this day is sometimes regarded as the first day of spring, as it is roughly the time when the first spring lambs are born. It corresponds to the Welsh Gŵyl Fair y Canhwyllau and is also known as the `Cross Quarter Day’ or (my favourite) `The Quickening of the Year’.

I wrote a post about this time last year, on the day I gave my first ever lecture in Maynooth University, on Computational Physics, in a theatre called Physics Hall. That was on Thursday February 1st 2018. It’s hard to believe that was a full year ago. Time certainly has gone quickly this year.

Owing to the vagaries of the academic calendar we’re a week later getting back to teaching this year than last year so my first Computational Physics lecture won’t be until this Thursday (7th February) at 9am, but sadly it won’t be in Physics Hall, which I rather liked, but in Hall C – a much less atmospheric venue, but one rather closer to my office, which will be handy if I forget anything (which I am prone to do). There are about 25 students taking this module, a few down on last year, which means they should fit comfortably into our computer lab. I’m not surprised they moved the lecture, really. The capacity of Physics Hall is 90, and even last year I only had about 30 students. Still, it did have a piano (which Hall C does not):

Computational Physics doesn’t start until Thursday. Before that I have to start my other module: Engineering Mathematics II. This (what you would probably call a `service course’) covers a mixture of things, mainly Linear Algebra but with some other bits thrown in for fun, such as Laplace transforms. Interestingly I find the Mathematical Physics students do not encounter Laplace Transforms in the first year, but perhaps engineers use them more often than physicists do? I think I’ve written only one paper that made use of a Laplace transform. Anyway, I have to start with this topic as the students need some knowledge of it for some other module they’re taking this semester. I reckon six lectures will be enough to give them what they need. That’s two weeks of lectures, there being three lectures a week for this module.

By coincidence rather than good planning, the timetable for this module is quite nice. I have lectures on Monday, Tuesday and Wednesday and then the students have a choice of tutorial (on either Thursday or Friday). That means I can get through a decent amount of material each week before each tutorial. I don’t do the tutorials, by the way: that’s left to one of our PhD students, who gets paid for doing that and correcting the weekly coursework. There are about 50 students on this module, divided into two courses: Electronic Engineering and Robotics and Intelligent Devices. We don’t have Civil or Mechanical or Chemical Engineering, etc at Maynooth.

Campus has been very quiet for the last week or so. The exam period finished in late January but lectures don’t start until tomorrow morning (Monday 4th February) so there have been few students around. No doubt it will be a different story tomorrow. I’ve done my first week’s notes and compiled my first problem set so I’m more-or-less ready to go. First lecture at 2pm tomorrow in Hall H, which is one of the rooms I taught in last term so at least I know where it is!

 

Meaningful Betting

Posted in Biographical, mathematics, Politics with tags , , , , on January 15, 2019 by telescoper

I’ve now finished my first batch of marking (Astrophysics & Cosmology) and I now have a few days to do other things until the next (larger) set of scripts arrives from Vector Calculus and Fourier Series which takes place on Saturday.

Before going home I turned my attention to the news of tonight’s “Meaningful Vote” in the House of Common’s about whether or not to accept the Withdrawal Agreement negotiated between PM Theresa May and the European Union. The Government is widely expected to lose the vote but it’s not clear by what margin. Interested to see what the betting markets think I had a look just now at the Betfair exchange and found this, on the basic question of whether the vote will pass or not:

(You might want to click on the image to make it clearer.) You can find the live odds here.

You will see that this is quite an active market – with over €700K being wagered. Odds of greater than 40 to 1 against a `yes’ vote but most of the action is on `no’. Note that quite a few punters are laying on this outcome, with odds of about 25-1 on. (The way odds are shown in the second row is that 1.04 means you with 4p for every £1 staked).

Clearly, therefore, the markets think the vote will fail. What is less clear is how many MPs will vote in favour of the Government. Here is the corresponding Betfair market:

The shortest odds are for the first option, i.e. for 199 or fewer voting with the Government, but there is activity across the whole range of possibilities. The press are talking about a defeat by 225 votes, but for what it’s worth I don’t think it will be such a large margin. I’m not going to bet on it, but I expect it to be defeated by less than 150 votes.

UPDATE: Not for the first time I was wrong. The vote was Ayes 202 Noes 432, a majority of 230 against. That means there were no abstentions.

Bill Bonnor on Cosmology with Negative Mass

Posted in mathematics, The Universe and Stuff with tags , , , , , on December 10, 2018 by telescoper

My post from Friday about negative mass in cosmology reminded me of my days at Queen Mary and discussions I had at that time with Bill Bonnor, who retired in 1985 but was a regular visitor to the weekly Relativity Seminars. I was sad to discover just now that Bill actually passed away in 2015 (at the age of 94) so I thought I would post a little note as a short tribute.

Bill Bonnor was an old-school mathematical relativist, which I definitely am not, but I recall talking to him quite a lot in the coffee room because we had a shared interest in gambling games. He had a liking for the fixed-odds competition in the football pools, which he played with considerable success.

Anyway, Bill Bonnor published a paper in 1989 about Negative Mass in General Relativity. It’s not all about cosmological implications of negative mass, but I’ve just typed up a quick summary. In fact I used some of this in a university examination question many moons ago!

Before reading this, you might wish to look up active the terms gravitational mass, passive gravitational mass, inertial mass and equivalence principle, which you can find discussed here (for example).