Archive for the mathematics Category

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.

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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).

Acute Geometry Problem

Posted in Cute Problems, mathematics on December 3, 2018 by telescoper

I saw a plea for help on Twitter from Astronomer Bryan Gaensler who is stuck with his son’s homework.

So please give him a hand by solving this to find a, b and c.

Your time starts now.

Halloween in LaTeX

Posted in History, mathematics on October 21, 2018 by telescoper

I forget where I found this list of spooky LaTeX commands but, with the dreaded Halloween coming up soon, I thought I’d share it here.

Anyway, it reminded of the mathematical curve known in English as The Witch of Agnesi, the witch of which is a mistranslation of the Italian versiera meaning a ‘sheet’ (ie the rope connecting to a ship’s sail) rather than a shortened version of ‘avversiera’ meaning ‘a female devil’ or ‘witch’.

Vector Calculus Weather

Posted in mathematics, The Universe and Stuff with tags , , on October 12, 2018 by telescoper

As it happens I did a lecture today about vector fields as part of my module on vector calculus. Whenever I did similar lectures in the past I used the day’s weather map as an illustration, so this morning I downloaded what turned out to be a particularly dramatic example. The curl of the velocity field around the weather system off the west coast of Ireland this morning was definitely non-zero…

Storm Callum turned out to be not as damaging as feared. Apparently it was rather windy in Maynooth overnight, but I slept right through it.

The Problem of the Moving Triangle

Posted in Cute Problems, mathematics with tags , , on August 16, 2018 by telescoper

I found this nice geometric puzzle a few days ago on Twitter. It’s not too hard, but I thought I’d put it in the `Cute Problems‘ folder.

In the above diagram, the small equilateral triangle moves about inside the larger one in such a way that it keeps the orientation shown. What can you say about the sum a+b+c?

Answers through the comments box please, and please show your working!

The Problem with Odd Moments

Posted in Bad Statistics, Cute Problems, mathematics with tags , , on July 9, 2018 by telescoper

Last week, realizing that it had been a while since I posted anything in the cute problems folder, I did a quick post before going to a meeting. Unfortunately, as a couple of people pointed out almost immediately, there was a problem with the question (a typo in the form of a misplaced bracket). I took the post offline until I could correct it and then promptly forgot about it. I remembered it yesterday so have now corrected it. I also added a useful integral as a hint at the end, because I’m a nice person. I suggest you start by evaluating the expectation value (i.e. the first-order moment). Answers to parts (2) and (3) through the comments box please!

Answers to (2) and (3) via the comments box please!

 

SOLUTION: I’ll leave you to draw your own sketch but, as Anton correctly points out, this is a distribution that is asymmetric about its mean but has all odd-order moments equal (including the skewness) equal to zero. it therefore provides a counter-example to common assertions, e.g. that asymmetric distributions must have non-zero skewness. The function shown in the problem was originally given by Stieltjes, but a general discussion can be be found in E. Churchill (1946) Information given by odd moments, Ann. Math. Statist. 17, 244-6. The paper is available online here.